electron-phonon interaction and charge instabilities in strongly correlated … · 2017-03-14 ·...

Scuola di Dottorato “Vito Volterra”

Dottorato di Ricerca in Fisica

Electron-phonon interaction

and charge instabilities in

strongly correlated electron systems

Thesis submitted to obtain the degree of

“Dottore di Ricerca” – Philosophiæ DoctorPhD in Physics – XXI cycle – October 2008

by

Andrea Di Ciolo

Program Coordinator Thesis Advisors

Prof. Enzo Marinari Dr. Jose G. Lorenzana

Prof. Marco Grilli

Contents

List of the main abbreviations 1

Introduction 3

1 Charge Density Waves and electron-lattice coupling 51.1 Density waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Phenomenology of the cuprates . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Stripes in LSCO . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Experimental phononic features in the cuprates . . . . . . . . . . . . . 201.4 Scattering experiments and determination of the phonon dispersion . . 251.5 Anomalous phonon softening . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Gutzwiller Approximation and Random Phase Approximation 332.1 The Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 The Holstein interaction in the adiabatic limit. The charge susceptibility 35

2.2.1 Exact relation between electron-phonon instabilities and chargesusceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 The Gutzwiller Approximation . . . . . . . . . . . . . . . . . . . . . . . 392.4 The Gutzwiller + RPA method . . . . . . . . . . . . . . . . . . . . . . 432.5 The GA+RPA approach to the paramagnetic homogeneous state . . . . 50

2.5.1 Half-filling case . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.5.2 Arbitrary filling case . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 The GA+RPA approach to inhomogeneous states . . . . . . . . . . . . 58

Appendix A - The GA+RPA approach to the homogeneous state - othercomputations 60

3 Charge instabilities of a homogeneous ground state 633.1 The Gutzwiller method in infinite dimensions: the charge susceptibility 653.2 Charge susceptibility in one dimension . . . . . . . . . . . . . . . . . . 69

3.2.1 Reliability test of the Gutzwiller approximation . . . . . . . . . 693.2.2 Renormalization of the electron-phonon interaction . . . . . . . 73

CONTENTS iii

3.3 Charge susceptibility in two dimensions. . . . . . . . . . . . . . . . . . 753.4 Phonon softening: the Kohn anomaly with strongly correlated electrons 833.5 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Charge stripes and optical phonon softening 904.1 Magnetic charge stripes in two dimensions . . . . . . . . . . . . . . . . 90

4.1.1 Realistic parameters and saddle-point solutions . . . . . . . . . 934.2 Charge response of the stripes . . . . . . . . . . . . . . . . . . . . . . . 964.3 Normal modes of a crystal with a superlattice . . . . . . . . . . . . . . 994.4 Softening of the optical phononic branches . . . . . . . . . . . . . . . . 103

Conclusions 111

Appendix B - The GA+RPA approach to broken-symmetry states 115

List of Publications and Pre-prints 140

Bibliography 141

List of the main abbreviations

LSCO La2−δSrδCuO4

LBCO La2−δBaδCuO4

BSCCO Bi2Sr2CaCu2O8+δ

YBCO Y Ba2Cu3O7−δ

e-e electron-electron

e-ph electron-phonon

ARPES Angle Resolved PhotoEmission Spectroscopy

INS Inelastic Neutron Scattering

FL Fermi Liquid

AF AntiFerromagnet

CDW Charge Density Wave

SDW Spin Density Wave

DOS Density Of States

GZW Gutzwiller

GA Gutzwiller Approximation

RPA Random Phase Approximation

2 CONTENTS

SB Slave Boson

QMC Quantum Monte Carlo

BZ Brillouin Zone

EBZ Extended Brillouin Zone

RBZ Reduced Brillouin Zone

PS Phase Separation

Introduction

In this Thesis I will study normal state properties of strongly correlated materials whichdisplay charge density waves (CDW’s).

Many experimental evidences in these compounds reveal that electron-lattice cou-pling and electron-electron correlations are in competition. Sometimes their interplaycontributes to make the phononic effects unclear: in some materials there are measuredquantities which can exhibit strong phononic effects, whereas in other measurementsphononic signatures can be virtually absent. This is a stimulating picture for the sci-entific curiosity behind a Ph-D Thesis: we want to investigate the relevance of thecorrelation between lattice ions and electrons in the physics of strongly correlatedmaterials and how this coupling is reduced and eventually suppressed by the strongelectronic repulsion.

A part of our study will be explicitly dedicated to one family of strongly correlatedmaterials: the cuprates. These high-temperature superconductors have a great scien-tific interest, due to their very complex phase diagram originated by the competitionof several orderings (charge, spin, superconducting ...) and then microscopically bythe interplay of electronic and phononic couplings. In particular, we will consider theresults of scattering experiments which show an anomalous softening of their higher-energy longitudinal optical phonon branch: this is strongly anisotropic and exhibits adip occurring at a particular wave vector. We think that this behaviour can be ex-plained via a relevant electron-phonon coupling and considering that the material hasan anisotropic electronic structure associated to a form of charge order: the “stripes”.They are one-dimensional hole-rich structures separated by antiferromagnetic hole-poordomains in a periodical spatial modulation of the electronic density.

For our theoretical studies we use the Hubbard-Holstein model, which is the min-imal, yet highly non-trivial, playground to consider the physics of coupled electronsand phonons with strong electronic correlations. For our computations we use the GA+ RPA (Gutzwiller Approximation + Random Phase Approximation). The GA is avariational mean-field approach which has given reliable results for the properties bothof homogeneous and inhomogeneous systems, at least in the low-energy sector of theirelectronic excitations. On top of the GA ground state we perform RPA fluctuations tocompute the response of the system; in particular, for our purposes we are interestedto obtain the charge susceptibility.

4 CONTENTS

This is the general plan of our work.

• In Chapter 1 we give a brief phenomenology survey of CDW’s in crystallinesolids. Then we present the phenomenology of the cuprates, in particular theevidences of stripes whose properties will be invoked to explain the anomalousphonon softening. Other evidences of the phononic effects in the cuprates arealso shortly reviewed.

• In Chapter 2 we describe the Hubbard-Holstein model and present a completelygeneral derivation of the GA+RPA formalism. Then we use this approach forhomogeneous systems: we report most computations except for some steps givenin Appendix A. These computations for the homogeneous case lead to the resultspresented in Chapter 3. For the case of an inhomogeneous ground state wereport only very selected fundamental steps sending the more interested readerto Appendix B at the end of this Thesis. All these other computations representthe analytical framework for the results presented in Chapter 4.

• In Chapter 3 we study how the electron-electron interactions renormalize theelectron-phonon coupling and determine the instabilities of a homogeneous sys-tem where phonons are adiabatically coupled to the electrons. In particular, fromthe instabilities of the charge susceptibility we recognize new incipient charge-ordered states evolving from the homogeneous one. After testing the reliabilityof our approach, we perform a methodical investigation, varying the doping, thelocal Coulomb repulsion and the momentum transferred in the electron-phononscattering on a wide range of values. In general, we explore analogies and dif-ferences among the trends of the charge susceptibility in one, two and infinitedimensions. We present the phase diagram and dedicate a particular attentionto the two-dimensional case, because it could give more relevant information forthe cuprates which are quasi-two dimensional layered compounds. In the lastSection we investigate how the Kohn anomaly in the phonon spectrum is renor-malized with strong electronic correlations in one and two dimensions. This studyis a natural trait d′union with the problem of the renormalization of the opticalphononic spectrum in the cuprates which gives the ultimate results of our inquiryin Chapter 4.

• In Chapter 4 we consider magnetic stripes and compute their charge response. Us-ing the obtained charge susceptibility, we solve the problem of normal modes as-sociated to the superlattice, thereby finding the optical phononic branches whichare to be compared to the experimental dispersion reported in Chapter 1.

• In the Conclusions we summarize the results presented in the previous Chaptersand discuss how our Thesis contributes to the research on strongly correlatedmaterials.

Chapter 1

Charge Density Waves andelectron-lattice coupling

In this mainly phenomenological Chapter we aim to justify on which experimentalgrounds the motivations of our Ph-D Thesis are founded. We are interested to studythe charge density waves (CDW’s) in strongly correlated materials and the competitionbetween electron-lattice coupling and electronic correlations. Therefore in this Chapterwe provide examples of CDW patterns in crystalline materials, with or without siz-able electron-electron correlations. Then we focus our attention on the physics of thecuprates which show charge- and spin-modulation patterns in their normal phase atintermediate doping. Moreover, these materials offer a natural playground to see whathappens when electron-phonon coupling competes with strong e-e interactions: someobserved quantities exhibit strong evidence of phononic effects, others quite a negligi-ble one. Within this controversial picture on the relevance of phonons, we dedicatea particular attention to the anomalous softening of the longitudinal optical phononbranch.

In the last years the general issue of electron-phonon (e-ph) coupling in the presenceof strong electron-electron (e-e) correlations has been raised in a variety of contexts.For instance, in the high-temperature superconducting (HTSC) cuprates recent pho-toemission experiments [1, 2, 3] indicate a sizable coupling of electrons with collectivemodes, possibly of phononic nature. The softening of a phonon peak in inelastic neu-tron scattering (INS) experiments [4, 5, 6], as well as features in tunnelling spectra [7],also suggests that electrons are substantially coupled to the lattice in these mate-rials. At the same time optical and transport experiments do not display a stronge-ph coupling except at very small doping, where polaronic features have been ob-served [8, 9, 10, 11]. In novel FeAs superconductors the role of e-ph coupling is stillunclear: LDA (Local Density Approximation) calculations predict a small coupling [12]but the first resistivity measurements show the evidence of a strong e-ph coupling for

6 Charge Density Waves and electron-lattice coupling

the superconducting samples, with a saturation-like trend at higher temperatures [13].An intriguing interplay between lattice and electrons has also been invoked to explaintransport in manganites [14], in single-molecule junctions [15] and in fullerenes [16],where a correlation-enhanced superconductivity has also been proposed [17].

All these examples show that the issue of e-ph coupling in the presence of stronge-e correlation is generally relevant and it translates into several related issues. Firstof all, the fact that various physical quantities appear to be differently affected byphonons indicates that the energy and the momentum structure of the e-ph couplingis important. In turn this emphasizes the role of e-e interactions as an effective mech-anism to induce strong energy and momentum dependencies in the e-ph coupling.Secondly, phonons may be responsible for charge instabilities. One possibility is thatthey mediate interactions between electrons on the Fermi surface giving rise to charge-density waves or Peierls distortions. It has also been proposed [18, 19] that a phonon-induced attraction gives rise to an electronic phase separation (PS), although in realsystems the latter is ultimately prevented by the long-ranged Coulombic forces withthe formation of nano- or mesoscopic domains, the so-called “frustrated phase sep-aration” [19, 20, 21, 22, 23, 24, 25, 26]). In fact, the formation of domains in realspace would lower the elastic stress associated to the electron-lattice coupling, but it ishindered by the electrostatic energy cost due to the charge imbalance. Typically, in asystem with substantially local interactions a macroscopic charge imbalance is allowed,and the onset of PS takes place. On the other hand, if long range interactions aresizable, PS is frustrated. However, 2d-systems are proner to the electronic PS thansystems in 3d (HTSC cuprates are layered materials and thus “quasi-2d”), as discussedin Refs. [24, 25].

1.1 Density waves

In this Section, I schematically illustrate the rich variety of charge and spin superstruc-tures detected in many classes of materials. For a general phenomenological treatmentof density waves in weakly correlated materials, one can see Ref. [27]. An updatedgeneral review of CDW’s in strongly correlated materials does not exist, to my knowl-edge. Since several evidences of charge patterns come from tunnelling experiments,one can consider Ref. [28], which is a review of the STM/STS (Scansion TunnellingMicroscopy/Spectroscopy) investigations on the HTSC’s with a dedicated part to localelectronic modulations observed over last years. Experimental evidences of stripes inthe cuprates are given in Ref. [29], within a more general theoretical approach to thetopic of charge order with strong electronic correlations.

Density waves are broken-symmetry states brought about by e-ph and/or e-e in-teractions. The CDW-material displays a periodic spatial modulation of the electroniccharge or spin density. Charge Density Wave (CDW)-ground states were first discussed

1.1 Density waves 7

Figure 1.1: Crystal structure of K0.30MoO3 [38].

by Frolich [30] and Peierls [31], whereas Spin Density Wave (SDW)-ground states werefirst postulated by Overhauser [32, 33]. Experimental evidences came later than the-oretical intuition and showed also that CDW’s and SDW’s can coexist in the samematerial.

We can distinguish into “classical” CDW’s and “strongly correlated” CDW’s: ingeneral, the mechanisms which lead to their formation are quite different. The for-mation of classical CDW’s is due to the Fermi surface topology (nesting and otherinstabilities). In corrispondence with these instabilities, the bare charge susceptibilityexhibits divergences at the typical CDW wave vector in 1d, and well pronounced peaksalso in 2d. On the other hand, the correlated CDW’s are the effect of an energy balance,typically between phase separation and electrostatic repulsion. In fact, as we will alsodiscuss later in our Thesis, phase separation can be frustrated by long range Coulombinteractions and in that case a charge-ordered state with a characteristic finite wavevector can arise. Short range magnetic correlations and proximity to an insulatingstate may also play a role.

Usually, a ground state with density waves is associated to a highly anisotropicband structure. In fact, these periodic patterns are found in organic and inorganic ma-


terials with ”quasi-one dimensional” or ”quasi-two dimensional” electronic structures:linear chains, layers ... The presence of 1d-structures can lead to stronger electronicinstabilities at momentum q = 2kF , kF being the Fermi momentum. Since in our The-sis we focus on charge-ordered systems, we dedicate our attention essentially to chargedensity waves in this brief survey on density waves in solids.

In weakly-correlated materials, the charge modulations come together with a strongelectron-lattice coupling and e-e interactions play definitely no role: we call these mod-ulations “classical CDW’s”. The mechanism driving to density waves in these com-pounds is substantially known. The electronic system presents charge/spin responsefunctions with divergences or huge maxima at some special values of the momentumq, due to the peculiar topology of their Fermi surfaces if compared to 3d-isotropic sys-tems. For example, the bare charge susceptibility of a 1d-system diverges at q = 2kF

and thus the formation of an electronic charge modulation with wave vector q = 2kF

and period L = π/kF is favoured with respect to a homogeneous configuration: theelectrons originates a “Peierls CDW”. Since in real materials electrons are coupled tothe ion lattices, such a deep reorganization of the electronic carrier distribution altersthe ionic lattice and the latter allows for a small deformation to lower the total elec-trostatic energy of the electron-phonon system. In fact, the ions move towards newequilibrium positions, in order to be at a shorter distance from the electronic wavepeaks: the “Peierls deformation” takes place. The ionic displacements are limited inextent not to exceed in the elastic energy cost. This is a simple scenario to illustratewhat happens in solids with the formation of these electronic CDW’s together withthe lattice Peierls transition1 and this is the starting point for the study I will providein Chapter 3: how sizable electronic correlations will change this picture in weaklycorrelated regime.

Now for illustrative purposes we mention some classes of materials which presentwell estabilished examples of “classical CDW’s”. Among inorganic linear chain com-pounds, there are three main groups: mixed valence platinum chain compounds,transition-metal chalcogenides and transition metal bronzes.

• Platinum chain complexes are composed of columnar arrays with Pt atoms withstrongly overlapped d orbitals; their most known compound isK2Pt(CN)4Br0.3×3.2H2O, also called “Krogmann’s salt”.

• Metachalcogenides combined transition metals M such as Nb, Ta, T i and Mowith chalcogen atoms C such as S or Se. They can present various characteristic

1Yet lately it has been argued by Johannes and Mazin that in real materials the lattice structuraltransitions should be treated less naively to give account accurately for the experimental results inclassical-CDW materials [34]. We will present and discuss their claims in our Conclusions becausefirst we need to present what happens to Peierls CDW with strong correlations, and these our resultsare in Chapter 3.

1.1 Density waves 9

substructures: a triangular prism with MC6 units and one or more linear chains.Based on similar chemical constituents, dichalcogenides, instead, represent themost important class of quasi-two dimensional materials with Peierls CDW.

• Metal bronzes, also known as “blue bronzes”, are compounds of formula A0.3MoO3

where A is an alkali metal such as K, Rb or T l. These compounds present rigidunits of clusters of MoO6 octahedra (Fig. 1.1).

Because of their complicated crystal structures, these materials present a wide va-riety of phonon modes, not so easily obtainable in band-calculations. Inelastic neutronscattering (INS) experiments can determine which phonon modes are coupled to elec-tronic degrees of freedoms. Then one can study the softening of the spectrum occurringto phonons when coupled to electrons. This softening is maximum at q = 2kF : theKohn anomaly [35]. In § 3.4 we study the phonon softening of systems with PeierlsCDW also in the presence of strong electronic correlations.

For the particularly interested reader, several reviews are available concerning low-dimensional inorganic conductors: one can see for example Refs. [36, 37, 38]. Alsoorganic linear compounds exist, for example charge-transfer salts with methyl groupsand large π-orbital overlap along the chain direction.

The charge (and spin) modulation in strongly correlated materials may be at-tributed to electronic correlations alone or to both correlations and coupling to thelattice; then the electronic modulation will induce lattice distorsions easy to be ob-served in diffraction experiments such as neutron scattering. The nature of the chargeorder in correlated systems is quite different from classical CDW’s: it is not directlyrelated to a Fermi Surface instability (due to nesting or van Hove singularity) andCoulomb effects are faintly screened. In § 1.2.1 we will report several experimentalevidences of stripes in the cuprates obtained using the INS technique.

In strongly correlated materials the formation of these charge patterns generallyoccurs in association with spin patterns. In the cuprates there are many evidencesof charge superstructures: stripes, first detected and recognized by Tranquada et al.in Ref. [39], and checkerboards [40] (see Fig. 1.2). Charge inhomogeneities are alsocommon to many other classes of strongly correlated oxides than the cuprates, such asthe nickelates [41, 42], the manganites [43, 44, 45, 46] and the cobaltites [47].

Now we want to illustrate the concepts of stripes and checkerboards better. We willconsider stripes with more attention because in our Thesis we are interested essentiallyto their evidence in cuprates.

”Stripes” are unidirectional density wave states with charge modulations, usuallyassociated to spin modulation. They are broken-symmetry states, in that they breaktwo lattice symmetries: the translational one and the invariance under 900-rotations inthe orthogonal direction. In the materials the stripe charge modulation can be eithervertical or diagonal with respect to the lattice axis and the positions of the lattice ions.


Figure 1.2: STM results on a sample of Ca1.90Na0.10CuO2Cl2: real space conduc-tance map (left) and Fourier transform map (right). A periodic 4a× 4aelectronic modulation is detected [40]. Notice that the map on the leftis properly a modulation of the density of states. There is yet no fullconsensus that it implies a real charge modulation.

”Checkerboards” (also known as “eggboxes”) are two-dimensional CDW’s which canbe visualized with the superposition of orthogonal stripes in the same domain, withoutbreaking the mentioned 900-rotational symmetry.

Stripes and checkerboards are examples of charge order in strongly correlated ma-terials: their electronic modulations can be static/quasistatic/dynamical, sometimesassociated with some type of disorder and impurities. In superconducting oxides, theyhave been clearly detected in the normal state.

As partly foretold, in recent years many evidences of charge order in the cupratesand in the oxychlorides have come from tunneling experiments (consider for exampleRefs. [40, 48, 49, 50, 51, 52, 53, 54]). In fact, STM probes can resolve spatial chargemodulations especially if these are static or slowly fluctuating. And actually periodicspatial modulations of the local DOS with a periodicity of about 4a have been de-tected (a being the lattice parameter). The detection of these weak periodic electronicmodulations in the superconducting state is made difficult by the presence of other in-homogeneities (pinning due to impurities, domain frustration ...) which tend to mask

1.2 Phenomenology of the cuprates 11

them. In general, in order to discriminate relevant and spurious signals, the experimen-tal analysis is performed not directly on the real-space images but on Fourier-transform(FT) maps of them. Using this approach, Howald et al. found four spots as dominantfeatures in their FT maps and conclude they could not be attributed to any otherelectronic effects but one dimensional stripe-like patterns [50].

There are also examples of weakly correlated CDW-materials which exhibit spectralproperties with unexpected similarities to strongly correlated materials, for instancedichalcogenides. These compounds present an anomalous metallic phase [55] with aphenomenon of “stripe formation” [56]. However, their filamentary stripe phase has aFermi-surface origin: their CDW’s are incommensurate with the lattice and phase fluc-tuations of the density are allowed energetically. Moreover, in the 2d-CDW dichalco-genide 2H-TaSe2 an anisotropic gap, the so called “pseudogap”, was observed [57].This normal state-pseudogap seems to resemble what seen in HTSC cuprates.

On the other hand, in strongly correlated oxydes in their striped phase relevantfeatures in the detected phonon branches can be associated to the presence of these 1d-electronic substructures which could determine Kohn-like anomalies in the spectrum,as we will show in our results in Chapter 4. Such experimental evidences are discussedin Section § 1.5.

1.2 Phenomenology of the cuprates

The High Tc (critical temperature) superconductivity represents one of the most inter-esting phenomena in Condensed Matter Physics. The cuprates are the most studiedmaterials with this property: they exhibit the highest Tc (138 K) at ambient pres-sure [59].

These materials can be either hole-doped or electron-doped by chemical substi-tution: in this Thesis we will consider only hole-doped compounds and the relatedevidence. For a broader review of the earliest experimental results and theoreticalissues, one can see Ref. [58].

After more than twenty years of investigations, a theory which explains the veryrich phase diagram of the cuprates is still lacking. These materials are characterised bystrong electron-electron (e-e) correlations. The undoped compounds should be metalsaccording to the standard Bloch’s band theory because they have a half-filled electronicband; on the contrary they are Mott insulators because a large e-e repulsion reduces thecharge mobility. Such a strong electronic correlation decreases under doping but it isstill basic to explain the peculiar physics of these compounds at least for small dopings.Upon increasing the doping, the doped Mott insulator evolves into a superconductorwith anomalous metallic properties in the normal state. We distinguish three rangesof doping regimes: “underdoping” if Tc is increasing with doping towards its maximumvalue; “optimal doping” if Tc is the highest; “overdoping” if Tc is decreasing. In the


underdoped regime there is the opening of a pseudogap below a crossover temperatureT ∗. The pseudogap is a depression of the spectral weight at the Fermi level in theabsence of an obvious broken-symmetry state, which shows up in different physicalquantities and gives the anomalous metallic behaviour. Also around optimal dopingthere are strong signatures of a non Fermi-liquid (FL) behaviour; for example theresistivity is linear in T up to high temperatures. In the overdoped regime a seeminglyFL behaviour is recovered. Experiments provide clear evidence that the cuprates aresusceptible towards the formation of electronic charge and spin inhomogeneities whichsometimes acquire long range order [39, 54, 60, 61, 62, 63, 65, 66] (see for exampleFigs. 1.5, 1.8 and 1.9).

In recent years, beyond the effects of e-e correlation, relevant fingerprints of phononshave emerged, particularly from the experiments on underdoped samples. For examplethere is a possible evidence of a notable electron-phonon (e-ph) coupling and of astrong isotope effect in photoemission spectra [1, 3, 67, 68], although others try toexplain these experiments mostly in terms of an incipient charge or spin order. On thecontrary, other quantities seem to be almost unaffected by the e-ph coupling, such asthe resistivity. To understand this puzzling behaviour, at least two issues have to beconsidered:

• these materials present several energy scales, associated with the quasiparticlecoherent scale, the Mott-Hubbard gap, the magnetic excitations ... ;

• the phononic features are renormalized in presence of the Coulomb repulsion.

Therefore many observed effects could indicate the presence of a relevant e-ph couplingin these materials. However the e-ph interaction has a different behaviour than in con-ventional weakly-correlated materials, due to the strong electronic screening action.In particular, the renormalization of the electron-lattice coupling and its connectionswith charge inhomogeneities are at the core of this Thesis, as it will be quite clearespecially from the reading of Chapters 3 and 4. In this Chapter we want to discusshow and to which extent the strong correlation and the electron-phonon coupling arerelevant for the physics of the cuprates, on the basis of the experimental evidences.Moreover we will underline the signatures of charge superstructures in these materi-als. In particular, in this Section we are focusing especially on the consequences ofstrong correlations, while in Sections § 1.3 and § 1.5 we will discuss better the exper-imental evidences about the electron-phonon interaction in the cuprates. In Fig. 1.3we show a schematic phase diagram of the cuprates, namely of the monolayered com-pound La2−δSrδCuO2 (LSCO). The antiferromagnetic order of the undoped insulatoris rapidly destroyed with increasing doping until the superconducting dome appears.The highest temperature Tc is achieved for a value of the doping δ = 0.15 ÷ 0.20.The physics of the cuprates is strongly influenced by their highly anisotropic structure,formed by two-dimensional copper-oxide (CuO2) planes separated by layers of other


Figure 1.3: Schematic phase diagram of the cuprates [69]. In this figure the dopingis indicated with x and not with δ.

atoms (Ba, La, Sr, O) (see Fig. 1.4). The most relevant physical processes are ex-pected to take place in the CuO2 planes where there are the charge carriers, partlycoming from the intermediate reservoir layers. One can also recognize perovskite-likeCuO6 octahedra where the Cu atom is at the center, four oxygens are planar and theother two oxygens are apycal. A large local Coulomb repulsion is present on the Cusites. At half-filling, the localized spins on the Cu sites align with antiferromagneticlong-range order, as obtained using a single-band model: the Hubbard model [70] (seeChapter 2 for a longer discussion). When these materials are doped by chemical substi-tution, the added holes are mainly distributed on the oxygen sites, and thus one mightquestion the validity of a single-band model. Nevertheless, as showed by Zhang andRice [71], the more realistic three-band model, in which the oxygens px and py orbitalsare considered along with the Cu d orbital, can be reduced, away from half-filling, to aneffective single-band one. Thus many studies focused on the Hubbard model, viewedas an effective single-band model for the cuprates [this is also our case], and also onits large-U limit for finite doping, the t-J model. In fact, in the original Anderson’sintuition [72], the t-J model is the ideal framework to explain the compresence of theMott insulating state and of the superconducting state in the cuprates. His idea isbased on the concept of spin-charge separation and the existence of a resonating va-


Figure 1.4: Crystal structures of La2−δSrδCuO4 (left) and Y Ba2Cu3O7−δ (right) [58].

lence bond (RVB) state made of spin singlets between neighbouring sites. Given thatin the whole our Thesis we study anomalies of the normal state, we will not discuss thepossible nature of the superconducting state and its connections to the insulator. Yetthis is still a very relevant open issue in the physics of these compounds: one can seefor example the extremely recent Ref. [78] concerning Bi2Sr2CaCu2O8+δ (BSCCO).

All the anomalous characteristics of the cuprates we have mentioned present cleardeviations from the FL paradigm which rules the physics of standard correlated metalsand this shows how much the physics of the cuprates is influenced by the presenceof a sizable Coulomb repulsion. Nevertheless a theory entirely relying on electroniccorrelation can not explain all the physics of these compounds.

In fact, in the oncoming Section § 1.3 we will show that purely electronic interactionsare not the only ingredient in the physics of cuprates. Pronounced phononic effectshave also been detected by different experiments. However, the evidence of the e-phcoupling does not appear so univocal:

• substantial isotope effects are visible in quantities like the pseudogap temperatureT ∗ and the superfluid density,


• while signs of phononic scales are invisible in resistivity and the superconductingtransition temperature Tc is only weakly dependent on the mass of the ions atoptimal doping.

1.2.1 Stripes in LSCO

Figure 1.5: (a) Momentum space. Large full disks are for the fundamental Braggmagnetic peaks; small full disks indicate the superlattice peaks in theLTT phase; empty disks and squares mark the magnetic superlatticepeaks originated by two different stripe domains; diamonds and trianglesare for the charge superlattice peaks due to the two stripe domains. (b)Real space. Stripe order model with holes and spin density schematizedfor a system with δ = 1/8. (c) Real space. Domains with orthogonalstripes on nearby CuO2 planes for a material in the LTT phase [60].

In 1989 Zaanen and Gunnarson first predicted the stripe formation in doped Mottinsulators [73] but only in 1995 Tranquada and his coworkers could interpret INS datameasurements in the cuprates consistently within a stripe picture [39]. Using a sampleof Nd-LSCO (La2−x−yNdySrxCuO4) with y = 0.4 and x = 0.12, they observed thatthe commensurate magnetic peak Q = (π/a, π/a) was shifted by a quantity ǫ = π/d,giving rise to four incommensurate peaks: ǫ is the (magnetic) incommensurability.In addition, new Bragg peaks emerged at the points (±2ǫ, 0) and (0,±2ǫ). These


Figure 1.6: Charge superlattice peaks for La1.48Nd0.4Sr0.12CuO4 [60]. Notice thathere and in other figures following the momenta are given in reciprocallattice units (r.l.u.), i.e. dropping the dimensional prefactor 2π/a.

results suggest that charges form domain walls separated by a distance d and thatthe staggered magnetization undergoes a π-phase shift while crossing them. The peakpositions indicate that the stripes are along the vertical and the horizontal direction,with a density of one hole every two Cu atoms (see Figs. 1.5 and 1.6). These staticstripes can be detected in these co-doped compounds thanks to a structural transitioninduced by the Nd atoms: they produce a buckling of the oxygen octahedra aroundthe Cu atoms and then the transition from the LTO (Low-T Orthorombic) to the LTT(Low-T Tetragonal) phase. Note that the chemical susbtitution which favours thestripe formation suppresses the superconducting state of the crystal and this indicateswhy it so difficult to achieve evidences of stripes and superconductivity in the samesample.


Another experimental condition which favours the stripe formation in the cupratesis the so called ”1/8 anomaly”, first observed in LBCO (La2−xBaxCuO4) systems [74],but later found to be common to all the cuprates. At the ”magic” doping δ = 1/8 theresistivity measurements signal a strong reduction of the superconducting transition Tc.In the light of the stripe picture provided before, one can explain this anomaly thinkingthat the Ba-substitution is responsible for a structural transition which eventuallypins the stripe order and reduces the superconducting phase to lower T . Then the”1/8 anomaly” can be enroled as one more evidence that static charge stripes andsuperconducting state compete. The interplay between fluctuating charge order andsuperconductivity is under an open debate, but we are not going to discuss it, since wewill be mostly concerned with (quasi)static stripes (see also Section § 1.5).

Insofar we have reported evidences of stripes in LSCO family compounds for δ = 1/8but there is a large amount of experiments on samples with different dopings between0.03 and 0.18 which confirms the presence of stripes. Evidences are not limited to theincommensurate peaks detected in neutron scattering experiments, but also come fromNuclear Magnetic Resonance, Nuclear Quadrupole Resonance, Muon Scattering Res-onance, Hall transport (for detailed references see Refs. [29]) and especially ARPES(Angle Resolved PhotoEmission Spectroscopy) starting from Refs. [75, 76]. We willconsider photoemission results on LSCO samples in Section § 1.3, within a wider dis-cussion on the relevance of e-ph interactions to explain some peculiar spectral featuresobserved in these materials. After methodical scattering experiments on underdopedsamples, it has been realized that the incommensurability ǫ is a linear function ofthe carrier concentration δ for 0.05 < δ < 0.12; for δ > 1/8 the curve ǫ(δ) tends tosaturate [62]. This trend has been reported in Fig. 1.7. This behaviour has been soexplained: for δ < 1/8 the stripe filling is fixed at one hole per two Cu sites and thisimplies that by increasing the amount of the charge carriers in the system upon doping,the stripe number increases and their average separation d(δ) reduces. In other words,the system finds the configuration most convenient energetically by keeping half-filledstripes and drawing up the domain walls. For larger dopings, the stripe filling increasesand the wall distance is kept at d = 4a, signalling the transition to a more homogeneouselectronic phases, as expected. Theoretically, the behaviour of ǫ(δ) and its saturationis one of the early findings of the studies with the Gutzwiller approximation (the mean-field method we use and describe in Chapter 2) applied to the stripes, as one can seein Ref. [64].

Scattering experiments have detected stripes also in lightly underdoped lanthanumcuprates but with different appearance (see for example Ref. [77]). In fact, for 0.02 <δ < 0.05 stripes are “diagonal”, in that incommensurate peaks in the Fourier mapsare rotated by 45o. Then these half-filled stripes would show analogies with the onesobserved in the nickelates. Alternatively the magnetic spots could be interpreted asthe evidence of a spiral magnetic phase.


Figure 1.7: Magnetic incommensurability. This is the Yamada’s plot which gathersdata coming from different experiments on Sr-, O- and Nd-doped sam-ples, on Zn-substituted samples and on O-reduced ones [62]. Be awarethat the notation used in this figure is misleading with respect to theconvention we adopt in the text. In fact, in the figure above δ is ourincommensurability ǫ and xeff is our chemical doping δ.

I would like to make an overall remark. The connection between stripes and otheranomalies is not clear at present. However, it is interesting that the three differentbehaviour of incommensurability and stripes correspond to three different regimes:

• spin-glass for 0.02 < δ < 0.05;

• “strange metal” for 0.05 < δ < 0.13;

• metal with reduced anomalies with respect to the FL scheme for δ > 0.13.

In summary, at present the presence of stripes is firmly estabilished in lanthanumcuprates and their nature have been revealed: charge stripes are static in Nd-LSCO(in general, in LSCO co-doped with rare earth atoms) and dynamical in pure LSCO.In other cuprates, such as BSCCO and YBCO, we are far to draw a similarly completepicture: we know that there are evidences of charge order, of fluctuating and statically


Figure 1.8: Stripe orbital pattern derived using the GA+RPA calculation applied tothe three-band Hubbard model in Ref. [64]: peak and ’trough’ locationsare indicated [65]. Occupancies consistent with the experiment performedon LBCO at δ = 1/8 are shown. For our purposes these results arevery important because they show that the charge profile predicted withGA+RPA method fit the experimental one. The GA+RPA is the methodwe will use in this Thesis and it is described in Section § 2.4.

disordered stripes, but many issues of these phases are still unclear. This is the reasonwhy we concentrate on experimental results attributed to stripes in LSCO and try toreproduce them within a stripe picture in Chapter 4.


Figure 1.9: LDOS (Local Density of States) map for Bi2Sr2Dy0.2Ca0.8Cu2O8+δ; inthe inset its Fourier transform is displayed [66]. The charge spatial ar-rangement forms a Cu-O-Cu bond-centered electronic pattern withoutlong-range order but with 4a-wide unidirectional electronic domains dis-persed throughout (a is the Cu-O-Cu distance). For our computationsin Chapter 4 we will consider a bond-centered stripe state, i.e. with thecharge imbalance centered in its bonds

1.3 Experimental phononic features in the cuprates

In this Section we will briefly describe the phononic evidences in the cuprates: they canindicate that the electron-lattice coupling in these materials is deeply affected by theelectronic correlations. Standard treatments of the electron-lattice correlation fail andmost experimental results can not be reproduced satisfactorily unless one considers thestrong e-e correlations. These evidences come from different experiments performed onvarious cuprate compounds.

The first ones concern the isotope effect on the transition temperature Tc: all thestrongly correlated superconductors exhibit an effect significantly small if compared tothe predictions of the BCS theory [79, 80, 81]. Other quantities are more sensible tothe isotope substitution than Tc, for example the in-plane penetration depth, as shown

1.3 Experimental phononic features in the cuprates 21

with different experimental techniques [82, 83, 84].A dependence on the mass of the oxygen atoms has been observed also in the

pseudogap temperature T ∗, which shows a larger effect compared to that of Tc and in-terestingly with a shift of opposite sign with respect of the shift of Tc: T

∗ increases a lotwhile Tc decreases marginally [85]. A justification was given to this last behaviour [86],but there is not a general explanation on the wide gamma of phononic effects dependingon the observed quantities.

Figure 1.10: Dependence of the planar resistivity with temperature for optimallydoped crystals. In particular, Bi2201 and Bi2212 indicate two samplesof BSCCO with one or the other isotope; 214 LSCO is for measure-ments on LSCO at δ=0.15 and 123 90 K for measurements on YBCOat δ=1 [87].

Also some anomalous trends of the resistivity can reveal signatures of the electron-phonon interaction in these materials [87]. We are considering the planar resistivityρab which is extremely low in comparison with the orthogonal axis resistivity ρc. Ina conventional metal with negligible e-e interactions one expects ρ ∼ A + BT 5 forTc < T < TD, being Tc and TD the superconducting temperature and the Debye tem-perature, respectively. This behaviour is explainable within a framework of electron-phonon scattering; for higher T , the normal resistivity is linear with T . In a correlatedmetal there is also a T 2-contribution due to the e-e interaction. This overall picture issubstantially inadequate to give account for the experimental evidence in the cuprates,in which ρ is strongly doping-dependent, shows a linear variation with T well belowTD and on the other side ρ exhibits high T -power laws T a where a 6= 1 in general (see


Fig. 1.10). Therefore the behaviour of the resistivity is far from being understood, butit reveals much weaker or even apparently absent phononic effects for a wide range oftemperatures [88].

Figure 1.11: Evidence of the LE kink (indicated with an arrow) in the dispersion ofLSCO and BSCCO samples for different dopings and temperatures. Allthe panels show the sudden change in the nodal direction (0, 0)− (π, π)except for the inset of the panel b which reports the sharpening of kinkmoving away from the nodal direction. In particular, one can see thedoping-dependence of the kink in LSCO (panel a) and in BSCCO (panelb-c), then the temperature dependence in LSCO (panel d) and BSCCO(panel e) [1].

In this decade many interesting results come from photoemission ARPES experi-ments on different cuprates. They evidenced the presence of a Low-Energy (LE) kinknear 70 meV in the electronic dispersion, both in the nodal direction [1, 67, 68] andin the antinodal one [89] (see Fig. 1.11). Similarly, optical measurements detected asharp feature in the self-energy [90]. In the photoemission spectra, the position of thekink is almost fixed independently of the doping level.

This could be interpreted as the signature of a coupling to the same half-breathingin-plane Cu-O bond-stretching phonon detected by INS experiments. Preliminary

1.3 Experimental phononic features in the cuprates 23

computations indicate that this effect is compatible with a phononic origin [91], thoughother descriptions using spin degrees of freedom [92, 93] and charge excitations [94] havealso been given. Within the explanations with phonons, some issues have not beencompletely clarified. In fact, a strong anisotropy of the electron-phonon interaction isrequired to account for the different behaviour between the nodal and the anti-nodaldirection and it is difficult to reconcile with the fact that the High-Energy (HE) partof the dispersion depends on the doping in an anomalous way while, closer to Fermilevel, the band remains almost untouched.

Recently Giustino et al. made a considerable effort to justify the observed pho-toemission spectra of LSCO, using first-principle calculations to describe the electron-phonon interaction [95]. Actually, before Ref. [95], other calculations had found arather weak e-ph coupling in Ca0.27Sr0.63CuO2 [96] and in Y Ba2Cu3O7 [97], but thosestudies were not specifically addressed to phononic signatures in the photoemissionspectra in LSCO, as instead Giustino et al. did. In fact, they looked for quite anaccurate quantitative accord with the the detected LE kink within their framework.They found that the phononic contribution to the renormalization of the electronicenergy is too small (about 6 times less) to reproduce the experimental dispersion. Thismeans that phonons play not an exclusive role to renormalize the cuprate bands, asclaimed first in Ref. [1]. However, Giustino and coworkers do not analize alternativescenarios, for example involving magnons. Other recent LDA computations performedby Heid et al. on superconducting YBCO find that the electron-lattice coupling is notstrong enough to reproduce the well pronounced LE kink [98]. They drawed this con-clusion, using a realistic phonon spectrum and comparing the momentum dependenceof the e-ph coupling and of the self-energy. Yet, the debate on the effective role of thephonons is far to be concluded while we are writing our Thesis. In fact, in October2008 D. Reznik et al. questioned the validity of these density functional theory calcu-lations [99]. They remarked that such approach fails to reproduce other huge phononiceffects known beyond photoemission results and is not so accurate to reproduce thefeatures of phonon softening in the experiments we discuss in next Section § 1.5. Thisis a definite problem for an “ab initio” computational framework. Ref. [95] treats elec-trons and phonons as independent entities which scatter each other and does not useany adjustable parameter: Reznik and coathours objected that this scheme seems inad-equate to give account of relevant strong correlation effects on phonons. More recentlynew anomalous features have arisen interestingly in the photoemission spectra of thecuprates. These evidences make theorists try to elaborate a proper physical frameworkto investigate the role of e-ph coupling and charge modulation but also of magneticcouplings and other effective interactions. Namely, one must consider the evidence ofthe above discussed LE kink in the electronic dispersion (around 0.05-0.08 eV ) and thepresence also of a HE kink (around 0.35-0.40 eV ), first detected by Lanzara’s team [100]and later seen by other experimentalists [101, 102, 103, 104, 105, 106, 107]. In associa-tion with the HE kink, another anomaly is observed: the “waterfall”, i.e. an extended


Figure 1.12: ARPES intensity maps of BSCCO samples for three different dopings.The thin arrow indicates the LE kink position at energy E0 and thethick arrow the HE kink position at energy E1. Below the HE kink, onecan see the vertical dispersion of the waterfalls [100].

vertical part of the quasiparticle dispersion whose tail ends around 0.7 eV (Fig. 1.12).The existence of these kinks at two different energy ranges is very important becauseit signals two distinct mechanisms of the band renormalization, in addition with thenarrowing effect due to e-e correlations. The typical energy of the kinks results to bea crucial property to identify the physical coupling. The LE kink is the signature ofa bosonic mode strongly coupled with the electrons, as we mention above. Also thephysical interpretation of the HE kink and the waterfall structure of the electronic dis-persion is presently under a strong debate. Researchers tend to rule out the hypothesisof a phononic effect because it would be at a too high energy. Instead, the kink energywould suggest a magnetic origin. Clearly, the possible magnetic nature of the couplingand the associated charge-spin textures will require further investigation to best fit theexperimental data.

1.4 Scattering experiments and determination of the phonondispersion 25

1.4 Scattering experiments and determination of

the phonon dispersion

There are many experimental methods to study the lattice-dynamical properties of asolid; the most powerful and direct ones involve the scattering (or the absorption) ofradiation by the crystal. Among these methods, Inelastic Neutron Scattering (INS)and Inelastic X-Ray Scattering (IXS) can provide the most detailed information todetermine the phononic dispersion curves of a solid. In this Section we will presentsome features of these experimental methods before describing some recent results ofscattering experiments in Section § 1.5.

In the INS experiments the energy of the beam neutrons is very low if compared tothe X-Ray energy: the scattering neutrons are responsible mostly of one-body emissionsfrom the crystal, for example one-phonon emission. Since we are interested preciselyto determine the phonon dispersion, this is a great advantage because the one-phononcontribution is the most important one in the total INS cross section and this makesthe mode analysis easier. On the other hand, in X-Ray scattering there are severalrelevant contributions to the total cross section other than the one-phonon processes,due to multiphonon emissions, Compton scattering ... This causes larger uncertantiesin the data analysis. However, also IXS experiments are performed to detect thephonon dispersion because generally their experimental setup allows a better resolution,especially in the momentum, in comparison to the resolution of the spectrometerstypically used in neutron scattering experiments.

Since in Section § 1.5 we will describe several relevant INS experiments showingphonon anomalies in the cuprates, here we intend to discuss some selected issues specif-ically about this method; for further reading one can see Refs. [108, 109, 110, 111], forexample.

When the neutron beam interacts with the crystal, the neutrons scatter with thelattice atoms. According to the energy and the momentum transferred in the scattering,the crystal can emit phonons, magnons ... This illustrates why neutron scatteringcan probe collective excitations in solids. Typically experimentalists use thermal ormoderately epithermal neutrons, with energy up to 0.1 eV : in this case the maincontribution to the total cross section and then to the intensity of the detected signalis given by one-phonon events, i.e. with the emission of a phonon. Multiphonon eventsand other processes play quite a marginal role and can be considered as backgroundevents.


The differential cross section for the coherent one-phonon inelastic scattering canbe presented in the following form:

(d2σ

dΩdE

)(Q, E) ∝

∑

qQn

|F nqQ|2

〈νn + 1〉Ωn

q

δ(Q − q − Q)δ(E − Ωnq) (1.1)

where F nqQ is the one-phonon structure factor and |F n

qQ|2 provides the intensity SnqQ of

the n-th phononic mode (its “spectral weight”):

SnqQ = |F n

qQ|2 ∝∣∣∣∣∣∑

iα

Q · enqαe

iQ·riα

∣∣∣∣∣

2

(1.2)

Let us specify the meaning of the quantities in the Equations above.riα indicates the position of the α-atom on the lattice site i.νn is the occupation factor for the n-th phonon mode.Q and E are respectively the momentum and the energy transferred in the neutron-

atom scattering. Due to the energy-momentum conservation laws determined by thedelta functions in Eq. (1.1), in these scattering events there is the emission of a phononof (pseudo)momentum q and energy ~Ωn

q; in addition, Q = q + Q, with Q belongingto the EBZ (Extended Brilluoin Zone), q to the RBZ (Reduced Brillouin Zone) andQ to the reciprocal lattice. Upon modifying the experimental setup, one can obtainΩn

q for several q and thus determine all the phonon dispersion curves. Then Eq. (1.2)discriminates the contribute of each phonon branch to the overall intensity.

Finally, enqα is the polarization vector for the n-th phonon mode.

The previous considerations together with Eqs. (1.1) and (1.2) will be helpful tointerpret the INS (and IXS) experimental results which we will report and discuss inthe following Section.

In particular, Eq. (1.2) indicates how to evaluate the intensities associated to thephonon branches determined in a neutron scattering experiment. Essentially, we willexploit this Equation in our computations in Chapter 4 in order to reproduce theexperimental results.

1.5 Anomalous phonon softening 27

1.5 Anomalous phonon softening

Figure 1.13: Displacement pattern of the O ions for the half-breathing longitudinaloptic phonon with wave vector q = (π/2a, 0, 0). The ion displacementsare indicated with arrows; the small full circles represent the O atomsand the large circles the Cu atoms. One possible explanation for theanomalous phonon softening is that there are dynamical stripes in thematerial and the phonon propagates perpendicularly to them, as shownin the picture above where open large circles denote hole-poor antiferro-magnetic regions and filled circles represent the hole-rich stripe lines [4].In our Thesis we will also consider the possibility that the phonon prop-agates parallel to the stripes.

The crystal structure of the cuprates allows for several phonon modes, both “out-of-plane” and “in-plane” with respect to the CuO2 plane. The most important out-of-plane lattice vibration is represented by the motion of the apycal oxygens perpen-dicularly to the CuO2 plane. Among the plane-polarized modes, one can consider themotion of all the four oxygen atoms (“full-breathing”) or the one involving only two ofthem (“half-breathing”) and due to the bond-stretching. This lattice vibration givesrise to higher energy longitudinal optical (LO) phonon and is illustrated in Fig. 1.13.

The experimental investigation of the former phonon branch resulted to be veryimportant to understand the role of the electron-lattice correlation in strongly corre-lated materials. In fact, the bond-stretching spectra detected in scattering experimentsshow peculiar anomalies, both in the large softening and in their broadening. Bond-stretching modes have long been known to show anomalies related to doping [112] andwhose origin is still an open issue. The mechanism claimed to explain this anoma-


lous softening involves relevant e-ph coupling and most likely the formation of stripedomains.

Figure 1.14: Dispersion of the highest longitudinal optic phonon mode in LSCO (toppanel) and linewidths determined by fitting a Gaussian line shape tothe observed profiles (bottom panel). The ellipse illustrates the instru-mental resolution, as in Fig. 1.15 [116].

Usually, phonon branches measured by neutron scattering can be quite well re-produced by (shell) models based on the interaction potential and including screen-ing effects [112, 113, 114], but for the higher energy (70-80 meV ) in-plane bond-stretching mode, which softens anomalously upon doping along the [100] direction [4,5, 6, 112, 115, 116, 117, 118] in LBCO and in LSCO. These spectra exhibit a verystrong asimmetry for q ∼ (π/2a, 0, 0) (for δ ∼ 1/8) both in the dispersion and inthe width. In Figs. 1.14, 1.15 and 1.16 we report this anomalous dispersion as mea-sured in Refs. [4, 116]. Other observations in optimally-doped YBCO show that inthis compound the in-plane oxygen bond-stretching mode has a very steep dispersionalong the [100] and [010] directions, while the [110] direction is much less anomalouslyrenormalized [119]. Therefore, this giant phonon anomaly seems to be common to theCuO2 planes of the cuprates, with a strong anisotropy between the [100] and the [110]directions.

The theoretical reproduction of these phononic evidences is not an easy task: thee-ph coupling is anomalously strong and the standard LDA calculations which usu-ally describes phonons very well seem not to work properly. A first explanation has


been given within the t-J model with e-ph interactions derived from a three-bandmodel [120]. In general, the interpretation of the experimental bond-stretching branch

Figure 1.15: Contour plot of the scattering intensities observed in LBCO at δ = 1/8and T = 10 K. The intensities below 60 meV are associated withthe plane-polarized Cu-O bond-bending vibrations and the intensitiesfor higher energies reveal the bond-stretching phonon branches. Thedashed line determines the position of the charge order wave vector [4].The experimental setup was performed to maximize the structure factorsand thus the intensities of the bond-stretching phonon modes.

has been an open debate for ten years. Such a strong renormalization of the phononfrequency of an undoped system upon adding free carriers denotes a strong coupling tothe correspondent phonon mode which would favour charge ordering, as first suggestedin Refs. [115, 116]. In particular, stripe patterns with periodicity of 4a would be relatedto the lattice distortion with q = (π/2a, 0, 0). McQueeney and collaborators claimed


that the key role is played by quasistatic stripes, i.e. with electronic fluctuations slowerthan the bond-stretching mode dynamics. Within this scenario, a second phonon modeappears at lower energy: at q = (π/2a, 0, 0) the bond-stretching mode is weakened andundergoes a spreading over an energy window of 10 ÷ 15 meV . This behaviour hasbeen elaborated in analogy with what happens in some charge-ordered bismuthatesand nickelates. Pintschovius and coworkers analyzed their data in a different way:they found a smooth dispersion with an anomalous high slope and broadening onlyaround the charge propagation vector, as one can see in Fig. 1.14.

Ref. [4] represents a step forward, because Reznik and coworkers could performscattering experiments with a better resolution. Using LBCO at δ = 1/8, a materialwith “quasistatic stripes” (have in mind Ref. [65] and thus Fig. 1.8), they find a strongenhancement of the phonon broadening and assign it to a sharp softening at the stripewave vector qc. The anomaly is strongest at lower temperatures and can not be at-tributed to anharmonic vibrations or to impurity effects. They conclude that also theanomalies observed in other cuprates are reminiscences of this sharp softening, due toincipient dynamical stripe order. In particular, they observe that the double degener-acy of the bond stretching mode is lifted in their results: one component follows theexpected cosine-like dispersion, while the other deviates so presenting a much sharperdip, a low-energy tail near qc (see Figs. 1.15 and 1.16). They interpret this behaviouras a Kohn-like anomaly due to stripes which lift the degeneracy: the splitting of thephonon mode is compatible with the crystal symmetry in the ordered stripe phase.

The scenario proposed by Reznik is gaining consensus, yet more direct measure-ments of the dip would be useful to clarify the problem. In fact, in Ref. [4] the dip is notdirectly visible in the data: the authors deduced its existence from the broad shoulderon the low-energy side of the phonon mode. To attain more direct evidences Inelas-tic X-ray Scattering (IXS) experiments have been performed recently for underdopedsamples (Ref. [117]) and very recently (Ref. [118]) for optimally doped samples. TheIXS experiments have an energy resolution comparable with the one of INS ones, butthe momentum resolution is higher. These recent works fully confirm the picture givenby Reznik and collaborators, with the direct observation of the dip in the phononicdispersion. The data are consistent with the two-phonon mode hypothesis and inter-estingly the effect persists in the superconducting state, despite the charge stripes tendto delocalize.

The interpretation of these phonon spectra within a stripe picture is what we aim todo in Chapter 4. It sounds a reasonable idea, due to the strong evidence of the presenceof stripes in LSCO family compounds, as discussed in § 1.2.1 and above all thanks tothe direct measurements of quasistatic charge stripes in LBCO using resonant soft X-rays scattering [65]. According to some recent experiments, stripes could even survivein the superconducting state; yet they would be hardly detectable because no longerstatic [121, 122]. The route to explain these experiments with stripes has already beenused by Mukhin et al. in Ref. [123], using a semiclassical toy model. They suggest to


explain the observed strong anomalies in the high-energy LO phonon spectrum as dueto the enhanced electronic polarizability associated with self-organized 1d-stripes. Theanomaly should occur at momenta parallel to the stripes. Also Citro et al. attribute thebond-stretching phonon anomalies to incommensurate CDW instabilities [124], using acumulant-RPA (Random Phase Approximation) approach which generalizes a previoustheoretical study [125].

In this Chapter we have described various evidences about a substantial electron-phonon coupling in the cuprates and the difficulties to arrange them in a consistentframework. Indeed, a full theoretical comprehension of the coupled electron-phononsystems in the presence of strong electronic correlations has not reached yet. Thephysics of e-e interactions and phononic degrees of freedom considered separately hasbeen widely studied and many key certainties have been obtained (Chapter 2). Instead,the behaviour of these interactions once coupled leaves several open issues, both fortheoretical models (Chapter 3) and for their signatures in real materials (Chapter 4).Therefore the study of their combined action rightly constitutes the most importantmotivation for our Thesis.


Figure 1.16: Dispersions of the anomalous branch in LSCO and of the [010] bond-stretching branch along the chain in YBCO, once subtracted by thedownward cosine dispersions, fitted to the zone centre and the zoneboundary for each sample. From top to bottom, the curves are forLSCO at δ = 0.07, 0.12, 0.15, 0.3 (being the compound at δ = 0.12codoped with Nd) and for YBCO at δ = 0.4 and 0.05 [4].

Chapter 2

Gutzwiller Approximation andRandom Phase Approximation

In this Chapter we describe how we modelize the competing e-e and e-ph interactions:we adopt the Hubbard-Holstein model in the adiabatic limit. Then we present the methodused to solve the problem: the GA+RPA (Gutzwiller Approximation + Random PhaseApproximation). In particular, we adopt the GA as a well-tested approximation forthe ground state energy of strongly correlated electron systems. Then, on top of thesaddle point solution we perform RPA fluctuations. In these computations we expandthe electronic energy up to second-order in the density and double occupancy deviations:a central role is played by the interaction kernel which provides the residual interactionbetween the quasiparticles.

2.1 The Hubbard model

In our computations for the e-e interactions, we will adopt the one-band Hubbardmodel with the electronic hopping tij between two neighbour atoms i and j:

He =∑

ijσ

tijc†iσcjσ +

∑

i

Uni↑ni↓ (2.1)

where c†iσ (ciσ) are creation (annihilation) fermionic operators; niσ=c†iσciσ is the densityoperator associated to these fermionic operators and U is the on-site Hubbard repulsion.One can work with a simpler version of Eq. (2.1), considering hopping only betweennearest-neighbour sites with tij = −t.

The idea behind this model dates back to Peierls [126] and Mott [127] and consiststo explain the metal-insulator transition of half-filled band systems as driven by stronglocal e-e repulsion. In fact, considering paramagnetic states, one can imagine a compe-tition between a kinetic term which tends to delocalize electrons favouring the metallic

34 Gutzwiller Approximation and Random Phase Approximation

solution and an intrasite repulsion term which tends to localize electrons, limiting theirhopping and favouring the insulating state. When the effect of U prevails over t, theMott transition takes place. The formal setting of the model and thus Eq. (2.1) wasformulated independently by Hubbard [70, 129], Gutzwiller [130] and Kanamori [131].Despite the apparent simplicity of such Hamiltonian, this model is not exactly solvablein general, but for 1d-systems [132]. Exact properties of the model in infinite dimen-sions have been obtained using Dynamical Mean Field Theory (DMFT) and numericalapproaches [133].

The Mott transition can be hidden by an antiferromagnetic (AF) instability only fornearest-neighbour hopping, unless the long range AF order is frustrated by the topologyof the lattice or for a sufficiently strong next-to-nearest neighbour hopping. One canunderstand why the AF order arises in the insulating state, considering the case oftwo sites, each one occupied by an electron. In the presence of a strong Coulombrepulsion (t/U≪1) and if and only if the electron spins are antiparallel, the systemallows a double-hopping process, associated to an intermediate virtual state of energyU . At the second perturbative order in t/U , this process brings an energetic gain ofthe order t2/U and therefore determines an AF coupling. This simple picture can givethe physical insight of the mappings which the Hubbard model undergoes in the strongcoupling limit. Specifically, for half-filled band systems, Eq. (2.1) is mapped into aHeisenberg model with coupling J = 4t2/U and with the Hilbert space restricted tosingly occupied states. For the partially-filled band systems the Hubbard model canbe projected into a t-J model where the fermion hopping is allowed only if the finalsite is vacant. For more information see for example Ref. [128].

Earliest approaches to Eq. (2.1) were within an effective scheme: some theorists firstelaborated a picture appropriate either to describe the insulating state [70, 129], othersfor the metallic state [134, 135, 136] in the paramagnetic sector. Then they moved fromthe limiting case towards the metal-insulator transition, introducing perturbatively themissing physical features. These methods failed to give reliable results when used toofar from their starting point but they were useful to provide a partial physical insightof the transition.

Hubbard started considering the insulator, whose density of states (DOS) is givenby two delta-like peaks at ±U/2. Then he introduced an effective-band picture evolvingfrom the spectral features of the insulator, considering two bands centered at ±U/2and with a finite width [70, 129]. One band is associated to hole states (lower band)and the other to double occupied states (upper band) and they are separated by a gap.This picture is accurate in the strong coupling regime t/U≪ 1: obviously it correctlyreproduces the atomic limit (t=0) and it is accurate for the high-energy electron physics.Upon decreasing U , the gap reduces: the onset of the metal is associated to the mergingof the two original bands. Yet for half-filled systems Hubbard found an ungappedsolution only for U = 0.

On the other side, Brinkman and Rice considered a Fermi-liquid metal composed

2.2 The Holstein interaction in the adiabatic limit. The chargesusceptibility 35

of quasiparticles whose mass m∗ is renormalized by the interaction U . They usedthe Gutzwiller method [130, 135, 136] and thus associated the Mott transition to thevanishing of the site double occupancy. This happens for U ≥ Uc together with thedivergence of the effective mass. This approach is intrinsically more accurate for thelow-energy electron physics. In fact, the Gutzwiller Approximation (GA) finds theinsulating ground state, but describes it inaccurately, neglecting the double-occupancyfluctuations associated to the high-energy processes: these ones are small but nonzeroin the insulator, unlike the fluctuations associated to the coherent processes. Giventhat the GA is our mean field-problem solver, we will dedicate § 2.3 to its discussion.We adopt it because we are essentially interested to study what happens close to theFermi energy and thus the accurate treatment of the low-energy quasiparticle physicsin the GA is adequate for our purposes.

In more recent times, slave-boson studies [139, 140] and DMFT investigations haveclarified many properties of the half-filled band system upon increasing U . They bothfind that in the DOS the spectral weight transfers from the low-energy quasiparticlepeak in weak coupling regime to the high-energy Hubbard bands in the strong couplingone. In particular, the DMFT studies give the following scenario at T=0: for U<Uc2

the ground state is a metal and for U>Uc2 the ground state is an insulator; interestingly,for Uc1<U<Uc2 the metal coexists with a metastable insulating state.

2.2 The Holstein interaction in the adiabatic limit.

The charge susceptibility

The Holstein interaction is the simplest possible way to consider electron-lattice cou-pling [137, 138]:

Heph +Hph =∑

iσ

g(a†i + ai)(niσ − 〈niσ〉) +∑

i

ω0a†iai (2.2)

where a†iσ (aiσ) are creation (annihilation) bosonic operators, ω0 is the optical phononfrequency and g is a constant e-ph coupling between the local electronic density andthe ionic displacement.

Considering Eq. (2.1) for the electronic energy and Eq. (2.2) for the e-ph interac-tion and the phonon energy, one actually deals with the Hubbard-Holstein model. Inparticular, we can consider a single-band system with e-e interaction and e-ph couplingon a lattice:

Htot = He +Heph +Hph (2.3)

In our Thesis we start from this model and study its adiabatic limit when the “adiabaticparameter” γ = ω0/D ≪ 1 (D is the electronic half-bandwidth). In fact, γ measures


the ratio between the phonon frequency and the electron kinetic energy: if γ is small,the typical phonon timescales are very long if compared to the electronic ones.

From the quantistic treatment of the harmonic oscillator, one knows that a†i =√1

2Mω0(Mω0xi − iPi) e ai =

√1

2Mω0(Mω0xi + iPi), where ~ = 1. Pi, xi and M are the

momentum, the displacement and the mass of the lattice ions respectively.Using this canonical transformation, we can write the Holstein interaction in the

form

Heph +Hph =∑

i

βxi(ni − n) +∑

i

(P 2

i

2M+

1

2Kx2

i

)(2.4)

where β = g√

2Mω0 and K = Mω20 . n = 〈ni〉 =

∑σ〈niσ〉 = Np/N is the average

density on a lattice with N sites and Np particles.We treat Eq. (2.4) in the extreme adiabatic limit (M→∞), where, using the

Born-Oppenheimer’s principle, we can represent the ground state of the system as|ψtot〉 = |ψe〉|χph〉. The electronic wave function |ψe〉 depends parametrically on theionic displacement xi. We assume that the ground state in the absence of e-ph cou-pling is uniform [i.e., no static CDW state is present]. For a fixed configuration ofdisplacements xi, the e-ph term of Eq. (2.4) acts as an external field on the elec-trons, producing a density deviation δni, being δni = 〈ni〉 − n. The exact adiabaticlimit is approached for γ → 0, keeping the e-ph coupling constant.

These considerations imply that for practical purposes we will not use Eq. (2.4)which still embodies all the quantistic effects in phonons. Instead, we will use:

Heph +Hph =∑

i

βxiδni +∑

i

1

2Kx2

i (2.5)

In this approach the electronic degrees of freedom of Eq. (2.3) are quantistic while thephonons in Eq. (2.5) can be handled as classical static fields. This is why one can solvethe electronic problem at fixed phonon coordinates and then minimize the energy withrespect to them.

2.2 The Holstein interaction in the adiabatic limit. The chargesusceptibility 37

2.2.1 Exact relation between electron-phonon instabilities andcharge susceptibility

Now we want to introduce the renormalization of the e-ph coupling and explain how therenormalized coupling is related to the electronic susceptibility which is the physicalquantity at the core of our studies in Chapters 3 and 4. The derivation we present hereis adequate in the case of homogeneous states. The appropriate extension to the caseof inhomogeneous states will be given in Chapter 4 and in Appendix B.

The total energy in the adiabatic limit has the form

Etot = Ee[δn] + Eeph[δn, x] + Eph[x] (2.6)

where Ee = 〈ψtot|He|ψtot〉, Eeph = 〈ψtot|Heph|ψtot〉 and Eph = 〈ψtot|Hph|ψtot〉; δn andx stand for the sets δni and xi respectively. We move to momentum space andexpand Ee[δn] up to second order in the density deviation δn:

Ee[δn] = E(0)e +

∑

q

(∂Ee

∂nq

)

0

δnq +1

2

∑

q

(∂2Ee

∂nq∂n−q

)

0

δnqδn−q, (2.7)

where the label “0” indicates that the derivatives must be evaluated within the uniformelectronic state (in the absence of the e-ph coupling); E

(0)e is therefore the ground state

energy in the absence of the e-ph coupling. The first order term vanishes identically.For the q 6= 0 terms this arises because stability requires that odd powers in the δnq

expansion must vanish (otherwise the systems would lower its energy by creating aCDW state). Instead the q = 0 term with ∂Ee/∂nq=0 vanishes because we are workingat a fixed particle number (δnq=0 = 0). Therefore the electronic ground state energyis quadratic in the density deviation:

Ee[δni] = E(0)e +

1

2

∑

q

κ−1q δnqδn−q (2.8)

where we customarily define κ−1q ≡

(∂2Ee

∂δn2q

)

0with κq being the static charge suscepti-

bility of the electronic system in the absence of the e-ph coupling.We now minimize the Holstein energy

EH = Eeph[δn, x] + Eph[x] =∑

i

(βxiδni +

1

2Kx2

i

)(2.9)

with respect to the ionic displacements xi at fixed δni finding xi=−βδni/K. Replacingthis expression in Eq. (2.9) and introducing the adimensional e-ph coupling

λ ≡ χ00β

2

K(2.10)


[χ00 is the density of states (DOS) of the non-interacting electron system], we find the

following expression for Etot:

Etot = E(0)tot +

1

2

∑

q

(κeph

q

)−1δnqδn−q (2.11)

with

κephq =

κq

1 − λκq/χ00

=κq

1 − λq

(2.12)

where we introduced the renormalized coupling

λq ≡ λκq

χ00

. (2.13)

Eqs. (2.11)-(2.13) provide the exact second-order expansion of the total energy in theadiabatic limit and establish a relation between the electronic charge susceptibilities,κq and κeph

q , in the absence and in the presence of the e-ph coupling respectively. By

construction λq is defined in such a way that λq = 1 indicates an instability, in analogywith the non-interacting e-ph coupling for which λ = 1 indicates a q = 0 instability.

We can write λq = βΓqκ0q/K, using the charge vertex [141, 142, 143, 144, 145, 146]

which acts as a renormalized quasiparticle-ph coupling

Γq ≡ βκq

κ0q

(2.14)

(κ0q is the non-interacting charge susceptibility). Then we find:

λq

λ=κ0

q

χ00

Γq

β(2.15)

Eq. (2.15) has been introduced to separate in the e-ph coupling the effects of finiteq from those of the e-e renormalization. Specifically, the factor κ0

q/χ00 contains the

effects of finite momentum, and is present even for non-interacting electrons, while thee-e interaction acts on the e-ph coupling via a modification of the electronic chargesusceptibility, given by Γq/β. In this exact adiabatic derivation both these effects actin a simple multiplicative manner on λ.

A second feature of the result in Eq. (2.12) is that the system can become unstableif, upon increasing λ, it meets the condition λ = λc with

λ = λc = χ00/κqc

(2.16)

for some qc. In this case, if no other (first-order) instabilities take place before, thesystem undergoes a transition to a charge-ordered state with a typical wavevector qc.

Now the whole issue to study the effects of e-e interactions on the e-ph couplingand the related electronic charge instability is reduced to the study of the electroniccharge susceptibility (in the absence of the e-ph interaction). This is the main goal ofChapter 3.

2.3 The Gutzwiller Approximation 39

2.3 The Gutzwiller Approximation

The Gutzwiller (GZW) variational method applied to the Hubbard model uses theGZW trial wave function which incorporates correlation effects beyond the HFA (Hartree-Fock Approximation) [130, 135, 136]. Since the Hubbard model describes the compe-tition between the hopping and the localization of the charge carriers as induced byCoulomb correlations, the idea is to apply a projector to a Slater determinant (Sd) inorder to reduce the number of doubly occupied sites. The optimum double occupancyprobability is then determined variationally. Best results are obtained if one allowsfor unrestricted charge and spin distributions, also to be determined variationally. Forexample, for the half-filled Hubbard model one recovers that a Sd with long rangeantiferromagnetic order is favoured [147]. A formal diagrammatic solution of the GZWproblem has been given by Metzner and Vollhardt [148, 149]. However, usually forpractical purposes one approximates the corresponding expectation values using theso-called Gutzwiller approximation (GA) [136].

In the first part of this Section we will illustrate the general ideas behind theGZW method and the results achieved by Brinkman and Rice at half-filling where thetreatment is completely analitical [134].

We consider the paramagnetic solution of Eq. (2.1) with nearest-neighbour hoppingtij = −t on a N -site lattice and we want to apply the GZW method in this case. Letus call the densities of doubly occupied sites and of sites singly occupied by electronswith spins ↑ or ↓ as D, n↑ and n↓ respectively. The uncorrelated state (U = 0) of thesystem is then |Sd〉.

For U = 0 the occupancies of a site due to ↑- and ↓-electrons are two independentevents, then D = n↑n↓; for a finite U , D decreases. The GZW method aims to find theground state of the system (and then the new value of D) through the minimization ofthe energy functional. To this purpose, one introduces the GZW trial wave function:

|ψ〉 = Pg|Sd〉 =∏

i

[1 − (1 − g)ni↑ni↓]|Sd〉 (2.17)

where Pg is the GZW projector; g is the original variational parameter for the mini-mization of the functional Ee[ψ] and it can assume values between 0 and 1. In general,Pg suppresses the weight of double occupancies: for U = 0, g = 1 and clearly theuncorrelated D is preserved; for U → ∞, g → 0 and all the configurations with doubleoccupations are ruled out due to the extremely large cost in the Coulomb repulsion.

Actually, one has to minimize the functional

Ee[ψ] =〈ψ|H|ψ〉〈ψ|ψ〉 =

−〈ψ|t∑

ijσ c†iσcjσ|ψ〉 + 〈ψ|U

∑i ni↑ni↓|ψ〉

〈ψ|ψ〉 (2.18)

Only in few cases Eq. (2.18) is exactly evaluable within the GZW method, thenGutzwiller elaborated the following ansatz, known as Gutzwiller approximation (GA):


distinct electronic configurations with the same densities of doubly and singly occu-pied states are assumed to be energetically equivalent. This statement clearly impliesan approximation because it neglects spatial correlations: for any given D the spatialpositions of the electrons really determine a more or less favoured hopping rate. Thiscomputational simplification also introduces an univocal relation between g and D, sothat the latter is used as the variational parameter in practical applications.

Starting from a system with density n = 1 − δ (δ is the doping) and introducingthe notation used by Vollhardt, one finds that the hopping factors are reduced by afactor z0 [161]:

z0 =

√2x2 − x4 − δ2

1 − δ2(2.19)

where

x =√

1 − n +D +√D (2.20)

For the GA energy one obtains

Ee0 = Nz20e

0 +NUD, (2.21)

e0 =

∫ µ

−∞

dω ωρ0(ω) (2.22)

where e0, ρ0, and µ denote the energy per site, the density of states and the Fermienergy of the non-interacting system respectively.

The minimization of Eq. (2.21) yields

x4(1 − x2)

x4 − δ2= (1 − δ2)

U

8|e0| ≡ u, (2.23)

which, by using Eq. (2.20), determines the double-occupancy parameter D0.1

For the homogeneous ground state, D is the only variational parameter for the GA,as it happens for mean-field computations before RPA performed in Chapter 3. Ingeneral, the electronic solution can present SDW, CDW, orbital ordering and morevariational parameters are required, also depending on different sites for supercell-lattice. This is the case of our computations in Chapter 4.

Eq. (2.19) provides the technical tool to realize the effective picture of stronglycorrelated systems in the GZW scheme. In fact, z0 is responsible for the band narrowingupon increasing U because it reduces the hopping kinetic term. The values of z0 variesbetween 0 (complete electron localization) and 1 (uncorrelated case). For U/t ≫ 1,

1We use the subscript “0”to indicate quantities evaluated at the saddle point, i.e. after the mini-mization process, and the superscript “0” to indicate uncorrelated quantities.

2.3 The Gutzwiller Approximation 41

z0 takes small finite values and is 0 only for doping δ = 0 and U ≥ Uc. This is thescenario of the half-filled case studied by Brinkman and Rice [134]. They describe thetrends of the energy and other quantities in the metallic state and the metal-insulatortransition. For n=1 Eq. (2.19) reduces into the relation z0 = 8D0(1 − 2D0) and forU < Uc the following results are achieved:

D0 =1

4

(1 − U

Uc

)(2.24)

z20 = 1 −

(U

Uc

)2

(2.25)

Ee0

N= e0

(1 − U

Uc

)2

, (2.26)

with Uc = 8|e0|. For U ≥ Uc, D0, z0 and Ee0 vanish. The insulating state in the GA istherefore characterized by the simultaneous vanishing of the energy and of the doubleoccupancy, while the quasiparticle mass m∗ ∝ 1/z0 diverges. As we have discussed in§ 2.1, the energy of the insulating ground state is not exactly 0 but ∼ −t2/U . Thisshows that the GA is able to obtain the Mott transition, but it is not reliable enoughto describe the Mott insulating properties.

One methodological remark is needed. In this treatment we have used the GZW trialwave function which is suitable for problems with relevant intrasite interactions and theGA which requires a quite crude ansatz to relate electronic energy and configurations.There are variational problems where it could be reasonable to use the GZW trial state,but not the GA.

In the second part of this Section we present the generalization of the previousresults for an arbitrary charge- and spin configuration.

We apply the Gutzwiller (GZW) variational method to Eq. (2.1) on d-dimensionalhypercubic lattices with lattice parameter a=1. We consider the GZW ansatz state|ψ〉 = P |Sd〉, where the GZW projector P (given in Ref. [162]) acts on the Slaterdeterminant |Sd〉. In general, |Sd〉 shall have enough degrees of freedom to give accountfor SDW- or CDW-broken symmetries. Even for the case of a paramagnetic uniformstate, this will be useful in order to determine the stability in the presence of anarbitrary perturbation of the charge.

The GZW variational problem can not be solved exactly except for particular cases,as for d=∞. Thus one uses the GA. In particular, we use the energy functional Ee[ρ,D]obtained by Gebhard [162] which is equivalent to the saddle-point energy found by


Kotliar and Ruckenstein in their slave-boson approach [152]:

Ee[ρ,D] =∑

ijσ

tijziσzjσρjiσ +∑

i

UDi (2.27)

with the GA hopping factors ziσ

ziσ[ρ,D] =

√(1 − ρii +Di)(ρiiσ −Di) +

√Di(ρii,−σ −Di)√

ρiiσ(1 − ρiiσ)(2.28)

being ρii=∑

σ ρiiσ. Here ρ is the single-fermion density matrix in the uncorrelated

state: ρjiσσ′ = 〈Sd|c†iσcjσ′ |Sd〉. D is the vector of the GA double occupancy parameters

Di = 〈ψ|ni↑ni↓|ψ〉.To consider arbitrary deviations, the charge and spin distribution ρ and the set of D

should be completely unrestricted. Since we will consider essentially charge deviationsonly the part diagonal in the spin indexes contributes, therefore we will use the notationρijσ ≡ ρijσσ.

One can show that the expectation value of the diagonal elements ρiiσ, calculatedfor the |Sd〉, coincides with the value of the density niσ, calculated for the GZW state|ψ〉 [162]. Namely,

niσ = 〈ψ|c†iσciσ|ψ〉 = ρiiσ = 〈Sd|c†iσciσ|Sd〉 (2.29)

Eq. (2.29) will permit us to express the charge density deviations δni =∑

σ δniσ asappearing in Eq. (2.6) via δρii =

∑σ δρiiσ.

We find the saddle-point solution minimizing Eq. (2.27) with respect to ρ and D.The variation with respect to ρ has to be constrained to the subspace of the Slaterdeterminants by imposing the projection condition ρ=ρ2:

δEe[ρ,D] − Tr[Λ(ρ2 − ρ)]

= 0 (2.30)

where Λ is the Lagrange multiplier matrix. Then it is convenient to define a GZWHamiltonian h[ρ,D] [157, 158]:

hijσ[ρ,D] =∂Ee

∂ρjiσ

(2.31)

The variation of Eq. (2.30) with respect to ρ leads to h−ρΛ−Λρ+Λ = 0. The Lagrangemultipliers can be eliminated algebraically [158]. Considering also the variation withrespect to D, we obtain the self-consistent GA equations

[h, ρ]− = 0, (2.32)

∂Ee[ρ,D]

∂Di

= 0. (2.33)

2.4 The Gutzwiller + RPA method 43

Eq. (2.32) implies that at the saddle point, h and ρ can be simultaneously diagonalizedby a transformation of the single-fermion orbital basis

ciσ =∑

ν

Ψνiσaν . (2.34)

leading to a diagonal h0: h0µν = δµνǫν . Moreover, the diagonalized ρ02 has an eigenvalue

1 for states below the Fermi level (hole states) and 0 for states above the Fermi level(particle states).

2.4 The Gutzwiller + RPA method

The GA can be recovered at the mean-field level (saddle-point) of the four-slave boson(SB) functional integral method introduced by Kotliar and Ruckenstein (KR) [152].In principle, the KR approach provides a scheme to include fluctuations beyond themean-field solution and in fact expansions around the slave-boson saddle point havebeen performed for homogeneous systems [153, 154]. However, the expansion of the KRhopping factor zSB is a highly nontrivial task with respect to the proper normal orderingof the bosons and to the correct continuum limit of the functional integral [155, 156].Substantially the KR computing approach is not easily eligible to deal with broken-symmetry states. The GA+RPA has been developed as a variational method whichcould be practically handled also for inhomogeneous systems [150, 151].

Seibold and Lorenzana introduced this scheme to compute fluctuation correctionson top of a GA state to dynamical and static correlation functions and to the groundstate energy. The method is called GA+RPA because it can be considered as a time-dependent GA, such HF+RPA can be viewed as time-dependent HFA in the limitof small amplitude oscillations [157, 158]. It also generalizes Vollhardt’s method ofRef. [161] to describe the low-temperature FL regime.

Since response functions are derived for systems with completely unrestricted chargeand spin distributions, GA+RPA is suitable also for the calculation of charge excita-tions on solutions with inhomogeneous textures. A key point of the GA+RPA approachis the proper determination of the time dependence of the variational double occupancyparameter. For this purpose Seibold and Lorenzana adopted an antiadiabatic approx-imation, in the sense that the double occupancy adjusts instantaneously to the timeevolution of the single particle densities. In principle, this approximation neglects thehigh-energy electronic dynamics. However this choice allows to handle practically thefluctuations on top of the GA ground state and to perform the appropriate calculationsalso for broken-symmetry states. The obtained results are expected to be physicallyreasonable at least for low-energy degrees of freedom (cf. Ref. [150]). This is a note-

2We recall that we use a subscript “0” to indicate the quantities evaluated at the saddle point.


worthy improvement if compared to the difficulties to include the fluctuations in theSB scheme.3

Following what presented in Ref. [151], we will perform a derivation of the RPAequation as follows from an energy expansion around the GA saddle point.

For some computations it can be convenient to introduce the effective quantity qijσso that qijσ = 1 if i = j and qijσ = ziσzjσ otherwise.

In order to derive the RPA equation we introduce a small time-dependent externalfield added to Eq. (2.1):

Hf(t) =∑

ijσ

(fijσ(t)c†iσcjσ + h.c.) (2.35)

with fijσ(t) = fijσ(0)e−iωt. As a consequence |ψ〉, |Sd〉 and the variational parametersacquire a time dependence and an additional term appears in the energy functional inEq. (2.27):

Ef [ρ,D](t) = 〈ψ(t)|Hf(t)|ψ(t)〉 =∑

ijσ

(fGAijσ ρjiσe

−iωt + h.c.) (2.36)

where fGAijσ = qijσfijσ.

The time-dependent field induces small amplitude oscillations of D and ρ aroundthe GA saddle point:

D = D0 + δD(t),

ρ = ρ0 + δρ(t).

The density and double occupancy fluctuations are constrained by the following re-quirements:

i) At all times ρ is constrained to be the one-body density matrix associated witha Slater determinant. This can be achieved by imposing:

ρ = ρ2 (2.37)

ii) The double occupancy is assumed to have a much faster dynamics than thedensity matrix so that it can be treated antiadiabatically. As a consequence, δDadjusts instantaneously to the evolution of the density matrix via the condition

∂Ee[ρ,D]

∂δDi

= 0 (2.38)

3To our knowledge, the state of art for the KR computations of fluctuations has not substantiallychanged for ten years. Insofar there is not an analog of the antiadiabatic approximation, i.e. atechnical tool which helps with practical computations. This is partly due to the difficulties to find acorrispondent trick for the double-occupancy boson in the integral formalism, partly to the intrinsiclarger number of the SB degrees of freedom (slave bosons and Lagrange multipliers) with respectto the GA variational parameters: probably more than one ansatz or working hypothesis would benecessary to achieve an adequate simplification [159].


We stress that Eq. (2.38) constitutes the basic work hypothesis of the GA+RPA ap-proach to Eq. (2.1) and it is necessary in order to derive an effective Gutzwiller in-teraction between particles, as one will obtain below in Eq. (2.44). The antiadiabaticapproximation is expected to be accurate for sufficiently low-energy excitations. Athigher energies, a comparison with exact diagonalization tests its validity, showingnotably that it is still accurate at least up to energies of the order of the Hubbard gap.

At this level one has to expand the GA energy [Eqs. (2.27) plus (2.35)] around thesaddle point. Eq. (2.27) is considered up to second order in the density and doubleoccupancy deviations:

Ee[ρ,D] = Ee0 + Tr[h0δρ]

+∑

ijσ

tij[ziσδ1zjσ + zjσδ1ziσ]δρjiσ

+∑

ijσ

tijρjiσδ1ziσδ1zjσ

+∑

ijσ

tijρjiσ[ziσδ2zjσ + zjσδ2ziσ]. (2.39)

where Ee0 is the saddle-point energy and the trace includes sum over spins. We haveused the following abbreviations for the z-factor expansion:

δ1ziσ ≡ ∂ziσ

∂Di

δDi +∑

σ′

∂ziσ

∂ρiiσ′

δρiiσ′ , (2.40)

δ2ziσ ≡ 1

2

∂2ziσ

∂D2i

(δDi)2 +

∑

σ′

∂2ziσ

∂Di∂ρiiσ′

δDiδρiiσ′

+1

2

∑

σ′σ′′

∂2ziσ

∂ρiiσ′∂ρiiσ′′

δρiiσ′δρiiσ′′ . (2.41)

To proceed further it is convenient to cast the second order expression in matrix form

Ee[ρ,D] = Ee0 + Tr[h0δρ] +1

2δρjiσL

σσ′

ijklδρlkσ′

+1

2δDiKijδDj + δDiSiklσδρlkσ (2.42)

where the Einstein sum convention has been used and the definitions for the matricesL, K and S follow immediately from Eqs. (2.39)-(2.41). The nonzero matrix elementsare given in Ref. [151].

We can integrate out the D fluctuations using the antiadiabaticity condition, i.e.Eq. (2.38). Expressing δDi in terms of the density fluctuations via

δDi = −(K−1)ijSjklσδρlkσ, (2.43)


one finally obtains an expansion of the energy as a functional of δρ alone, i.e.Ee[ρ] ≡ Ee[ρ,D(ρ)]:

Ee[ρ] = Ee0 + Tr[h0δρ] +1

2δρjiσ[L0 − S†

0K−10 S0]

σσ′

ijklδρlkσ′ (2.44)

The matrix (L0 − S†0K

−10 S0) can be considered as an effective interaction kernel W

between particle-hole excitations in the GA. For the paramagnetic regime at half-fillingW reduces to the quasiparticle kernel of Vollhardt’s FL analysis [161]. The off-diagonalelements of the matrices Kij, L

σσ′

ijkl and Siklσ can induce intersite interactions betweenthe GA quasiparticles: this is in contrast with conventional HF theory of the Hubbardmodel which is purely local.

The expansion of Ef [ρ,D] (Eq. (2.36)) is needed up to first order only, being alreadylinear in the external field f :

Ef [ρ,D] = F0 + Tr[fGA0 δρ]

+∑

ijkσσ′

ρ0jiσfijσ

∂qijσ∂ρkkσ′

δρkkσ′ +∑

ijkσ

ρ0jiσfijσ

∂qijσ∂Dk

δDk (2.45)

where F0 = fGA0 ρ0 is the energy contribution to the system frozen at the saddle-point

level and we used the fact that qijσ depends on diagonal densities only. As before, thedouble-occupancy fluctuations can be eliminated through Eq. (2.43) and thus we can

define Ef [ρ] ≡ Ef [ρ,D(ρ)] and

fijσ ≡ ∂Ef [ρ]

∂ρjiσ

. (2.46)

Now, we proceed in analogy with the nuclear physics treatment of effective mean-fieldtheories in which the interaction potential is density dependent [157, 158]. Indeed,Eq. (2.44) can be viewed as the energy expansion of an effective mean-field theorywhere part of the density dependence is due to the GA hopping factors ziσ in thekinetic term of the Hamiltonian. The GA+RPA method has an important advantagewith respect to other methods (e.g., equation of motion or diagrammatic methods): thepresent derivation is solely based on the knowledge of an energy functional associatedwith a Slater determinant, precisely what the GA provides.

The density matrix of an effective mean-field theory of this kind obeys this equationof motion [157, 158]:

i~ρ = [h[ρ] + f(t), ρ], (2.47)

where we have defined the effective Gutzwiller Hamiltonian

hijσ[ρ] =∂E

∂ρjiσ

, (2.48)


which depends on densities only. At the saddle point, we have h0 = h[ρ0] = h0. TheRPA is obtained by considering the limit of small amplitude fluctuations in Eq. (2.47).

It is convenient to define the four subsectors of the density matrix fluctuations usingthe projector properties of the density matrix:

δρhh ≡ ρ0δρρ0, (2.49)

δρpp ≡ σ0δρσ0, (2.50)

δρhp ≡ ρ0δρσ0, (2.51)

δρph ≡ σ0δρρ0. (2.52)

where σ0 = 1 − ρ0. The Slater determinant condition in Eq. (2.37) implies that thefluctuations given in Eqs. (2.49)-(2.52) are not independent. In fact, in terms of thefluctuations, Eq. (2.37) reads:

δρ = ρ0δρ+ δρρ0 + (δρ)2. (2.53)

The projection of Eq. (2.53) onto the hh and pp sector of the saddle-point Slaterdeterminant yields:

δρhh = −(1 + δρhh)−1δρhpδρph ≈ −δρhpδρph (2.54)

δρpp = (1 − δρpp)−1δρphδρhp ≈ δρphδρhp (2.55)

where the result on the right side is valid in the small amplitude limit. From Eqs. (2.54)and (2.55) we learn that pp and hh density projections are actually quadratic in the

ph and hp matrix elements. Therefore, on computing h from Eqs. (2.44) and (2.48)one should consider the fact that the term Tr[h0δρ] =

∑µ ǫµρµµ (which is first order

in the pp and hh density projections) yields a quadratic contribution in the ph and hpmatrix elements:

Tr[h0δρ] =∑

p

ǫpδρpp +∑

h

ǫhδρhh =∑

ph

(ǫp − ǫh)δρphδρhp. (2.56)

In addition, one can neglect the pp and hh matrix element in the last term of Eq. (2.44).Thus, up to second order in the particle-hole density fluctuations, one obtains for theenergy expansion Eq. (2.44):

Ee[ρ] = Ee0 +1

2(δρhp, δρph)

(A BB∗ A∗

)(δρph

δρhp

). (2.57)

Here the so called RPA matrices A and B are given by

Aph,p′h′ = (ǫp − ǫh)δpp′δhh′ +∂hph

∂ρp′h′

(2.58)

Bph,p′h′ =∂hph

∂ρh′p′(2.59)


where A contains matrix elements between particle-hole excitations and B is composedof matrix elements between the ground state and two particle-hole excitations. A andB are related to W ≡ (L0 − S†

0K−10 S0) via

Aph,p′h′ = (ǫp − ǫh)δpp′δhh′

+∑

ijσ,nmσ′

(Ψpiσ)∗Ψh

jσWσσ′

ij,nm(Ψh′

nσ′)∗Ψp′

mσ′

Bph,p′h′ =∑

ijσ,nmσ′

(Ψhiσ)∗Ψp

jσWσσ′

ij,nm(Ψh′

nσ′)∗Ψp′

mσ′

where the transformation amplitudes Ψνiσ have been defined in Eq. (2.34).

To lowest order, we can now linearize Eq. (2.47) retaining only ph and hp matrixelements:

i~δρ = [h0, δρ] + [∂h

∂ρδρ+ f , ρ0], (2.60)

where we use the shorthand notation

∂h

∂ρδρ =

∑

ph

(∂h

∂ρhp

δρhp +∂h

∂ρph

δρph

).

Then from Eqs. (2.48), (2.57) and (2.60) one obtains the following linear responseequation:

(A BB∗ A∗

)− ~ω

(1 00 −1

)(δρph

δρhp

)= −

(fph

fhp

). (2.61)

This inhomogeneous equation can be solved by inverting the matrix on the left-handside which yields a linear relation between the external field and the density rearrange-ment:

δρ = R(ω)f . (2.62)

Thank to Eq. (2.62), we can compute the response function of a single particleobservable O:

O =∑

ijσ

(oij,σc†iσcjσ + h.c.),

In fact, in analogy with Eqs. (2.35) and (2.36), the time evolution of O is given by

〈Ψ(t)|O|Ψ(t)〉 =∑

ijσ

(oGAijσρjiσ(t) + h.c.). (2.63)


and the time evolution of ρ is found using Eq. (2.62).The linear response matrix R(ω) has poles at the eigenfrequencies of the eigenvalue

problem corresponding to Eq. (2.61) with f = 0:(

A BB∗ A∗

)− ~Ωn

(1 00 −1

)(X(n)

Y (n)

)= 0. (2.64)

Here ~Ωn ≡ En − E0 denote the excitation energies of the system. Similarly to theHF+RPA method, the vacuum of these excitations is not the old starting GA state |Ψ0〉but a new state with both GZW-type correlations and RPA ground-state correlations.We denote this state by |Φ0〉 and the corresponding excited states by |Φn〉.

The response matrix R can be written in the Lehmann representation:

R(ω)ph,p′h′ =∑

n>0

[Xn

phXn∗p′h′

ω − Ωn + iǫ−

Y np′h′Y n∗

ph

ω + Ωn + iǫ

](2.65)

In analogy with the HF+RPA method, the following notations are introduced:

〈0|a†hap|n〉 ≡ Xnph, (2.66)

〈0|a†pah|n〉 ≡ Y nhp. (2.67)

The states |n〉 are not true excitations of the system but represent auxiliary nota-tional objects. Roughly speaking, they can be thought of as RPA states without theGutzwiller projector. For example |0〉 is the analog of the state |Sd〉 but at RPA level,in that it contains RPA ground-state correlations but lacks GZW correlations. Theyare “unprojected RPA states”. The eigenvector (Xn

ph, Ynhp) can be identified with the

particle-hole and hole-particle components of the unprojected RPA excited state |n〉with respect to the unprojected RPA ground state |0〉.

Schematically the GZW states and the unprojected RPA ones are related as follows:

|Sd〉 P

−→|Ψ0〉

RPA ↓ ↓ RPA|0〉 P

−→|Φ0〉

where P indicates the GZW projection.

All the machinery presented in this Section allows us to compute the response ofone-particle observables. In particular, this formalism is appropriate for the calculationof charge excitations in inhomogeneous doped systems which will be presented in Chap-ter 4. In the following Section and in Chapter 3 we will consider the charge responseof homogeneous systems. The formal derivation here provided is general and suitableto work with the dynamical GA+RPA method. Instead, in our practical computationsin Chapters 3 and 4, we will perform a static approach of GA+RPA.


2.5 The GA+RPA approach to the paramagnetic

homogeneous state

In this Section we present our static approach to the homogeneous case. Therefore,a large amount of the following derivation can be seen as an illustration of the generalformalism already given in the § 2.4. Nevertheless, we have decided to report ourcomputations here because they could help the reader gain a more physical insightof the formalism. Thus, for example, here we can treat more explicitly particle-holeelements because there is a well defined quantum number and we can more easilyidentify particle (hole) states as fermionic states with the momentum higher (lower)than the Fermi momentum. In Subsections § 2.5.1 and § 2.5.2 we also characterizesome typical features of our approach to the homogeneous state, giving analyticalresults especially for the half-filled band system (for any U) and for the U = ∞ case(for any fillings).

In the absence of an external field, we consider the paramagnetic homogeneousstate as the saddle-point solution, i.e. we expand the energy around the paramagneticsaddle point. Then we will consider the charge response of the homogeneous groundstate [167].

In order to compute the static charge susceptibility, we evaluate the electronicenergy in the presence of an external field f :

Hf =∑

ijσ

fijσc†iσcjσ (2.68)

where fijσ = f ∗jiσ.

Within the subspace of the Slater determinants, we now consider small amplitudedeviations of the density matrix ρ due to Hf , given in Eq. (2.68). This leads to anadditional contribution Ef [ρ] to Eq. (2.27)

Ef [ρ] =∑

ijσ

fijσρjiσ (2.69)

The field f produces small amplitude deviations δρ, δD around the unperturbed saddle-point density, i.e. δρ = ρ− ρ0 and δD = D −D0: δρ and δD are both linear in f . Inthe presence of the external field f , Eq. (2.32) will turn into

[h+ f, ρ] = 0 (2.70)

which is the static version of Eq. (2.47). We will expand EGAe [ρ,D] = Ee[ρ,D] +Ef [ρ]

around the saddle point Ee0 up to the second order in δρ and δD [150, 151]:

EGAe [ρ,D] = Ee0 + δE(1)

e + δE(2)e (2.71)

2.5 The GA+RPA approach to the paramagnetic homogeneousstate 51

δE(1)e (δE

(2)e ) contains first (second) order derivatives of the GA energy. We will first

consider δE(1)e in this Section and then δE

(2)e in the two following subsections.

The expression for δE(1)e is

δE(1)e = Tr[h0δρ] + Tr[fδρ] (2.72)

It is convenient to work in the momentum space, where h0kσ,k′σ′ = δkk′δσσ′ǫkσ. Inaddition we restrict the external perturbation to a local field on the charge sector:fijσ = fiδij with

∑i fi = 0 so that Eq. (2.69) becomes

Ef [ρ] =∑

q

f−qδρq (2.73)

where we introduced the Fourier transform of the density deviation δρq:

δρq =∑

kσ

δρk+q,kσ. (2.74)

Unoccupied states are named particle states (p) and we will use the shorthand notationk > kF for the restriction in the momentum. Analogously hole (h) states are occupiedstates with k < kF , kF being the Fermi momentum.

The matrix elements of the δρ are not all independent since ρ must fulfill theprojector condition ρ2 = ρ which we can write in terms of δρ:

δρ = ρ0δρ+ δρρ0 + (δρ)2

which is Eq. (2.53). Since ρ0kσk′σ′ = δkk′δσσ′ρk with ρk = 1 for k < kF and 0 otherwiseρ0 projects onto occupied states.

Taking matrix elements of Eq. (2.53), one finds for hh density deviations (k, k′ < kF )

δρkσk′σ′ = −∑

k′′>kF ,σ′′

δρkσk′′σ′′δρk′′σ′′k′σ′ (2.75)

and for the pp density deviations (k, k′ > kF )

δρkσk′σ′ =∑

k′′<kF ,σ′′

δρkσk′′σ′′δρk′′σ′′k′σ′ . (2.76)

Then the hh and pp matrix elements are quadratic in the ph and hp δρ matrix elements.Therefore in Eq. (2.72) the term Tr[h0δρ] =

∑k ǫkδρkk is first order in the hh and pp

matrix elements but yields a quadratic contribution in the ph and hp matrix elements.The density deviations that are off-diagonal in the spin index contribute to the magnetic


susceptibility [164] but not to the charge susceptibility, therefore in the following theywill be neglected. One obtains:

Tr[h0δρ] =∑

k>kF ,σ

ǫkσδρkkσ +∑

k′<kF ,σ

ǫk′σδρk′k′σ

=∑

k>kF ,k′<kF ;σ

(ǫkσ − ǫk′σ)δρkk′σδρk′kσ (2.77)

Notice that in the GA ǫkσ is renormalized by interactions and is related to the baredispersion ǫ0kσ through the relation ǫkσ = z2

0ǫ0kσ. Eq. (2.77) shows that the first nonzero

contribution beyond the saddle-point energy is of second-order in the particle-holedensity deviations which are our independent variables.

Now we proceed by considering δE(2)e separately for n = 1 and general n.

2.5.1 Half-filling case

Closed formulas can be obtained at half-filling which illustrate the physics in a sim-ple manner. This generalizes to arbitrary momenta the computation done by Voll-hardt [161]. The second-order energy contribution for the local charge deviations is

δEC(2)e =

1

2N

∑

q

Vqδρqδρ−q +1

N

∑

q

LqδρqδD−q +1

2N

∑

q

UqδDqδD−q (2.78)

being δDq=1

N

∑i e

−iq·riδDi and

Vq =e0z02

(z′′++ + 2z′′+− + z′′−−) +2(z

′

)2

N

∑

kσ

ǫ0k+q,σnkσ

Lq = 2e0z0z′′

+D +2z

′

z′

D

N

∑

kσ

ǫ0k+q,σnkσ

Uq = 2e0z0z′′D +

2(z′D)2

N

∑

kσ

ǫ0k+q,σnkσ (2.79)

where e0 =1

N

∑kσ ǫ

0kσnkσ; z′ and z′′ denote the derivatives of the hopping factors

given in Appendix A.Using Eq. (2.33), we can eliminate the δD deviations in Eq. (2.78) so that finally

the energy functional depends on δρ deviations alone, i.e. EGAe [ρ]=EGA

e [ρ,D(ρ)]. Wefind:

δD±q = −U−1q Lqδρ±q.


Thus the energy EGAe [ρ] is

EGAe [ρ] = Ee0 +

1

N

∑

kqσ

(ǫkσ − ǫk+q,σ)δρphkσ;k+q,σδρ

hpk+q,σ;kσ

+∑

q

f−qδρq +1

2N

∑

q

Aqδρqδρ−q (2.80)

where

Aq = Vq − L2qU

−1q (2.81)

is the GA residual interaction kernel. We have introduced the notation δρphkσ;k′,σ to

indicate that only ph elements should be taken into account, i.e. sums are restrictedto k > kF and k′ < kF . The density deviations can be decomposed in ph, hp, pp andhh contributions. ph and hp matrix elements contribute quadratically to Eq. (2.80)while pp and hh are higher order and one should replace

δρq =∑

kσ

δρphk+q,kσ +

∑

kσ

δρhpk+q,kσ (2.82)

Minimizing Eq. (2.80) with respect to the deviations δρ and considering the constraintson the momenta, one finds the following equation for δρq:

δρq = −χ0qfq − χ0

qAqδρq. (2.83)

Here χ0q is the static Lindhard function, that is the charge susceptibility of the non-

interacting quasiparticles:

χ0q = − 1

N

∑

kσ

nk+q,σ − nkσ

ǫk+q,σ − ǫkσ

(2.84)

ǫkσ = z20ǫ

0kσ. In fact, we evaluate RPA fluctuations not for free electrons, but for free

quasiparticles whose mass and electronic band are renormalized by the GA factor z20 .

From Eq. (2.83), we obtain the linear response equation [157, 160]:

δρq = −κqfq (2.85)

with the GA+RPA static response function

κq =χ0

q

1 + χ0qAq

(2.86)

κq=0 is the charge compressibility studied by Vollhardt for n = 1 (see Ref. [161]). ForAq=0 and z0 = 1, one will recover κq=χ

00q for the non-interacting electrons.


0 0.2 0.4 0.6 0.8 1 1.2 1.4UUc

0

10

20

30

40

50

Aq

n=0.9, q=Π

n=0.9, q=Π2

n=0.9, q=0

n=1, all q

Figure 2.1: 1d-Aq as a function of U/Uc for n = 1 and n = 0.9. At half-filling Aq isindependent from the momentum and diverges at U = Uc (see Eqs. (2.89)and (2.90)); for finite dopings the momentum dependence is recovered.This behaviour occurs in any dimension.

In the case of nearest-neighbour hopping on a d−dimensional cubic lattice, we canuse the relation:

1

N

∑

kσ

ǫ0k+q,σnkσ =e0

d

d∑

ν=1

cos qν = e0ηq

where

ηq =1

d

d∑

ν=1

cos qν (2.87)

with ν = 1, ..., d. Using Eqs. (2.79) and (2.81), Aq takes the following form:

Aq

e0=z02

(z′′++ + 2z′′+− + z′′−−) + 2(z′

)2ηq − 2(ηqz

′z′D + z0z′′+D)2

ηq(z′D)2 + z0z′′D(2.88)

Starting from Eq. (2.88) we find an analytical expression of the effective interactionAq in term of the Coulomb interaction U for U < Uc:

Aq =U(Uc + U)(U − 2Uc)

4Uc(U − Uc)(2.89)

being Uc = 8|e0n=1|. Thus we find that at half-filling Aq is independent from themomentum q. In the weak coupling limit, we recover the HF-RPA result Aq ≈ U/2.


Aq is an increasing function of U , diverging at U = Uc:

limU→U−

c

Aq =U2

c

2(Uc − U)(2.90)

Then at the Mott transition not only the charge compressibility vanishes [134], butalso the susceptibility κq for any momentum (see Eq. (2.86)).

Fig. 2.1 illustrates the analytical results given in Eqs. (2.88)-(2.90).

The half-filling case is particularly favourable to show the equivalence of the quan-tities evaluated in our formalism and the correspondent ones obtained in the Landautheory [160] and in particular in the Fermi-liquid analysis perfomed by Vollhardt [161]and Lavagna [154] for q = 0. For illustrative purposes, we can consider Eqs. (2.86) and(2.89).

For q = 0, Eq. (2.86) gives the interacting charge compressibility. In the GA+RPAapproach to the homogeneous half-filled system, Eq. (2.86) is a scalar relation: in thiscase the general matricial RPA equation (later given in Eq. (2.99)) decouples into inde-pendent scalar ones because both the general kernel interaction and the susceptibilitymatrix are diagonal. Now we intend to compare our Eq. (2.86) to the result for thecharge compressibility obtained in Landau theory:

κq=0 =χ0

q=0

1 + F s0

=χ00

q=0/z20

1 + F s0

(2.91)

where χ00q=0 is the density of states of the non-interacting electrons, χ0

q=0 the one of thenon-interacting quasiparticles and F s

0 is the spherosymmetrical Landau parameter forthe quasiparticle interaction. Eq. (2.91) is substantially Eq. (1.58) of Ref. [160].

As already discussed, the form of Eq. (2.89) is indipendent from the momentumand clearly provides the dependence Aq(U/Uc) also in the case of q = 0. ComparingEq. (2.86) with (2.91), one obtains the relation between our quasiparticle residual termAq=0 and the parameter F s

0 used customarily in the literature:

F s0 = χ0

q=0Aq=0 = (χ00q=0/z

20)Aq=0 (2.92)

Eq. (2.92) will help the reader find the equivalences between our results and the onesobtained in other works with GA or SB approaches for q = 0 and n = 1. For examplewith our Eq. (2.89) we reobtain the result given in Eq. (54) of Ref [161] and thereby inEq. (18a) of Ref. [154] (remember also the dependence z2

0(U/Uc) at half-filling whichwe report in Eq. (2.26)).


2.5.2 Arbitrary filling case

The full derivation of δE(2)e in real and momentum space is given in Appendix A.

In addition to δρq, we find that it is convenient to introduce the quantity

δTi =∑

jσ

tij(δρjiσ + δρijσ)

and its Fourier transform

δTq =∑

kσ

(ǫ0kσ + ǫ0k+q,σ)δρk+q,σ;kσ (2.93)

δTq corresponds to the intersite charge fluctuations, while δρq describes the local ones.The second-order energy expansion for the charge deviations is given by

δEC(2)e =

1

N

[1

2

∑

q

Vqδρqδρ−q + z0z′D

∑

q

δDqδT−q

+1

2z0(z

′ + z′+−)∑

q

δTqδρ−q

+∑

q

LqδρqδD−q +1

2

∑

q

UqδDqδD−q

](2.94)

with the following definitions:

Vq =e0z02

(z′′++ + 2z′′+− + z′′−−) +(z′ + z′+−)2

2N

∑

kσ

ǫ0k+q,σnkσ

Lq = e0z0(z′′+D + z′′−D) +

z′D(z′ + z′+−)

N

∑

kσ

ǫ0k+q,σnkσ

Uq = 2e0z0z′′D +

2(z′D)2

N

∑

kσ

ǫ0k+q,σnkσ (2.95)

where z′ and z′′ denote derivatives of the hopping factors which are given in Ap-pendix A.

Using Eq. (2.33) one can eliminate the double occupancy deviations and arrive atthe following functional which only depends on the local and intersite charge deviations:

δEC(2)e =

1

2N

∑

q

(δρq

δTq

)(Aq Bq

Bq Cq

)(δρ−q

δT−q

)(2.96)

where

Wq ≡(Aq Bq

Bq Cq

)(2.97)


is the interaction kernel. The elements of Wq are given by

Aq = Vq −L2

q

Uq

Bq = z0(z′ + z′+−) − 2z0z

′D

Lq

Uq

Cq = − (z0z′D)2

Uq

Since the energy expansion in Eq. (2.96) is a quadratic form in δρq and δTq (see alsoEq.(2.93)), it is useful to introduce the following representation for the static Lindhardfunction χ0

q of the non-interacting quasiparticles:

χ0q = − 1

N

∑

kσ

(1 ǫ0kσ + ǫ0k+q,σ

ǫ0kσ + ǫ0k+q,σ (ǫ0kσ + ǫ0k+q,σ)2

)nk+q,σ − nkσ

ǫk+q,σ − ǫkσ

. (2.98)

The calculations proceed analogously to the half-filling case. We find that the RPAseries for the charge excitations then corresponds to the following Bethe-Salpeter equa-tion:

χq = χ0q − χ0

qWqχq (2.99)

For general fillings the response function χq is given by a 2 × 2 matrix whose element(χq)11 is the charge susceptibility κq.

For U = ∞ we have derived an analytical expression for Aq valid for any fillingn < 1 and any dimension:

Aq = (−e0) 5 − 4n− ηq(1 − n)(2 − n)3

(2.100)

We notice for later use that, since e0 < 0, the interaction has a minimum at q = 0, amaximum at q = (π, π, ...) and diverges as n→ 1.

In § 2.2.1 we have introduced the charge vertex Γq representing the renormalizationof the e-ph coupling. Within the GA+RPA Γq is evaluated as

Γq = β(χq)11

(χ0q)11

(2.101)

We stress that Γq is the renormalized quasiparticle-ph coupling. The renormalizedcoupling for the electrons instead is gq = ZqΓq, with Zq the quasiparticle weight, givenby z2

0 in the GA. In Chapter 3 we will compute Γq and gq, using β = 1.


2.6 The GA+RPA approach to inhomogeneous states

Even explicit basic steps of the computations we did for inhomogeneous systems wouldbe too long to be presented in this Section. Hereby we stress the key-points along thescheme described in § 2.4 and we address the interested reader to Appendix B.

In the general framework developed, we study RPA fluctuations on top of a GAinhomogeneous ground state, i.e. with charge and spin superstructures.

We consider scattering processes from a state |i, α〉 to another state |j, β〉: i andj are cell indexes, α and β intracell indexes. Then, given a N -site lattice with Nc

supercells composed of Na atoms, one works with i, j = 1, ..., Nc and α, β = 1, ..., Na.Each site is univocally determined by the position riα = Ri + δα. Considering the

hopping parameter tijαβ, if i 6= j the hopping involves two different cells, else it is anintracell hopping; the condition α = β identifies two equivalent sites.

The interaction kernel

The basic ingredient of this approach is the interaction kernel, obtained with an ex-pansion up to second order in the density deviations. In particular, we expand the GAenergy up to second order in ρ and D:

Ee =∑

ijαβσ

tijαβziασzjβσ〈c†iασcjβσ〉 +∑

iα

UDiα = Ee0 + δE(1)e + δE(2)

e (2.102)

The single-particle density matrix is ρjiβασ = 〈c†iασcjβσ〉0. We determine the form of

δE(2)e in real space:

δE(2)e =

∑

iαβσσ′

Xiiαβσσ′δρiiασδTiασ′ +∑

ijαβσσ′

1

2Vijαβσσ′δρiiασδρjjβσ′

+∑

ijαβσ

LijαβσδρiiασδDjβ +∑

iαβσ

ZiiαβσδDiαδTiασ

+∑

ijαβ

1

2UijαβδDiαδDjβ (2.103)

with complete definitions given in Appendix B.

2.6 The GA+RPA approach to inhomogeneous states 59

Then we move to the momentum space using the following Fourier transform laws:

δρiiασ =1

N

∑

qQ

ei(q+Q)·δαeiq·Riδρq+Q,σ

δTiασ =1

N

∑

qQ

ei(q+Q)·δαeiq·RiδTq+Q,σ

δDiα =1

N

∑

qQ

ei(q+Q)·δαeiq·RiδDq+Q

where q belongs to the Reduced Brillouin Zone (RBZ); Q and Q′ are superlatticevectors. Exploiting the antiadiabatic condition, one can cast Eq. (2.103) in the generalform:

δE(2)e =

1

2

∑

qQQ′ηη′

(δρq+Q,η

δTq+Q,η

)WqQQ′ηη′

(δρ−q−Q′,η′

δT−q−Q′,η′

)

=1

2

∑

q

δRq Wq δR−q (2.104)

where η, η′ = ± and

δRq+Q,+ =∑

σ

δRq+Q,σ δRq+Q,− =∑

σ

σδRq+Q,σ

WqQQ′++ =1

4

∑

σσ′

WqQQ′σσ′ WqQQ′+− =1

4

∑

σσ′

σ′WqQQ′σσ′

WqQQ′−+ =1

4

∑

σσ′

σWqQQ′σσ′ WqQQ′−− =1

4

∑

σσ′

σσ′WqQQ′σσ′

with σ, σ′ = ±1.As one can see from Eq. (2.104), Wq is the interaction kernel.

When we consider the expansion from a charge-ordered state without spin order orin the case of a collinear spin solution, Wq presents zero off-diagonal blocks due to the

decoupling of the charge and the spin sector. We then have a complete separation of thecharge and the spin sector within Wq, so that we can extract a purely charge WqQQ′,++

and a purely spin WqQQ′,−−, being the off-diagonal blocks WqQQ′,+−=WqQQ′,−+=0.In these cases, Wq is decoupled in a charge block and a spin block, both along the

diagonal with size 2Na × 2Na. In more general cases Wq is a non-sparse matrix with

size 4Na × 4Na.The same observations on the matrix elements and blocks suit the susceptibility

χ0qQQ′ηη′ and χqQQ′ηη′ as well.


The susceptibility

In order to evaluate the RPA series, we first have to evaluate the susceptibility χ0q on

the GA level:

δRq = −χ0q fq (2.105)

where fq is an arbitrary external field. Using Eq. (2.104) for the RPA interaction

WqQQ′ηη′ and Eq. (2.105) for the GA susceptibility χ0qQQ′ηη′ , we can find the superlattice

GA+RPA susceptibility χqQQ′ηη′ . Formally the RPA summation can be written as

χq = χ0q − χ0

q Wq χq (2.106)

Upon inverting this equation, we finally obtain the charge correlation function (χqQQ′,++)ρρ,as explained in Appendix B.

Appendix A - The GA+RPAapproach to the homogeneous state- other computations

In this Appendix we complete the derivation of the second-order term δE(2)e in the

energy expansion of Eq. (2.71).

Real space energy expansion

In real space we obtain:

δE(2)e =

∑

ij

tij 1

2z0[z

′

(δρiiδTCij + δmiδT

Sij ) + z

′

+−(δρiiδTCij − δmiδT

Sij )]

+1

8Tij0[(z

′

+ z′

+−)2δρiiδρjj + (z′ − z

′

+−)2δmiδmj

+ z0(z′′

++ + 2z′′

+− + z′′

−−)(δρii)2 + z0(z

′′

++ − 2z′′

+− + z′′

−−)(δmi)2]

+1

2Tij0[z

′

D(z′

+ z′

+−)δρiiδDj + z0(z′′

+D + z′′

−D)δρiiδDi]

+ z0z′

DδTCij δDi +

1

2Tij0[(z

′

D)2δDiδDj + z0z′′

D(δDi)2] (2.107)

with the variations corresponding to the charge density ρii = ρii↑ + ρii↓ and to themagnetization density mi = ρii↑ − ρii↓. Further on, we have defined the transitivedeviations:

Tij = Tij0 + δTCij + δT S

ij

which in the charge- and spin sector read as

δTCij =

∑

σ

(δ〈c†iσcjσ〉 + δ〈c†jσciσ〉)

δT Sij =

∑

σ

σ(δ〈c†iσcjσ〉 + δ〈c†jσciσ〉).


Since we study a paramagnetic system, it is convenient to define the following abbre-viations for the z-factors and its derivatives:

ziσ ≡ z0,∂ziσ

∂ρiiσ

≡ z′

,

∂ziσ

∂ρii−σ

≡ z′

+−,∂ziσ

∂Di

≡ z′

D

∂2ziσ

∂ρ2iiσ

≡ z′′

++,∂2ziσ

∂ρiiσ∂ρii−σ

≡ z′′

+−,∂2ziσ

∂ρ2ii−σ

≡ z′′

−−

∂2ziσ

∂D2i

≡ z′′

D,∂2ziσ

∂ρiiσ∂Di

≡ z′′

+D,∂2ziσ

∂ρii−σ∂Di

≡ z′′

−D

For the half-filled paramagnetic state we have z′

= z′

+− and z′′

+D = z′′

−D.

Momentum space energy expansion

We transform Eq. (2.107) into momentum space. For the paramagnetic system the

expansion separates into the charge- and spin sector δE(2)e = δES

e + δECe .

In the spin sector we find:

δESe =

1

N

∑

q

[1

2z0(z

′ − z′

+−)(δSzqδT

S−q + δT S

q δSz−q) +NqδS

zqδS

z−q ] (2.108)

with the following definitions:

δmi =1

N

∑

q

eiqi·riδmq =2

N

∑

q

eiqi·riδSzq

Nq =1

N[(z

′ − z′

+−)2∑

kσ

ǫ0k+q,σnkσ + z0(z′′

++ − 2z′′

+− + z′′

−−)∑

kσ

ǫ0kσnkσ ]

The form of δECe in momentum space is slightly more complicate:

δECe =

1

2Nz0(z

′

+ z′

+−)∑

q

δρqδTC−q

+1

4N2

∑

kqσ

[(z′

+ z′

+−)2ǫ0k+q,σnkσ + (z′′

++ + 2z′′

+− + z′′

−−)z0ǫ0kσnkσ]δρqδρ−q

+1

N2

∑

kqσ

[z′

D(z′

+ z′

+−)ǫ0k+q,σnkσ + z0(z′′

+D + z′′

−D)]δρqδD−q +1

N

∑

q

z0z′

DδDqδTC−q

+1

N2

∑

kqσ

[(z′

D)2ǫ0k+q,σnkσ + z0z′′

Dǫ0kσnkσ]δDqδD−q (2.109)

From Eq. (2.109), we finally recover Eq. (2.94) by using the definitions of Lq, Uq andVq from Eq. (2.95).

Chapter 3

Charge instabilities of ahomogeneous ground state

In this Chapter we exploit the relation between the electronic charge susceptibility in thepresence and the one in the absence of the e-ph coupling, which in § 2.2.1 we showed tobe exact in the adiabatic limit of the e-ph interaction. This means that in this particularlimit we can determine the instabilities of the electron-phonon system studying thebehaviour of the susceptibility without the e-ph coupling. Therefore in this Chapter wepresent our results on the static charge instabilities of the homogeneous Fermi Liquid.We first discuss the reliability of the GA in the various dimensions: in particular, weconsider the solution in 1d where the exact ground state of the Hubbard model is known,and in infinite dimensions where the Gutzwiller method can be solved without the usualapproximation. In our study we obtain the phase diagram of the Hubbard-Holsteinmodel in the static case, exploring similarities and differences especially between theresults in 1d and in 2d. Upon varying the Coulomb repulsion, the e-ph coupling and thedoping, we present a systematic characterisation of the different possible phases of thesystem: homogeneous paramagnet, Peierls CDW and phase separation. In addiction,we consider the renormalization of the e-ph coupling due to the electronic correlationsand we explain the obtained trends [167]. In the last part of this Chapter we study thesoftening of the phonon spectrum, and namely the Kohn anomaly in the presence ofelectronic correlations. The static renormalization of the spectrum is studied both in 1dand in 2d. For 1d-systems we also perform an approximate evaluation of the phonondamping, valid in a quasistatic approach to the problem [168].

Phonons coupled to strongly correlated electrons have already been investigated bymeans of numerical techniques like Quantum Monte Carlo (QMC) [141, 169, 170, 171,172, 173], exact diagonalization [174, 175, 176, 177], Dynamical Mean Field Theory(DMFT) [178, 179, 180, 181, 182, 183, 184], and (semi)analytical approaches like slavebosons (SB) and large-N expansions [18, 142, 143, 144, 145, 146, 185, 186, 187]. De-

64 Charge instabilities of a homogeneous ground state

spite this variety of approaches, a systematic and thorough investigation within thesame technical framework is not yet available either due to the demanding characterof the numerical approaches or to the limited parameter ranges investigated so far.Therefore in this Chapter we study the renormalization of the electron-lattice couplingin the presence of strong e-e correlations systematically considering the momentum,doping and interaction-strength dependencies. In particular we want to elucidate howcharge density wave (CDW) or phase separation (PS) instabilities are modified in thepresence of e-e interactions. To this aim the GA+RPA is a technique which is notnumerically too demanding, but still provides a quantitatively acceptable treatment ofthe strongly correlated regime. In this regard the GA can represent a good compro-mise allowing extensive and systematic exploration of various parameter ranges whilekeeping a reliable treatment of the low-energy physics. It has been recently shownthat the Gutzwiller variational approach provides remarkably accurate positions of thecomplex magnetic phase boundaries in infinite dimensions [188]. This indicates thatthe Gutzwiller energy and its derivatives are quite accurate. In this Chapter we extendthese results to the charge channel also in infinite dimensions, where the GA to theGutzwiller variational problem is exact [149, 162]. In addition, in order to make con-tact with layered systems like the cuprates, we study the 2d-case where the GA is stillexpected to give an accurate estimate of the energy. This also gives us the opportunityto study the interplay between nesting in the presence of e-ph coupling, which favoursPeierls distortions, and strong correlation which favours PS.

To obtain the phase diagram in the presence of both e-e and e-ph interactions, inprinciple, one should compute the GA energy for every possible charge-ordered state.However, we will study the static response functions of the uniform state to an externalperturbation and locate the relevant instabilities. In this way we can not find first-ordertransitions but we are confident that we will not miss much of the physics of the model,gaining a considerable practical advantage. In § 2.2.1 we showed that for Holsteinphonons in the adiabatic limit the charge susceptibility in the presence of e-ph and e-einteraction is simply related to the charge susceptibility without phonons. Thereforeour work reduces to compute the latter which is done in the GA. The method we usehere corresponds to the static limit of the GA+RPA, rooted in Vollhardt’s Fermi liquidapproach [161]. As a by-product, our work generalizes Vollhardt’s computations of thezero-momentum and half-filled charge susceptibility to any momentum and filling. Thisapproach is not as accurate as QMC or DMFT studies as far as the electron dynamicalexcitations are concerned, because it inherently deals with the quasiparticle part oftheir spectrum. Nonetheless with relatively moderate numerical efforts it allows for asystematic analysis of momentum, doping and interaction dependencies of the screeningprocesses underlying the quasiparticle charge response and the related e-ph coupling.

It is worth mentioning that the dynamical version of our GA+RPA approach hasbeen tested in various situations and found to be accurate compared with the exactdiagonalization [150, 151, 164, 166]. Computations for realistic models have provided

3.1 The Gutzwiller method in infinite dimensions: the chargesusceptibility 65

a description of different physical quantities in accord with experiments [64, 163, 165].In this Chapter we consider a static expansion around the paramagnetic state and thuswe ignore antiferromagnetic states and related instabilities that arise as the system ap-proaches half-filling1. In Chapter 4 we will consider a “striped” state with a particulartype of broken-symmetry state with charge and spin order.

In the following Sections we present results with the electronic hopping restrictedto nearest-neighbour atoms (tij = −t) and put t = 1 which makes the energy and thecharge susceptibilities dimensionless. Occasionally we will explicitly rescale the inter-action by the Brinkman-Rice transition Uc leaving the susceptibility units untouched.

3.1 The Gutzwiller method in infinite dimensions:

the charge susceptibility

The study in d = ∞ is obviously the most suitable case for the GA+RPA formalismessentially because the GA corresponds to the exact evaluation of the GZW variationalwavefunction in this limit.

We consider the case of a hypercubic lattice in infinite dimensions with nearest-neighbour hopping, where the density of states per spin is given by

ρ0(ω) =

√2

π

1

texp

(− ω2

2t2

).

In this case a momentum dependence in the response is still present via the quan-tity [190]

ηq =1

d

d∑

ν=1

cos qν

which enters the interaction kernel Wq and the correlation functions χq. For example,the non-interacting susceptibility reads

χ0q = −4

∫ µ

−∞

dω′

∫ ∞

µ

dω′′

(1 ω′+ω′′

z20

ω′+ω′′

z20

(ω′+ω′′)2

z40

)

× Λq(ω′, ω′′)

ω′ − ω′′

1In Ref. [189], Sangiovanni et al. presented a methodical DMFT-based comparison between theparamagnetic and the AF solution of the Hubbard-Holstein model. In that paper they stressedsimilarities more than differences. Clearly AF correlations change a lot the physics of lightly dopedsystems, but on the other hand the suppression of the e-ph coupling is only quantitatively modifiedand no new trends arise.


0 0.5 1 1.5 2U/U

c

0

1

2

3

Kq

ηq=1

ηq=0.5

ηq=0

ηq=-0.5

ηq=-1

1 1.5 2U/U

c

0

0.05

0.1

Kq

n=0.99

Figure 3.1: d=∞ - Charge susceptibility as a function of U/Uc for n = 0.99. Hereand in Fig. 3.2 the inset shows an enlargement of the large-U region.

with

Λq(ω′, ω′′) =

1

2πz40t

2√

1 − η2q

×

exp

[− 1

4z40t

2

(ω′ − ω′′)2

1 − ηq+

(ω′ + ω′′)2

1 + ηq

]

and in the two limiting cases ηq=0 = 1 and ηq=Q = −1 one can give analytical expres-sions for the static susceptibility matrices of the Lindhard function:

χ00 = 2ρ(µ)

(1 2µ2µ 4µ2

)

χ0Q =

1√2πz2

0t

(1 00 0

)E1

(µ2

2t2z40

),

where Q is the momentum (±π,±π,±π...) and E1(x) denotes the exponential inte-gral [191].

As discussed in Ref. [190], for a generic q the corresponding ηq is trivially zero;only for special q, ηq takes nonzero values between -1 and +1. In particular for thehypercubic lattice the relevant q’s are the ones along the diagonal (0, 0, 0...)−(π, π, π...)(and the other equivalent directions of the hypercubic lattice, which form a set ofmeasure zero). This means that for this infinite-dimensional lattice, the study of themomentum depencence of the quantities is sensitive to effects in the (1,1,1...) direction

3.1 The Gutzwiller method in infinite dimensions: the chargesusceptibility 67

0 0.5 1 1.5 2U/U

c

0

0.5

1

1.5

Kq

ηq=1

ηq=0.5

ηq=0

ηq=-0.5

ηq=-1

1 1.5 2U/U

c

0

0.1

0.2

Kq

n=0.9

Figure 3.2: d=∞ - Charge susceptibility as a function of U/Uc for n = 0.9.

of the Brillouin zone, while it cannot access the other directions like, for instance(1,0,0,0...) or (...0,0,0,1).

In Figs. 3.1 and 3.2 we show how the charge susceptibility depends on U for selectedmomenta at n = 0.99 (almost half-filling) and n = 0.9 respectively. The curves corre-spond to different values of the interaction strength in units of the critical Uc = 8

√2/πt,

the interaction at which in the infinite-dimensional hypercubic lattice the electrons un-dergo the metal-insulator transition at half-filling (n=1) in the GA. At small U thesusceptibility has a strong enhancement at qc = Q (see dot-dot-dashed line in Fig. 3.1):this is due to the nesting of the Fermi surface and leads to the Peierls CDW instabilityin the presence of coupling to the lattice [Eq. (2.16)].

Starting from the small-U side, the charge susceptibility is suppressed upon ap-proaching Uc and then slightly increases again when U is further increased. The be-haviour of κq for U > Uc strongly depends on momentum and the suppression is mosteffective for large q and close to half-filling. This behaviour will appear to be commonto systems in every dimension.

Perfect nesting occurs only at half-filling due to the matching of the Fermi hy-persurface when translated by Q. In analogy with low dimensional systems one maywonder whether away from half-filling an incommensurate CDW is favoured. Figs. 3.3and 3.4 show the momentum dependence of the susceptibility for density n = 0.99 andn = 0.9 and various values of U/Uc, respectively.

At small U one finds the nesting-induced enhancement at q = Q for both fillingsindicating that incommensurate CDW formation is not favoured, i.e. there is no shiftof qc. The fact that for small U/Uc the instability momentum is pinned at Q is specific


-1 -0.5 0 0.5 1η

q

0

0.1

0.2

0.3

0.4

0.5

0.6

Kq

U/Uc=0.31

U/Uc=0.78

U/Uc=1.25

U/Uc=1.88

n=0.99

Figure 3.3: d=∞ - Charge susceptibility as a function of ηq for n = 0.99 for variousvalues of the interaction strength U/Uc.

to a high-dimensional system, where the effects of nesting of the Fermi surface areweak and the effects of doping in changing the Fermi surface are negligible. We willsee in Section § 3.3 that this is not the case in 2d, where upon doping qc moves awayfrom the (1,1) direction and shifts along the (1,0) direction. The topology of the Fermisurface is clearly very relevant in 1d, as we will show in Section § 3.2.

Now we switch to one of the most interesting results we have found: at large Uthe order of the dominant instability is reversed. In fact, the CDW is replaced bya q = 0 (ηq = 1) instability. This implies that including a Holstein phonon thesystem will undergo PS before the CDW instability arises. One qualitatively recoversa momentum structure similar to what is obtained within the large-N expansion ofthe Hubbard model for U = ∞ [192, 193]. Thus our results allow to understand thecrossover from the weak coupling behaviour to the strong coupling one. In this casethe residual repulsion between quasiparticles is most effective at large momenta leadingto a suppression of κq for ηq → −1. The momentum dependence of the susceptibilitybecomes weak for intermediate values of U , slightly below Uc. All these features aremost pronounced upon approaching half-filling.

From Figs. 3.3 and 3.4 we conclude that when U is increased beyond a value of about0.78 Uc the maximum in the charge response moves from large to small momenta. Thissignals that the inclusion of (momentum-independent) phonons would drive the systemtowards a PS instability at large U ’s, while at small U ’s the systems would undergo atransition to a CDW state. The behaviour of the charge response also allows to inferthat the e-e correlations suppress more severely the e-ph coupling at large momentumtransfer than at small transferred momenta. This is the reason why, upon increasingU , the system will undergo more easily a low-momentum instability (PS), rather thanbecoming unstable at finite (and large for δ ≪ 1) momenta. All these considerationson the renormalization of the e-ph coupling are possible using Eq. (2.13): λq ≡ λκq/χ

00.

3.2 Charge susceptibility in one dimension 69

-1 -0.5 0 0.5 1η

q

0

0.1

0.2

0.3

0.4

0.5

Kq

U/Uc=0.31

U/Uc=0.78

U/Uc=1.25

U/Uc=1.88

n=0.9

Figure 3.4: d=∞ - Charge susceptibility as a function of ηq for n = 0.9 for variousvalues of the interaction strength U/Uc.

3.2 Charge susceptibility in one dimension

In this Section we test our GA approach at work in 1d which is clearly the mostunfavourable case for a mean-field like FL approach. Besides the intrinsic physicalinterest of this study, the 1d-calculations allow a comparison with exact results tounderline advantages and drawbacks of our method. In order to highlight all the effectsof correlations, it is important to single out the features of momentum dependencieswhich are already present in non-interacting systems via the Lindhard function χ0

q. Wetherefore show in Fig. 3.5 the elements of the matrix χ0

q (see Eq. (2.98)) for U = 0 in1d. The diagonal (off-diagonal) elements of χ0

q are even (odd) functions of the dopingδ. (χ0

q)11 is always positive and presents two peaks, at q = 2kF and q = 2(π − kF );one peak at q = π at half-filling. −(χ0

q)12 is a decreasing function of the filling: it ispositive for n < 1, vanishes identically at n = 1 and becomes negative for n > 1. Both(χ0

q)12 and (χ0q)22 vanish at q = π, because ǫ0k+π,σ = −ǫ0kσ. The peaks of (χ0

q)11 and(χ0

q)22 coincide with the zeros of the denominator ǫk+q,σ − ǫkσ in Eq. (2.98).Then we consider how the structures in the non-interacting susceptibility χ0

q aremodified by the interactions (see Eqs. (2.97) and (2.99)). For n = 1 χ0

q and Wq arediagonal, then χq is diagonal in every dimension. For q = (π, π, ...) and n 6= 1 only χ0

q

is diagonal. In fact, we find that close to half-filling Bq ≪ Aq for finite U but not zeroand then χq is not diagonal.

3.2.1 Reliability test of the Gutzwiller approximation

To test the limits and the validity of the GA in 1d, we first compare our results tothe exact solution [132, 194, 195] and to a QMC study [171]. Then we will report allour GA results, noticing that in the q=0 limit the charge susceptibility coincides with


0 Π2

Π 3Π2

2Π

q

0

0.5

1

1.5

2

2.5

Χ110HqL

0 Π2

Π 3Π2

2Π

q

0

0.5

1

1.5

2

-Χ

120HqL

0 Π2

Π 3Π2

2Π

q

0

1

2

3

4

Χ220HqL

Figure 3.5: Diagonal (top and bottom-right panels) and off-diagonal (bottom-leftpanel) matrix elements of the 1d-Lindhard susceptibility as a functionof the momentum for U = 0 and n = 1 (full line), n = 0.8 (dashed line)and n = 0.7 (dashed-dotted line).

the compressibility. We have checked that indeed the RPA result coincides with thecompressibility obtained as the inverse of the second derivative of the GA energy.

In Fig. 3.6 we present the charge compressibility as a function of n for differentvalues of U/Uc, with Uc = 32t/π. Uc is the Coulomb repulsion at which the Brinkman-Rice transition takes place for n = 1 in the GA. In the exact solution the metal-insulatortransition already occurs for infinitesimal U , and so Uc is used only as an energyunit. In the top panel (a) we show κq=0 calculated using the exact 1d-ground state ofthe Hubbard model and in the bottom panel (b) we report our GA results. Naturaldifferences arise because we adopt a FL picture for the electrons, while the groundstate of Eq. (2.1) in 1d is a Luttinger liquid. A much better estimate of the energy andof the compressibility can be obtained within the GA using antiferromagnetic (AF)states [150]. This occurs because the AF broken-symmetry state is closer to the 1dLuttinger liquid with rather long-range AF correlations.

In Fig. 3.6a we see that the exact solution at n = 1 is insulating for any finite U . Onthe other hand, we remark that the GA compressibility has a jump discontinuity for


Figure 3.6: 1d-Charge compressibility as a function of n: calculated with the Betheansatz - exact 1d solution (a) and with the GA (b).


0 0.2 0.4 0.6 0.8 1qΠ

0.2

0.3

0.4

0.5

0.6Κ q

UUC=0.78 QMC

UUC=0.78 GA

UUC=0.39 QMC

UUC=0.39 GA

Figure 3.7: 1d-Charge susceptibility (n = 0.6) as a function of q/π: comparisonbetween GA and QMC results [171].

n = 1 and U > Uc. In fact, its left and right limits are finite, while its value computedin n = 1 is zero2. The GA does not find the divergences of the exact result for U = ∞,in the limits n → 0 and n → 1. Therefore the GA results appear to be more reliablefor fillings not too close to n=1 or to the n → 0 limit. Looking at Eq. (2.99) we canunderstand why this happens in 1d. In fact, for low fillings (dopings) the quantityχ0

0Wq=0 diverges due to the van Hove divergence of the DOS χ00. Since the product

χ00Wq=0 is large, the system is effectively at strong e-e coupling and then the GA is

inadequate. Away from these extreme cases, the quantity χ00Wq=0 is small and then

the GA works fairly well for intermediate fillings n ≃ 0.2 ÷ 0.8. Close to half-fillingat U = ∞ the exact solution shows a divergence that arises because the system isequivalent to free spinless fermions which have a divergent DOS at n = 1. Also herethe GA obviously fails because it represents a band of quasiparticles with divergingmass at half-filling and no van Hove singularity is present.

In Fig. 3.7 we report the charge susceptibility for a comparison to Hirsch andScalapino’s QMC results [170, 171]. Since their data are for n = 0.6 we expect the GAto give reasonable results. Although our formalism is at T = 0 and the QMC study atT 6= 0, the comparison is meaningful because their results are at T = 0.0690 t, quitelow if compared to the electronic energy scales. The QMC susceptibilities generallyagree with ours within ten-twenty percent deviations, and with larger deviations athigh q for U ∼ Uc. For large U , in fact, the QMC data exhibit the transfer of the peakfrom q = 2kF to q = 4kF , signature of the spin-charge separation of the 1d-Luttinger

2This characteristic behaviour of the GA charge compressibility holds in every dimension. See forexample also Fig. 3.12 in § 3.3 (compressibility in 2d).


0 2 4 6 8 10UUc

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Κ q

q=0

q=Π2

q=2kF=3Π4

q=Π

Figure 3.8: 1d-Charge susceptibility as a function of U/Uc for n = 0.75.

liquid, clearly absent in our 1d-FL. The QMC curves present also effects due to thefinite T .

0 0.2 0.4 0.6 0.8 1 1.2UUc

0

0.2

0.4

0.6

0.8

1

g q

q=0

q=Π2

q=2kF=7Π8

q=Π

Figure 3.9: 1d-Renormalized e-ph coupling as a function of U/Uc for n = 0.875.

3.2.2 Renormalization of the electron-phonon interaction

The above results indicate that our FL scheme works quantitatively rather well awayfrom half-filling (where a GA-AF state would do a better job) and provides reasonablemomentum dependencies (but for the 4kF -peak shift for large U). In Fig. 3.8 we showthe charge susceptibility κq as a function of U/Uc. For small deviations of the densityfrom half-filling the compressibility has a minimum close to U = Uc; for large q this


0 1 2 3 4 5 6q

0.2

0.4

0.6

0.8

1

Gq

Figure 3.10: 1d-Charge vertex as a function of q for n = 0.9 for U/Uc = 0.1 (dotted-dashed line), U/Uc = 0.46 (dashed line), U/Uc = 0.93 (dotted line) andU/Uc = 1.23 (solid line).

minimum becomes too shallow to be clearly seen in Fig. 3.8. At small momenta thecharge susceptibility is close to the compressibility. As the momentum approachesq = 2kF = nπ the charge susceptibility takes its highest values at small U , signallingthe Peierls instability for U = 0. This divergence however is strongly suppressed byU . At small doping, κq is finite but still shows a shallow minimum close to U = Uc.This behaviour is due to the proximity of the Mott phase, as will be clear below intwo dimensions. Therefore e-e interactions renormalize the Peierls instability whichneeds a finite λ to occur. Having Eq. (2.12) in mind, we now are interested to findthe momentum q at which the susceptibility κqc

has a maximum. This in fact willdetermine the corresponding instability in κeph

qc, the susceptibility in the presence of

the e-ph coupling: with a large enough coupling (λ > λc), the system will have atransition to an inhomogeneous state with that typical wavevector qc. We see in Fig. 3.8that for U & Uc the order of the instabilities reverses as in d = ∞ since the chargesusceptibility at q = 2kF becomes smaller than the susceptibility at q = 0 (related tothe phase separation). In other words, at large U and increasing the electron-phononcoupling λ the system becomes unstable towards phase separation (PS) before thePeierls instability arises. This result holds in all dimensions.

To complete our analysis of 1d-systems, we consider explicitly the renormalizede-ph coupling.

In Fig. 3.9 we show the behaviour of the quantity gq (introduced in § 2.5) withrespect to the electronic interactions. We see that the e-e interactions suppress thebare e-ph coupling for any transferred momentum q. A similar trend has been foundin higher dimensions too.

3.3 Charge susceptibility in two dimensions. 75

In Fig. 3.10 we present the quasiparticle vertex Γq (introduced in § 2.2.1; see alsoEq. (2.101)) for different U . In general, Γq signals a reduced coupling between electronsand phonons. This reduction is larger for intermediate-large transferred momentumq. This tendency is more pronounced at large U . An additional feature is the dipminimum corresponding to the peaks in the bare Lindhard susceptibilities at q = 2kF .This effect is sharpest for small U and shows that Γq has the strongest renormalizationfor the Peierls momentum. Thus, already for small U , the Peierls instability in thesusceptibility is removed and for larger U we find that the phonon-driven Peierls stateis suppressed, leaving the place to PS.

It is important to notice that the failures of the GA found in our comparison withthe exact results can be always traced back to specific features of the 1d-physics: thesusceptibility peak transfer to q = 4kF for large U , due to the spin-charge separation;the equivalence to free spinless fermions at U = ∞ and in general the divergences ofthe exact compressibility at half-filling and for n → 0. Therefore these failures shouldnot be attributed to the GA per se, but to the underlying assumption of a FL groundstate. This makes us confident that our results are reliable when this assumption ismore justified, and actually this will occur in higher dimensions.

3.3 Charge susceptibility in two dimensions.

H0,0L HΠ,0L HΠ,ΠLq

0.04

0.06

0.08

0.1

0.12

0.14

Κ q

H0,0L HΠ,0L HΠ,ΠL

Figure 3.11: 2d-Charge susceptibility for n = 0.9 along the open path q = (0, 0) −(π, 0) − (π, π) for U/Uc = 0.5 (solid line), U/Uc = 0.9 (dot-dashed line)and U/Uc = 2 (dotted line).

We start by characterizing the tendency to the Peierls instability in 2d. For smallU and n = 1 the charge susceptibility has the Peierls peak at q = (π, π), associated


with the Fermi surface nesting. For the doped system the response exhibits a peak fora qc close to (π, π), featuring the tendency to develop an incommensurate CDW in thepresence of phonons (see for example the full-line curve in Fig. 3.11). The momentumof the instability undergoes a shift in the (1, 0) direction of the Brillouin zone. Forsmall U the peak is located at qc = (α(δ)π, π) with

α(δ) = 1 − 0.46δ − 1.30δ2... ∼ 1 − δ/2.

This depends little on U at weak coupling due to the weak q-dependency of Wq closeto n = 1. A behaviour compatible to ours has been found also for the 2d-Holsteinmodel [197] with QMC and RPA calculations on a 8 × 8 lattice.

0 0.2 0.4 0.6 0.8 1 1.2 1.4UUc

0

0.1

0.2

0.3

0.4

0.5

0.6

Κ q=

0

n=1

n=0.9

n=0.7

Figure 3.12: 2d-Charge compressibility as a function of U/Uc for various fillings.

Along the (1,0) direction, κq exhibits another peak at q′c (the peak close to q = (0, 0)

in the full-line curve in Fig. 3.11). This corresponds to the scattering between states atthe (rounded) corners of the Fermi surface in adjacent Brillouin zones. Upon increasingU (cf. Fig. 3.11), the nesting-induced peak structure disappears. Simultaneously theresponse at large wave-vectors is suppressed and overcome by that at q = (0, 0). Thisindicates that the order of the instabilities is reversed like in the d = ∞ and d = 1cases: for large U the system undergoes phase separation before the CDW instabilityarises. This behaviour will be more clarified upon analyzing the charge susceptibilityas a function of filling and interaction.

In Fig. 3.12 we show the charge compressibility in 2d as a function of U/Uc, beingUc = 128t/π2. For U = 0 the compressibility is given by the non-interacting density ofstates at the Fermi energy. The latter diverges for n → 1 due to the logarithmic VanHove singularity. For U close to Uc one recovers a similar behavior as in d = ∞. Thecompressibility vanishes at the Mott transition point and has a minimum close to Uc

for n 6= 1.


0 0.2 0.4 0.6 0.8 1n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Κ q=

0

UUc= ¥

UUc=1

UUc=0.2

UUc=0

Figure 3.13: 2d-Charge compressibility as a function of n for various U/Uc.

In Fig. 3.13 we present the compressibility as a function of n for different valuesof U/Uc. For n → 0 the compressibility can be computed exactly using a low densityexpansion [198]. The ground state energy reads:

E

N= πtn2 − 2πt

n2

ln(n)+ ...

where the second term is the leading correction due to interactions. Computing thecompressibility as κ−1

q=0 = ∂2E/∂n2/N one finds that the zero density limit is given bythe non-interacting compressibility

κq=0(n→ 0) = (2πt)−1.

In GA+RPA we find instead a small suppression of the zero density compressibilitywith interaction. This is not surprising since RPA is expected to break down in thelow density limit. Still this dependence is quite small and we expect that our resultsare accurate at moderate densities. For small U the compressibility is an increasingfunction of n and reaches the maximum value for n = 1 as a consequence of thevan Hove enhancement. For U = Uc, κq=0 vanishes for n = 1; for larger U > Uc,κq=0 flattens, still exhibiting a smooth maximum for finite doping. Therefore the GAcompressibility has a jump discontinuity for n = 1 and U > Uc: its left and right limitsare finite, while it vanishes at n = 1.

The qualitatively different behaviour of the compressibility for small U and largeU is clear: for low fillings the system is weakly affected by e-e interactions and itscompressibility increases with n no matter how large U is. Approaching half-filling thecorrelated nature of the system becomes relevant and reduces the compressibility of


0 0.2 0.4 0.6 0.8 1 1.2UUc

0

0.5

1

1.5

2

Κ q

q=H0, 0L

q=HΠ, 0L

q=HΠ2, Π2L

q=qCDW=HΠ, ΠL

0 0.5 1 1.5 2UUc

0

0.1

0.2

0.3

0.4

0.5

0.6

Κ q

q=H0, 0L

q=HΠ, 0L

q=HΠ2, Π2L

q=qCDW=H0.942Π, ΠL

q=HΠ, ΠL

Figure 3.14: 2d-Charge susceptibility as a function of U/Uc for n = 1 (top) andn = 0.9 (bottom).

the electron liquid around n = 1. One should also keep in mind that close to half-fillingAF correlations (here neglected) will become relevant also in 2d.

In Fig. 3.14 we show the charge susceptibility κq as a function of U/Uc for n = 1and n = 0.9 and selected momenta.

At small momenta the charge susceptibility is close to the compressibility for bothfillings. As the momentum approaches q = qc the charge susceptibility takes its high-est values for small U . In particular for n = 1 (perfect nesting) and U = 0 the chargesusceptibility diverges indicating that an infinitesimal λ renders the system unstable.The susceptibility, however, is strongly suppressed by U and at half-filling for any mo-mentum goes to zero for U = Uc. Therefore, as for d = ∞, e-e interactions renormalizethe non-interacting CDW instability which thus needs a finite λ to occur.

At small doping, κq is finite and shows a shallow minimum close to U = Uc. As


H0,0L HΠ,0L HΠ,ΠLq

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

Κ q

H0,0L HΠ,0L HΠ,ΠL

Figure 3.15: 2d-Charge susceptibility for U/Uc = 100 along the open path q =(0, 0) − (π, 0) − (π, π) for n = 0.98 (dashed line), n = 0.9 (dotted line),n = 0.7 (solid line) and n = 0.4 (dotted-dashed line).

in d = ∞ and d = 1, we see from the larger κq that for U & Uc the PS instabilitybecomes dominant.

Our results are consistent with Ref. [19] where Castellani et al. find PS in a U = ∞slave boson (SB) investigation at T = 0. On the other hand, SB calculations for 2d-systems [143, 144] have found PS for T ≈ t even in the absence of phonons, while forlower T an homogeneous state is restored. To our knowledge, this reentrant behaviourhas not been confirmed by other techniques and should be taken with care due to thepoor performance of mean-field SB techniques at finite T .

The dominance of the PS instability at large U can be understood from the strongcoupling (U/t≫ 1) results close to n = 1. To illustrate the behaviour of the suscepti-bility in the very strong coupling case, in Fig. 3.15 we show the charge susceptibilityfor U/Uc = 100 along the open path q = (0, 0) − (π, 0) − (π, π).

If we consider fillings quite close to n = 1, the Coulombic repulsion completelysuppresses the Peierls peaks which upon doping become visible again for n ∼ 0.7.Clearly, more evident peaks appear for lower fillings; in fact, for n ≪ 1 the effects ofe-e interactions are weak even for very large U . For n < 0.5, we find that the groundstate is a CDW and that the susceptibility exhibits quite different features with respectto the fillings n & 0.7.

The most noticeable feature of Fig. 3.15 is the peak in κq centered at q = (0, 0): itgets narrower as the doping δ vanishes. Therefore upon reducing δ the charge responseto a local perturbation spreads more and more out in space and the small q peakwidth is a measure for the corresponding inverse screening length. We can give ananalytical interpretation of this behaviour, using the approximate relation κq ≈ 1/Aq


valid for δ ≪ 1 and adopting for Aq its U = ∞ form (see § 2.5.2). In Fig. 3.15 we see

0 2 4 6 8 10UUc

0

0.025

0.05

0.075

0.1

0.125

0.15Κ q

q=H0, 0L

q=HΠ, 0L

q=HΠ2, Π2L

q=HΠ, ΠL

0 2 4 6 8 10UUc

0

0.025

0.05

0.075

0.1

0.125

0.15

1A

q

q=H0, 0L

q=HΠ,0L

q=HΠ2, Π2L

q=HΠ, ΠL

Figure 3.16: 2d-Charge susceptibility and 1/Aq as a function of U/Uc for n = 0.98.

that for large U , the curves κq proceed separately, approaching a saturation value forU → ∞ [192]. For low doping the saturation value for κq can be estimated thanks toEq. (2.86). In fact, Eq. (2.86) has been derived as an exact relation for n = 1 but weexpect it to be a good approximation for fillings close enough to n = 1. In the case oflow doping and U ≫ Uc, κ

0qAq takes large values; then, using Eq. (2.86), we find simply

κq ≈ 1/Aq. Thus for low doping the saturation of Aq implies that κq also saturates,giving a maximum in κq at q = 0 (cf. Eq. (2.100) which is the U = ∞ form of 1/Aq).We have checked that for low doping Eq. (2.100) gives a quite accurate representationfor the whole κq curve (see Fig. 3.16). In particular, if we consider the low q expansionof 1/Aq, we observe that the q = (0, 0) peak of the susceptibility is well fitted by a

Lorentzian peak of half-width q =√

8δ in the x- and y-directions. It is worth notingthat the typical momentum associated to the peak depends only on the doping δ andnot on the energy e0. In the low q limit a relation q ∼

√δ can also be found using the

U = ∞ single SB quasiparticle interaction [193].


Now we can exploit Eq. (2.12): κephq =

κq

1 − λq

. In fact, using this equation, all the

previous considerations on the pure electronic response lead us to the phase diagram ofthe e-ph system in Fig. 3.17. The main outcome is that the large e-e interaction changes

Figure 3.17: Instability surface λ(n, U/Uc) of the paramagnet towards CDW and PS:the sphered-line marks the transition between the CDW state and thestate with PS. The picture above is essentially the phase diagram of theparamagnet in 2d.

the nature of the charge instability from an incommensurate CDW to a PS. At largeU & Uc and moderate doping this effect already occurs for small values of the bare e-phcoupling λ. This is the result of a compromise between a reduced quasiparticle kineticenergy (which renders the system prone to instabilities) and the modest screening ofthe e-ph coupling, when one is away from the Mott-insulating phase at half-filling.The fact that the screening is less important at small transferred momenta obviouslyfavours the occurrence of PS at qc = 0 with respect to the incommensurate CDW.


In the case of 2d-systems we now consider explicitly the renormalized e-ph couplinggq = z2

0Γq for the bare electrons. In Fig. 3.18 we show the behaviour of the quantity gqas a function of the e-e interaction. The trend is quite clear: the suppression of the bare

0 0.2 0.4 0.6 0.8 1UUc

0

0.2

0.4

0.6

0.8

1

1.2

g q q=HΠ4, Π4L QMC

q=HΠ, ΠL QMC

0 0.2 0.4 0.6 0.8 1UUc

0

0.2

0.4

0.6

0.8

1

1.2

g q

q=HΠ4, Π4L GA

q=HΠ, ΠL GA

0 0.2 0.4 0.6 0.8 1UUc

0

0.2

0.4

0.6

0.8

1

1.2

g q

q=HΠ4, Π4L SB

q=HΠ, ΠL SB

Figure 3.18: 2d-Renormalized e-ph coupling as a function of U/Uc for n ≈ 0.88 attwo different momenta. For comparison we report the results obtainedat finite T (T = 0.5) with SB (empty symbols) [see Fig. 3 of Ref. [143]]and QMC (filled symbols; QMC results are for finite Matsubara electronfrequency ω = π/2) [see Fig. 4a of Ref. [141]]. All the results areobtained for a 8 × 8 lattice.

e-ph coupling is stronger for large q. We compare our results at T = 0 with SB [143]and QMC [141] calculations at T = 0.5 (these latters are also performed at a finiteMatsubara frequency of the incoming and outgoing fermions ω = πT ). The agreementis generically rather good. However, in the finite-T results an upturn of gq is alsopresent (more pronounced for small q); in Ref. [143] it was interpreted as the signatureof an incipient PS, which then disappears at zero temperature (besides our findings,for T = 0 other different treatments find that the ground state is homogeneous [141,143, 144]). The nature of this reentrant behaviour is still unclear.

In Fig. 3.19 we show the vertex gq for different U along a triangular path. Thisquantity displays minima at wavevectors q = qc (and at q′

c, see also Fig. 3.11) thussuppressing the non-interacting instabilities. These minima arise because the baresusceptibility κ0

q is maximal at these wavevectors, while the corresponding quantity inthe presence of U , κq is small due to the suppressed scattering at large momenta whenthe interaction becomes sizable. On the other hand the charge susceptibility is reduced

3.4 Phonon softening: the Kohn anomaly with strongly correlatedelectrons 83

HΠ,ΠL H0,0L HΠ,0L HΠ,ΠLq

0.05

0.1

0.15

0.2

0.25

0.3

0.35

g q

HΠ,ΠL H0,0L HΠ,0L HΠ,ΠL

Figure 3.19: 2d-Renormalized e-ph coupling for n = 0.9 along the closed path q =(π, π) − (0, 0) − (π, 0) for U/Uc = 0.46 (dashed line), U/Uc = 0.62(dotted-dashed line), U/Uc = 0.93 (dotted line) and U/Uc = 1.23 (solidline).

less at small momenta and this gives rise to the pronounced maximum around q=(0,0)in the large-U case (U & 0.9 in Fig. 3.19). The shape of the curves given in Fig. 3.19is very similar to those obtained within a SB calculation at T = 0.002 [143]. However,this seeming agreement has to be taken with a pinch of salt since the results shown inRef. [143] are for the quasiparticle-phonon vertex while ours correspond to the vertexfor bare electrons and thus should differ by a factor z2

0 .

3.4 Phonon softening: the Kohn anomaly with strongly

correlated electrons

The Kohn anomaly consists in the extreme softening of the phonon spectrum at|q| = 2kF (kF is the Fermi momentum), occurring if phonons are coupled to theelectrons. This is a static exhibition of the enhancement of the Lindhard function (χ0

q)11

at that momentum [27, 35]. In weakly correlated materials this has been predictedlong time ago [200, 201] and confirmed experimentally [202, 203, 204, 205, 206]. Inthis Section we use our previous results to describe what happens upon increasing thee-e interaction. We will show the suppression of the Kohn anomaly for a fixed e-phcoupling and thus the shift of this phononic feature to larger e-ph couplings.

We consider the single-band system on 1d-chain and 2d-square lattice. From the


pole of the phonon propagator Dq [207]

Dq =2ω0

ω2 − ω20 + λω2

0(χq)11/χ00

, (3.1)

one determines the renormalized phonon frequency Ωq:

Ωq = ω0

√1 − λRe(χq)11/χ0

0 (3.2)

and the phonon damping γq:

γq = − 1

2Ωq

(λω2

0Im(χq)11/χ00

)(3.3)

Eqs. (3.2) and (3.3) are approximate under the assumption that Im(χq)11 is small(quasistatic case). In the case Im(χq)11 = 0, one will add an infinitesimal constantiǫ = i0+ to the denominator of Eq. (3.1) to regularize the electronic propagator Dq.

The static form of (χq)11 allows to reproduce adequately the softening associatedto the Kohn anomaly. In particular, within our framework, i.e. static evaluation ofthe electronic susceptibility and adiabatic limit for the e-ph coupling, we can recoverEq. (3.2), as we show in a derivation which holds for any type of e-e interactions.

The exact energy in the adiabatic limit for the e-ph interaction is

Etot[ρ, x] = Ee0 +∑

q

[1

2κq

δρqδρ−q + βxqδρ−q +1

2Kxqx−q

](3.4)

where Ee0 is the ground state energy in the absence of the e-ph coupling, and theparameters of the Holstein hamiltonian have been defined in Section § 2.2. In orderto obtain the adiabatic energy surface seen by the lattice, we minimize Eq. (3.4) withrespect to δρ, obtaining the condition relating the lattice configuration to a given chargedeviation δρ±q = −βκqx±q. We replace this expression into Eq. (3.4) and apply thesecond Hamilton equation: ∂Etot/δx−q = −Mxq.

Then one finds: xq +Ω2qxq = 0 with Ω2

q = ω20 −λκqω

20/χ

00, using λ = (β2χ0

0)/K andK = Mω2

0 . When Einstein phonons are coupled to electrons in a homogeneous state,the renormalized phonon frequency is

Ωq = ω0

√1 − λ

κq

χ00

(3.5)

In Fig. 3.20 we present our results for the phonon dispersion Ωq in 1d and 2d for amoderate coupling λ=0.5. At this coupling the non-interacting system has alreadybeen driven to a Peierls CDW and Ωq becomes an imaginary quantity close enoughto q = 2kF . Upon increasing U , the effective e-ph coupling is weakened, so that the

3.4 Phonon softening: the Kohn anomaly with strongly correlatedelectrons 85

system becomes homogeneous and the Kohn anomaly is well observed at q = 2kF .Already for small U the phonon softening and the dip minimum due to the Kohnanomaly is strongly suppressed with respect to the U = 0 case and for U/Uc = 0.5 theanomaly is practically removed for λ = 0.5 (Fig. 3.21a). Increasing λ, the anomalyreappears (Fig. 3.21b).

As one can see from Eq. (3.2), the specific differences between the 1d and the 2d caseare due to the behaviour of the susceptibility κq as a function of q, with the minima ofΩq corresponding to the maxima in κq. While in 1d Ωq has only one minimum at 2kF ,in 2d it shows a global minimum at qc = 2kF and other local minima depending onthe doping which substantially changes the shape of the Fermi surface from the nestedone for n=1.

In order to compute the damping, we need to consider a finite frequency ω in thesusceptibility χq. In fact, we must go beyond the static limit to evaluate Imχq; in fact,the static Imχq is 0 everywhere, then the damping is absent (see Eq. (3.3)).

We calculate the damping γq in 1d, after linearizing the electronic band close tokF and considering the expansion of the particle-hole (p-h) excitation energy aroundq = 2kF . Our minimal dynamical investigation then makes use of a quasistatic formof Imχq, physically reasonable for ω ∼ ω0 ≪ t. Under these assumptions, one findsthat the non-interacting Im(χ0

q)11 has a simple dependence in the momentum q: it isa finite constant within the “fan” region bounded by ω = |vF (q − 2kF )| (vF being theFermi velocity) and vanishes outside. This means that the Holstein phonon acquires afinite damping if and only if its renormalized frequency Ωq is in the energy-momentumregion with Im(χ0

q)11 6= 0 (see Eq. (3.3)). This is essentially due to the particular shapeof the particle-hole continuum in 1d which we linearize in the surroundings of q = 2kF

in our approximate calculations.The interacting Im(χq)11 exhibits a non-constant momentum dependence within

the fan, due to the RPA treatment with the residual quasiparticle interaction Wq:Imχq=(Imχ0

q)/[(1 +Reχ0qWq)

2 + (Imχ0qWq)

2].In Fig. 3.21 we show the damped Ωq in 1d. The damping is maximum in the

proximity of the Kohn anomaly, occurring for low values of the U/λ ratio: for small U atfixed moderate coupling (Fig. 3.21a) and for larger e-ph coupling at fixed intermediateU (Fig. 3.21b). However, the phonon broadening takes place for a larger extent of thephonon branch in the more correlated regime (U/λ≫ 1) where γq is less sizable. Thisbehaviour can shed some light on the scattering experiments of strongly correlatedmaterials with well-formed 1d stripe structures, described in § 1.5.

The main goal of this Section has been to illustrate the evolution of the Kohnanomaly in the presence of electronic correlations. Essentially the e-e interactionsweaken the effective e-ph coupling and thus shift the appearance of the phononicanomaly to higher values of the bare coupling. All the results and the argument pro-vided to obtain Eq. (3.2) within the adiabatic scheme for the e-ph interaction representalso another application of the approach shown in § 2.2.1.


3.5 Summary of Chapter 3

In this Chapter we have investigated the effects of strong electronic correlations onthe e-ph coupling; in particular we considered the case of phonons coupled to the localcharge density as described by the Hubbard-Holstein model. In § 2.2.1 we had exploitedthe adiabatic limit of the lattice degrees of freedom to show an exact result relatingthe screening of the e-ph coupling to the purely electronic static charge susceptibility.This result holds generically for any kind of e-e interaction (not only for the Hubbardone) and should also provide valuable information in the partially adiabatic case offinite phonon frequency (ω0 ≪ D). It is important to note that the analysis of thecorrelation-driven screening of the e-ph coupling can be performed by investigatingthe purely electronic problem. The latter was investigated within the static limitof the GA+RPA method. This technique assumes a Fermi-liquid ground state andconsiders the low-energy quasiparticle physics. Therefore our low-energy descriptionof the electronic liquid is appropriate in high dimensions, where the Fermi-liquid is agood starting point. Particularly favourable is the d = ∞ case, where the GA becomesthe exact solution of the GZW variational problem.

The main outcome is that (strong) correlations induce rigidity in the charge densityfluctuations thereby reducing the effective e-ph coupling when this is of the Holsteintype. More specifically the analysis of the momentum dependence shows that the e-phcoupling is more severely reduced in processes with large momentum transfer. Thisresult, which was already known in large-N approaches to the infinite-U Hubbard-Holstein models [18, 142, 186], is considered here within a systematic variation of thecorrelation strength. In particular, from Figs. 3.18 and 3.19 one can see that, while atsmall U ’s the effective e-ph vertices at small and large transferred momenta differ atmost by 40 percent, at large U ’s the e-ph coupling at large momenta can be five or moretimes smaller than the couplings at low momenta. The fact that the e-ph coupling isscreened less for small transferred momenta has important consequences as far as thecharge instabilities of the model are concerned. Indeed, the e-e interactions not onlygenerically reduce the effect of the phonons and enhances the minimum strength ofe-ph coupling to drive the system unstable, but also introduces a momentum depen-dence, which changes the nature of the instability upon increasing the strength of thecorrelation. While for small e-e interaction the leading instability is of the Peierls type,with the formation of CDW at momenta |qc| = 2kF , upon increasing U , the scatteringprocesses at small transferred momentum become comparatively stronger and lead toa PS instability at vanishing qc. Our technique allows a systematic investigation ofhow the CDW instability transforms into the PS instability leading to a phase diagramlike the one shown in Fig. 3.17 for the 2d-system.

Of course the PS instability is specific to the short range nature of the model.When the long range Coulomb interaction is included, the large-scale PS of chargedholes is prevented and a frustrated PS occurs with the formation of various possible

3.5 Summary of Chapter 3 87

textures [19, 21, 23, 24, 25, 26, 193].We also notice that the bare e-ph coupling λ needed to drive the systems unstable

are rather small (of order one or less) at large U and moderate doping (see Fig. 3.17).A good compromise is indeed reached in this region, where the quasiparticles have asubstantially reduced kinetic energy (the effective mass is 3-5 times larger than the bareone), but the system is not too close to the insulating phase, where the interaction wouldscreen too severely the e-ph coupling. Therefore, in this rather metallic regime the(frustrated) PS instability is quite competitive with respect to the polaron formation,which could instead be favoured by the stronger correlation effects occurring in theantiferromagnetic region of the phase diagram [199].

The general interest of the above findings and the encouraging reliability test ofthe GA technique discussed in the present Chapter are a stimulating support for theextension of the present work to the dynamical regime. Future natural extensions willalso consider the analysis of the correlation effects on the phonon dynamics and theirinvestigation in broken-symmetry states, like the stripe phase, as we start to do inChapter 4.


0 0.5 1 1.5 2 2.5 3q

0

0.2

0.4

0.6

0.8

1W

qΩ

0

UUc=0.5

UUc=0.2

UUc=0.12

UUc=0

Figure 3.20: Phonon dispersion at different U/Uc for n = 0.8 and λ = 0.5 in 1d (toppanel) and along the closed path (0, 0) − (π, 0) − (π, π) − (0, 0) in 2d(bottom panel). Inset - the critical e-ph coupling λc as a function ofU/Uc (see Ref. [167]). For λ < λc the system is homogeneous and theKohn anomaly is more evident upon decreasing the ratio U/λ.

3.5 Summary of Chapter 3 89

Figure 3.21: 1d-Phonon dispersion with the quasistatic damping for n = 0.8 andω0/t=0.1 at different U/Uc with λ = 0.5 (panel a) and at different λwith U/Uc = 0.5 (panel b). The curves are drawn as sequences of diskscentered in Ωq and with area proportional to γ2

q. The area bounded bythe equation ω = |vF (q − 2kF )| gives the linearized p-h spectrum closeto q = 2kF : out of this area Imχq = 0 (vF is the Fermi velocity).

Chapter 4

Charge stripes and optical phononsoftening

This Chapter required a considerable effort to generalize the GA+RPA computationsfor inhomogeneous systems. In fact, in Chapter 3 we performed a methodical study fora homogeneous paramagnetic system. Here we present our findings for systems on a su-perlattice, with an inhomogeneous ground state, both with CDW and spin order. Even-tually we are interested to compute the static susceptibility of such broken-symmetrystates. In particular, in this Chapter we concentrate on the electronic response of themagnetic stripe phase. Indeed, we think that a coupling between stripes and phononscould be responsible for the anomalous softening of the bond-stretching phonon branchin LSCO, as described in Section § 1.5. Thus we characterize the obtained susceptibil-ity and show the peak response associated to the Kohn anomaly of the metallic stripes.Then we develop a proper theoretical framework to determine the superlattice phononmodes and the renormalization of the optical spectrum herewith. In this Chapter wegive the results of our study in order to reproduce these experimental phononic fea-tures. In Appendix B we report the full formal derivation to compute the interactingsusceptibility matrix in broken-symmetry states.

4.1 Magnetic charge stripes in two dimensions

The experimental and theoretical background about the existence and the propertiesof the stripes in the cuprates (especially in LSCO) was reviewed in § 1.2.1.

In this Chapter our investigations are based on the one-band Hubbard model withhopping restricted to nearest (tij = −t) and next-nearest (tij = −t′) neighbours (seeEq. (2.1)).

Previous papers have shown that with realistic parameters for LSCO (U/t = 8and t′/t = −0.2) at δ=1/8 the GA+RPA approach leads to vertical metallic stripes

4.1 Magnetic charge stripes in two dimensions 91

in agreement with the experimental data [64, 209]. Here we perform calculations inlarge systems typically consisting of N ∼ 103 lattice sites in order to obtain sufficientmomentum resolution for the computation of charge excitations. For stripes orientedalong the y-direction this is most conveniently done by decomposing the lattice intoNc unit cells of size Na = d × 2 with N = NcNa. The locations of the unit cellsdefine a Bravais “superlattice”. The positions of the Bravais superlattice are given byR ≡ n1a1 + n2a2 with ni integer. For odd d the Bravais lattice is generated by theelementary translations a1 = (d, 0) and a2 = (0, 2), whereas for even d by a1 = (d, 1)and a2 = (0, 2). From the translation vectors we can obtain the generators bi ofthe reciprocal superlattice defined by the relation ai.bj = 2πδij and construct theassociated magnetic Brillouin zone. For odd d the latter is simply a rectangle spannedby the reciprocal superlattice vectors b1 = (2π/d, 0) and b2 = (0, π) whereas for evend the shape is more complex due to the non-orthogonality of the translation vectors(see for example Fig. 4.3).

Figure 4.1: Unit cell used for the d = 4 stripe computations, d being the interstripedistance (as in Ref. [210]).

More specifically, we consider charge stripes at a distance d=4, then with the 4× 2supercell schematized in Fig. 4.1 and we focus on a bond-centered (BC) stripe solution,i.e. with the charge imbalance centered in its bonds (cf. Fig. 4.2). All the results re-ported in this Chapter for interacting systems have been obtained using the parametersmore suitable to reproduce experimental data for LSCO: U/t=8, t′/t=-0.2 and δ=1/8.

At this doping the BC stripe is the stable solution of the one-band Hubbard modelobtained in the GA, while the site-centered (SC) stripe, with the charge imbalancecentered in the sites, has a little higher energy. In the three-band model the situationis interchanged. In general, the difference of their energy is very small in both modelsand they tend to become degenerate for larger dopings. In addiction, BC texturesturn out to be more stable also in first-principle computations [211]. Therefore wedecided to consider BC stripe solutions. Clearly, we decided to adopt the one-band

92 Charge stripes and optical phonon softening

-2 0 2 4 6 8

-4

-2

0

2

4

Figure 4.2: Bond-centered striped phase: charge pattern. The inner rectangle en-closes a unit cell, as given in Fig. 4.1. Each site is marked by a diskwhose radius is proportional to the local charge density ni. In this figurewe show the charge distribution associated to the GA ground state for a16× 100 lattice; the corresponding charges obtained for a 40× 40 latticeare equal within 0.1%. The charge density modulation between two notequivalent sites is rather weak: ∆n ∼ 0.1. These charge densities are ob-tained with the typical electronic parameter set adopted in this Chapter:U/t=8, t′/t=−0.2 and δ=1/8.

Hubbard model just because it provides reliable results, similar to the ones of the three-band model and, due to the simplification of the Hamiltonian, it allows to performcomputations on larger lattices with less effort, thereby obtaining a better resolutionin the momentum analysis and reducing the finite size effects. For a short review ofthe results obtained applyng GA+RPA to the single- and multi-band Hubbard model,one can see for example Ref. [212]. Experimental evidences of bond-centered chargemodulations in the cuprates have been presented in Fig. 1.9 (see also Ref. [66], forexample).

In Fig. 4.3 we show the first magnetic BZ and the extended Brillouin zone (EBZ)for d = 4 stripes. The momenta in the EBZ label the N plane wave states of thesystem. Alternatively one can label the plane wave states by a reduced momentum kin the first MBZ (the central polygon in Fig. 4.3) and a set of Na reciprocal superlatticevectors Q ≡ m1b1 + m2b2 with mi integer, the momentum of the plane wave state


-Π-Π2

0 Π2

Π

qx

-Π

-Π2

0

Π2

Π

q y

-Π-Π2 0

Π2 Π

-Π

-Π2

0

Π2

Π

Figure 4.3: The central exagon is the first magnetic BZ for d = 4 stripes. Thick linesenclose the EBZ. The dots are a set of magnetic reciprocal superlatticevectors that define a set of higher magnetic BZ’s with total volume equalto the EBZ volume (as in Ref. [210]).

being given by Q + k. The dots in the figure indicate a possible choice for this set.Each dot defines also a higher MBZ’s with the dot at the center.

In Fig. 4.4 we show the eight-band electronic dispersion of the stripe solution for a16×100 lattice. In fact, we have performed most RPA computations for the susceptibil-ity on this lattice to have a better momentum resolution in the y-direction, i.e. parallelto the metallic stripes. These results are shown in the following Section § 4.2. However,we underline that we found that the electronic properties and the GA parameters ofthe stripe solution are virtually identical by comparing results for 40×40 and 16×100lattices; also results on small lattices such as the 8 × 8 case are quite close. This hasbeen checked for the GA hopping factors, for the electronic double occupancies, andalso for quantities such as the charge densities and the electronic dispersion.

4.1.1 Realistic parameters and saddle-point solutions

In this Subsection we intend to explain in which sense U/t = 8 and t′/t = −0.2 are therealistic parameters to be used to reproduce the experimental results on lanthanumcuprates. A similar discussion can be also found in Refs. [210] and [213].


H0,0L H0,Π2L H0,ΠLk

-2

-1

0

1

2

3

4

5

Ε k

H0,0L H0,Π2L H0,ΠL

Figure 4.4: Band structure of the bond-centered stripe solution on a 16× 100 latticefor U/t=8, t′/t=−0.2 and δ=1/8: cut along the (0, 1) direction. Bandsare measured in units of t.

For these and similar parameters in a moderately wide range, the lowest-energysolutions of the Hubbard model are metallic stripes. As a possible procedure to fix thepresent values,

• one can fit the magnetic excitations of the undoped compound [214] and then

• require that the doping dependence of the stripe periodicity reproduces the mag-netic incommensurability determined experimentally in the LSCO-LBCO com-pounds [62]. In the one-band model, it turns out that site-centered (SC) andbond-centered (BC) stripes are substantially degenerate for δ=1/8. For our com-putations we restrict to the latter also because for the magnetic spectra they gaveslightly better agreement between theory and experiment [210].


To achieve these goals, one will determine a narrow window of admittable realisticparameters for the next-to-nearest neighbour hopping and for the Coulomb repulsion.In particular, within the present one-band description, we fix parameters practically inthe following way.

• The value of the nearest-neighbour hopping parameter was fixed to t′/t = −0.2according to Refs. [209, 215]. In particular, in Ref. [209] Seibold and Lorenzanafound that the stripe filling is very sensitive to t′/t and it turned out that a ratioof t′/t = −0.2 is necessary to describe half-filled stripes as inferred from theexperimental incommensurability behaviour [62] (see also Fig. 1.7). This value isquite close to t′/t = −0.17, the value obtained for the band structure of La2CuO4

in a LDA study [215].

• The spin-wave dispersion of the undoped system along the magnetic zone bound-ary poses a constraint on the value of U . In fact, the linear spin wave the-ory for the Heisenberg model with only nearest-neighbour exchange interactionsJ = 4t2/U does not yield any dispersion at all. Only higher order in t/U ex-change contributions start to produce a dispersion in the magnetic excitationsalong the magnetic zone boundary which is therefore very sensitive to the valueof U . After computing the magnon spectra for the half-filled system within theGA+RPA and fitting the spin-wave dispersion of the undoped La2CuO4 measuredat T = 10K [214], one sees that the best choice is U/t = 7.5 ÷ 8 which yields anexcellent agreement with the measured dispersion also along the magnetic zoneboundary [165, 210, 216]. For the quite close value of U/t = 10, the fitting in theinsulator is significantly worse [216]. We also mention that a similar estimate forU/t was obtained from QMC calculations of the magnon dispersion of undopedcuprates [217] and in a fit of the doping dependence of spectral weights in opticaldata, within an exact diagonalization approach [218]. Both in Ref. [217] and inRef. [218] the one-band Hubbard model is adopted to fit the experimental data.

Finally we note that although the present parameter set was originally optimizedconsidering especially magnetic properties, it also gives appropriate results for chargeexcitations. For example, within GA+RPA one obtains a gap in the optical excitationsof 1.8 eV which is close to the 2 eV value of reflectivity experiments [219].

The saddle-point solutions were characterized in Ref. [209] and also in the first partof this Section. We consider stripe textures oriented along the y-axis (vertical stripes)where SC and BC solutions have similar energies. Spin are ordered antiferromagneti-cally in the vertical direction whereas the ferromagnetic spin alignment between sites attransverse positions 0 and 1 indicates the presence of a domain wall where also dopedholes accumulate (see Fig. 4.1). In order to compare our results with experimentaldata in La2−δSrδCuO4 and La2−δBaδCuO4, we assume that the chemical doping δ ofthe samples coincides with the effective hole number of the system.


Once we have explained why we consider U/t = 8 and t′/t = −0.2 pre-fixed parame-ters in our study to reproduce the experimental data, it turns out that the only tuning(free) parameter is the e-ph coupling λ. This means that we will try to reproducethe observed branches reported in Section § 1.5, testing several values for λ withoutchanging the pre-determined electronic parameters U and t′. This is what we aim todo in Section § 4.4.

4.2 Charge response of the stripes

In this Section and in Section § 4.4 we present results for the correlated system fordoping δ = 1/8, hopping t′ = −0.2 and Coloumb repulsion U = 8.

In particular, here we report examples of the charge susceptibilities we have com-puted for the magnetic-stripe state. Such computations for inhomogeneous systemshave been performed for quite different purposes with respect to the correspondingones done for Chapter 3. In fact, for the homogeneous case we investigate methodi-cally the behaviour of the susceptibility to acquire the phase diagram of the system invarious dimensions. As an extra result, we used the computed charge susceptibility tostudy how the phonon spectrum softens with strong correlations.

In this Chapter we do not want to illustrate how the trends of the susceptibility ofbroken-symmetry states change upon varying the model parameters. Such a study ofthe charge excitations could be an interesting issue for the future, but here we computethe charge response of the stripe system specifically because we want to illustrate theexperimental findings presented in § 1.5. This is why we concentrate only on a specificset of realistic parameters suitable for LSCO/LBCO and ignore other choices.

Essentially, as already discussed, we are convinced that the strong anisotropy shownin the softening of the optical phonon branch is due to a sizable coupling betweenphonons and electronic stripe modulations. We can illustrate this idea with this rea-soning. Typically, in the absence of a relevant coupling between phonons and otherelectronic excitations of the system, the optical spectrum is isotropically smooth (andnearly flat) over all the Brillouin zone. In the cuprates (LSCO/LBCO and YBCO,at least) a strong deviation from this behaviour can be attributed to the presence ofa strong coupling between the lattice and the electronic modes: these materials havein fact highly anisotropic band-structures. In particular, in LSCO-materials the wellpronounced dip at half-Brillouin zone suggests that stripes can be the right electroniccandidates. In fact, they present half-filled one-dimensional metal structures and theircharge response has a peak just in correspondence with that particular wave vector.In this picture, the stripes effectively reduce the dimensionality of the system, so thatthe anomalous phonon softening can be seen as a manifestation of a Kohn anomalyparallel to their direction. To check the validity of these considerations, we first needto compute the susceptibility. Then we will use it to obtain the normal modes and the

4.2 Charge response of the stripes 97

renormalized phonon branches of the superlattice system, using the formalism devel-oped in Section § 4.3.

HΧqQQ',++0 LΡΡ

0.00.2

0.40.6

0.8

qx

0.0

0.5

1.0

1.5

qy0

2

4

H0,0L H0,ΠLq

0

1

2

3

4

5

HΧ

qQQ

',++

0L ΡΡ

H0,0L H0,ΠL

Figure 4.5: Non-interacting charge susceptibility on a 16×100 lattice: 3d-plot in theRBZ-1st quadrant (left) and cut along the (0, 0)-(0, π) direction (right).Notice that stripes are oriented along the y-direction. Here and in Fig. 4.6we have reported diagonal elements χqQQ, using Q=Q′=(0, 0) for the firstRBZ and Q=Q′=(π/4, π) for the upper-adjacent RBZ.

In Figs. 4.5 and 4.6 we report the behaviour of the susceptibility. In the top panelswe report 3d-plots on the first quadrant of the first RBZ: the complete one can beobtained simply through a four-fold symmetrization around q = (0, 0) (the Γ point).In the bottom panels we present the dispersion along the (0, 1) direction, with theresponse peak at (0, π/2). Clearly, were the stripes horizontal, one would have obtainedexactly the same plots under a 90o-rotation and so for example exchanging qx and qyone will have exactly the same cuts of the bottom panels but along the (1, 0) direction.Notice that in LSCO-materials the dispersion along the (1, 0) and (0, 1) directionson the CuO2 planes are totally equivalent. This is because in real materials there arenanoscopic patches where stripes are “horizontal” and others where they are “vertical”:this is the twinning.

In Fig. 4.5 we show the sharp peak of the non-interacting charge susceptibility atq = (0, π/2) for vertical stripes. As it is evident from the cut along the direction(0, 0) − (0, π), the peak is essentially symmetrical in that direction and falls quiteabruptly away from q = (0, π/2) on either side.

In Fig. 4.6 we present how the response is modified for the interacting charge suscep-tibility. The peak at q = (0, π/2) is still present, although weakened by the Coulombrepulsion. This is what one can expect after the studies presented in Chapter 3. Inter-


HΧqQQ',++LΡΡ

0.00.2

0.40.6

0.8

qx

0.0

0.5

1.0

1.5

qy

0.30.40.50.60.7

H0,0L H0,ΠLq

0.3

0.4

0.5

0.6

0.7

HΧ

qQQ

',++L ΡΡ

H0,0L H0,ΠL

Figure 4.6: Interacting charge susceptibility on a 16×100 lattice for U/t=8, t′/t=−0.2and δ=1/8: 3d-plot in the RBZ-1st quadrant (top) and cut along the(0, 0)-(0, π) direction (bottom).

estingly, also its form is substantially changed by electronic correlations: it is stronglyasymmetrical. In particular, the susceptibility (χqQQ′,++)ρρ raises quite smoothly fromΓ to the peak, then it has a sudden fall before becoming nearly flat until the BZ-boundary. For our purposes, this is an encouraging behaviour: in fact, the observedphonon branches present a strongly asymmetrical dip.

4.3 Normal modes of a crystal with a superlattice 99

4.3 Normal modes of a crystal with a superlattice

We consider a multiband system with e-e interactions and e-ph coupling on a lattice:

Htot = He +Heph +Hph (4.1)

More specifically, electrons are placed on a superlattice with a Na-atom unit cell andare coupled to the lattice ions through a Holstein interaction. Then the total energyhas this form in real space:

Etot = Ee + Eeph + Eph = Ee0 +1

2

∑

ijαβ

(∂2Ee

∂niα∂njβ

)

0

δniαδnjβ

+1

2

∑

iα

(P 2

iα

M+Kx2

iα

)+∑

iα

βxiαδniα (4.2)

where Ee0 is the electronic energy of the ground state, δniα = niα − n is the electronicdensity deviation and n is the average density of the system. Piα, xiα and M arethe momentum, the displacement and the mass of the lattice ions1 respectively. Weremember that i and j are cell indexes; α and β label atoms in the unit cell (see

Section § 2.6). To shorten the notation, we will replace

(∂2Ee

∂niα∂njβ

)

0

with e′′ijαβ which

is symmetrical under the exchange α ↔ β. The latter is the inverse of a generalizedcharge susceptibility as shown below.

In this Section we intend to develop the formalism to obtain the normal modes ofthe superlattice ions. We will not use any further assumptions on the nature of theelectronic interactions.

In the adiabatic limit for the e-ph coupling, Eq. (4.2) can be cast as follows:

Etot = Ee[δn] + Eeph[δn, x] + Eph[x]

= Ee0 +1

2

∑

ijαβ

e′′ijαβδniαδnjβ +∑

iα

βxiαδniα +1

2

∑

iα

Kx2iα (4.3)

Then we Fourier transform Eq. (4.3) to formulate the energy in momentum space,defining the Fourier transform in the following way:

δniα =1

Nc

∑

q

eiq·(Ri+δα)δnqα

xiα =1

Nc

∑

q

eiq·(Ri+δα)xqα

1In our treatment we consider the ion masses equal: Mα=Mβ=M .


This Fourier transform is partial, because it is done only on supercells indicated by iand j and not also within the unit cells (see Appendix B for further details).

Thus the form of the energy in momentum space is

Etot = Ee0 +1

2Nc

∑

qαβ

e′′qαβδnqαδn−qβ

+1

Nc

∑

qα

βx−qαδnqα +1

2Nc

∑

qα

Kxqαx−qα (4.4)

with

e′′qαβ =∑

j

e′′ojαβeiq·(Ro−Rj)eiq·(δα−δβ) (4.5)

Notice that Eq. (4.4) gives the exact energy of the system in the adiabatic limit for thee-ph interaction.

One indeed can identify e′′qαβ with (κq)−1αβ , where κqαβ is the charge susceptibility

matrix. This identification is straightforward in the homogeneous case, when both thesequantities are scalar. In general, one can prove the statement within the generalizedlinear response theory, starting from the equation:

δnqα = −∑

β

κqαβfqβ (4.6)

where f is an external field acting on the electronic charge sector.We can write the electronic energy of the system in the presence of this perturbation:

Eef = Ee + Ef = Ee0 +1

2Nc

∑

qαβ

e′′qαβδnqαδn−qβ +1

Nc

∑

qβ

fqβδn−qβ (4.7)

Minimizing Eq. (4.7) with respect to the deviations δn, one finds:

δn−qβ = −∑

α

f−qα((e′′q)−1)αβ

δnqα = −∑

β

((e′′q)−1)αβfqβ (4.8)

By comparing Eq. (4.8) with Eq. (4.6), we conclude that

((e′′q)−1)αβ ≡ κqαβ (4.9)

4.3 Normal modes of a crystal with a superlattice 101

Now we minimize Eq. (4.4) with respect to δn: in this way we obtain the chargedensity for an arbitrary lattice configuration.

δnqα,+ = −∑

β

βκqαβxqβ

δn−qβ,+ = −∑

α

βx−qακqαβ (4.10)

Replacing Eq. (4.10) into Eq. (4.4), one finds:

Etot[x] = Ee0 +1

2Nc

∑

qαβ

Kqαβx−qαxqβ (4.11)

where the generalized spring constant matrix Kqαβ is

Kqαβ = Kδαβ − β2κqαβ (4.12)

Up to Eq. (4.12) we exploited the adiabatic limit for the e-ph interaction. Now weare interested to study the phononic dynamics, namely to determine the phononicbranches. Applying the second Hamilton equation to Eq. (4.11) yields:

∂Etot

δxqβ

=1

Nc

∑

α

x−qαKqαβ = −Mx−qβ

∂Etot

δx−qα

=1

Nc

∑

β

Kqαβxqβ = −Mxqα

A harmonic ionic displacement x can be written as xqα(t) = xqα(0)eiΩqt, Ωq being thefrequency of the vibration. Now we can present this eigenvalue problem:

1

Nc

∑

β

Kqαβxqβ = −Mxqα = MΩ2qxqα

∑

β

[Kqαβ −MΩ2

qδαβ

]xqβ = 0 (4.13)

Among the possible square frequencies Ω2q and the modes xq of the lattice vibrations,

one can find respectively the eigenvalues (Ω2)nq and the eigenvectors xn

q, solving theproblem given in Eq. (4.13), whose kernel operator is real and symmetrical.

Finally from Eq. (4.13) one recognizes that our generalized elastic tensor Kqαβ isalso the dynamical matrix of the eigenvalue problem which gives the normal modes ofthe superlattice vibrations. The square roots of the eigenvalues gives the Na renormal-ized branches Ωn

q of the phonon spectrum, with q ∈ RBZ. Then one can reformulate


Eq. (4.13) in order to work with the normalized modes eqα =√Mxqα:

∑

β

[Kqαβ

M− Ω2

qδαβ

]eqβ = 0

Actually we shall find eigenvalues and eigenvectors of the matrix Kqαβ/M . We re-member that β2 = (λK)/χ0

0, K = Mω20 and that χ0

0 is the DOS of the non-interactingelectron system. ω0 is the phonon frequency in the absence of coupling to the elec-trons: the non-interacting phonon spectrum is given by Na degenerate frequencies ofthe uncoupled normal modes of vibrations. Using Eq. (4.12) and the definitions of theconstants β and K, we can present Eq. (4.13) in the following form (cf. Eq. (3.5) forelectrons in a homogeneous state):

Kqαβ

M=

K

Mδαβ − β2

Mκqαβ

= ω20

[δαβ − λ

χ00

κqαβ

](4.14)

Then the Na normal phononic modes are determined within the RBZ. For a givenreduced wave vector q, they are characterized by their polarization vector en

q and bythe associated frequencies Ωn

q. Both the eigenvectors and the frequencies are periodicalfunctions of the momentum in the EBZ: en

q = enq+Q and Ωn

q = Ωnq+Q. The set of all the

branches constitutes the phonon dispersion.

In this Section we use the charge susceptibility matrix κqαβ and the local den-sity variations δniα. When we effectively perform our computations, we will use theGA+RPA method and for example we will identify the onsite charge deviations δniα

with the local charge deviations δρiiα,+2 and

κqαβ ≡ (χqαβ,++)ρρ (4.15)

Thus at the end we will compute the Na ×Na matrix (χqαβ++)ρρ.

We remark, however, that all the results obtained in this Section are exact withinthe adiabatic limit for the e-ph interaction, and then can be used both for other formsof e-e interactions and with other methods to compute the charge susceptibility.

2Indeed, we remember that in the GA the local density δniασ coincides with δρiiασ (see Eq. (2.29)).

4.4 Softening of the optical phononic branches 103

4.4 Softening of the optical phononic branches

In this Section we present our results for the phononic branches which we computedusing the formalism for the superlattice normal modes developed in Section § 4.3 (seeEqs. (4.13) and (4.14)). We also evaluated the intensities Sn

qQ of the phonon branchesfor neutron scattering experiments, as described in Section § 1.4, in particular usingEq. (1.2).

On the whole, Eqs. (4.14) and (1.2) allow to study the phonon softening in thestatic approximation for the susceptibility, so that the imaginary part of the latter iszero and the phonon damping is absent.

In the (e-ph) coupled case there areNa renormalized optical branches emerging fromthe Na degenerate original uncoupled branches ω0. For λ = 0, one of the degeneratebranches has relative intensity Sn

qQ/SqQ = 1 and all the other branches have no weight.[For a given momentum, the total one-phonon intensity is SqQ =

∑n S

nqQ.] Upon

activating the e-ph coupling, for small λ in principle one can identify a main branch withSn

qQ/SqQ∼ 1 and Na − 1 ”shadow” branches with SnqQ/SqQ≪ 1, or almost vanishing

intensities. For strong λ, the redistribution of the branch intensities with respect tothe uncoupled case is not easily predictable, in general.

In this Section we want to show that, once we have evaluated the contributionof each phonon mode to the total intensity, the set of our (appropriately weighted)branches can reproduce the observed phonon dispersion. In particular, in our investi-gation we will focus on the branch cut in the direction parallel to the stripes.

In Figs. 4.7-4.9 we report our theoretical dispersions along the (0, 1) direction. Wewill comment these calculated phonon branches and, for a straightforward comparisonto the experimental ones, in Fig. 4.10 we give the corresponding branch cut as obtainedin Ref. [4] (top panel) and in Ref. [118] (bottom panel). Finally we will conclude thisSection by discussing the other two recent theoretical works concerning the softeningof the bond-stretching phonon: the approach of Mukhin et al. [123] and the study ofCitro et al. [124], already mentioned in Section § 1.5.

In Figs. 4.7-4.9 the branch points are represented by ellipsoidal disks with their areaproportional to (Sn

qQ)2. Given that the relative intensity of the main branch points isorder ∼ 1 and the one of the shadow branch points can be order ∼ 10−3÷10−4, not allthese last points can be fully visible together with the first ones in the used scale. In ourdispersion plots we consider the adimensional ratio Ωq/ω0 to present directly the sizeof the phononic renormalization with respect to the uncoupled branch. This is what wehave already done in Section § 3.4, but in this case we want to compare our brancheswith particular experimental data and thus we need to fix the overall energy scale. Wecan consider t = 342 meV as in Ref. [210] and ω0 ∼ 0.25 t, so that we will obtain ω0

around 90 meV . This is the phonon frequency observed in the insulating compound:


in fact, ω0 can be obtained from undoped compounds, where the charge degrees offreedom are frozen and the contribution of the second term in Eq. (4.14) is negligible.Notice once more that in these experiments ω0 fully satisfies the adiabatic constraintto be considerably small with respect to the electronic bandwidth, confirming that weappropriately use the adiabatic limit for the e-ph interaction to study this experimentalproblem.

We make some comments about the dispersions we obtain upon increasing λ. Infact, we can identify important features common to all these spectra:

• the dip of the softening around q = (0, π/2);

• the strong asymmetry of the phonon softening: the phonon branch is faintlyrenormalized for small momenta, then has a steep descent and for large momentathe branch continues to be relevantly softened;

• the overall trend (and the two features mentioned above) are comprehensiblehaving in mind the behaviour of the charge susceptibility as shown in Fig. 4.6.

These general considerations are compatible with the experimental branches givenin Fig. 4.10. We perform this investigation varying λ to verify if in our approach we canobtain a dispersion in close accord to the observed one. In particular, we will determineat which λ one will obtain a softening resembling the experimental one whose entity isaround 20% of the uncoupled branch of the insulating compound.

In Fig. 4.7 we show the dispersion for low λ. The decrease of the isotropy isquite weak and a tendentially cosinusoidal trend can be seen close to the minimum(cf. Ref. [116], for example). The branches still have close energy and the entity oftheir softening is around few percent. It is evident that the dispersion for small e-phcoupling reproduces only qualitatively (and rather poorly) the trends of the spectra,but for the position of the shallow dip at q = (0, π/2).

In Figs. 4.8 and 4.9 we report our results for intermediate-strong e-ph couplingλ = 1 ÷ 2. These are the values of the coupling which allow us to find a better (andgenerically good) agreement with the experimental results. We find a strong asymmetryin the softening, with a wider separation of the branches. This is a nontrivial result,because the experimental evidence is well reproduced for quite large e-ph coupling.

Therefore we need a substantial e-ph coupling to recreate a well pronounced phononsoftening. This is due to the strong suppression of the effective electron-lattice couplingin correlated systems, and the weak coupling predicted by LDA calculations is notsufficient to fit the experimental data (see for example Refs. [96, 97]). Looking at theIXS (Inelastic X-Ray Scattering) measured spectrum, we can conclude that maybe λ =1.5 will best fit the experimental softening and the shape of the dip, also quantitatively.Larger e-ph couplings exceed the strength we need to reproduce quantitatively theanomalous phonon softening.


Thus we can reproduce the experimental anisotropic structure of the phononicbranches using the Hubbard-Holstein model. Hereby we would like to make an overallconsideration. Looking at Eq. (4.14), which indicates how to evaluate the frequencyrenormalization, one can see that the momentum dependence is entirely included inthe electronic charge susceptibility. In fact, in our theoretical framework the bareoptical phonon frequency ω0 is dispersionless and the e-ph coupling λ is constant uponvarying the wave vectors. Since we have found a satisfactory agreement with theexperimental asymmetrical phonon branches, as a by-product of our study we realizethat the momentum dependence in the charge degrees of freedom is the main ingredientto explain the observed phonon softening. The inclusion of a momentum-dependent e-ph coupling could just complicate our minimal working approach without too significantimprovements in the reproduction of the experimental evidence.

Let us comment briefly the other two theoretical works dedicated to this problem,though with rather different goals with respect to our approach.

Mukhin and coworkers aim to reproduce the qualitative features of the anomaloussoftening with the simplest approach. They consider an array of 1d-Luttinger liquidsembedded in 2d-optical phonons and moreover use a non-interacting charge suscepti-bility. Within such simplified picture, they succeed to determine the position of thepeak and the entity of the phonon linewidth in the peak surroundings. However theyconsider only a very narrow interval around q = 2kF of the stripes, without reproduc-ing the observed asymmetry of the dip and without trying to describe quantitativelythe overall trend of the dispersion, as we do in this Section.

Citro et al. perform a more complete study using a model with long-range Coulombinteractions but essentially they are interested to the doping dependence of the soft-ening, without referring to a specific experimental evidence. They affirm that, using astatic approximation for the charge susceptibility, they can capture the main featuresof the phononic softening but they do not achieve a satisfactory quantitative compari-son with the recent experimental observations. Including dynamical corrections to theform of the susceptibility, they reproduce the evolution of the softening upon varyingdoping in good quantitative accord with experiments. This represents a useful acquisi-tion on the role of the dynamical corrections beyond the static evaluation of the chargedegrees of freedom. As we have shown, using a fully static form of the susceptibility, wecan give account for the observed dispersion but clearly the inclusion of the dynamicaleffects will be helpful, especially because it will allow to describe the phonon damping.

In this Chapter we have studied the charge response of magnetic stripes. Then wehave showed that we can reproduce the experimental phononic dispersion fairly well,considering a strong coupling between phonons and charge degrees of freedom asso-ciated to the stripes. We have found a quite good quantitative agreement with the


experimental results, and a satisfactory qualitative one. In particular, the results pre-sented in Section § 4.4 show the level of accuracy we can obtain within a static approachand using the GA+RPA method to deal with this specific problem in the adiabaticlimit for the e-ph interaction: the last part of our Thesis is essentially dedicated toreproduce the anomalous phonon spectra observed in LSCO and LBCO.

For the future there are several natural extensions of our present work. They willregard the inclusion of the dynamical effects into the susceptibility and thereby thecomputation of the phononic damping. The evolution of the phonon softening upondoping is also an interesting issue for further investigations.

And more in general we could perform a study of the charge excitations uponvarying the interstripe distance, for example, in order to reproduce physical effects forunderdoped materials, with doping δ < 1/8.


H0,0L H0,Π2L H0,ΠLq

0.6

0.7

0.8

0.9

1

WqΩ

0



0.6

0.7

0.8

0.9

1

WqΩ

0


Figure 4.7: Phonon branches for λ = 0.2 (top) and λ = 0.5 (bottom) on a 16 × 100lattice. Results are for for U/t=8, t′/t=−0.2 and δ=1/8.



0.6

0.7

0.8

0.9

1

WqΩ

0



0.6

0.7

0.8

0.9

1

WqΩ

0


Figure 4.8: Phonon branches for λ = 1 (top) and λ = 1.5 (bottom) on a 16 × 100lattice. Results are for for U/t=8, t′/t=−0.2 and δ=1/8.



0.6

0.7

0.8

0.9

1

WqΩ

0


Figure 4.9: Phonon branches for λ = 2 on a 16 × 100 lattice. Results are for forU/t=8, t′/t=−0.2 and δ=1/8.


Figure 4.10: Top panel - This is the phononic dispersion for LBCO at δ = 1/8,already displayed in Fig. 1.15. Bottom panel - Phononic dispersion forLBCO at δ = 0.14 taken from Ref. [118]. Squares and circles representthe frequencies as obtained with two different model fits, with one andtwo modes respectively; the solid line correspond to shell-model latticedynamics calculations for the sixth longitudinal optical phonon mode,with the branch minimum found at the BZ boundary.

Conclusions

In this Thesis we have showed how the e-ph coupling is renormalized by e-e interac-tions in strongly correlated systems. We have studied the charge instabilities of thehomogeneous (paramagnetic) state and the charge response of magnetic stripes. Inboth cases we have also studied the phonon softening in the presence of a coupling tothe electrons.

In the adiabatic limit for the e-ph interaction, we have showed an exact relationbetween the charge susceptibility in the presence and in the absence of phonons, whichhas been one of the basis of our study and has allowed us to determine the chargeresponse of the Hubbard-Holstein model studying the response of the Hubbard model.

In order to compute the electronic response, we have applied the GA+RPA to theHubbard model, both for homogeneous and for inhomogeneous systems. In Chapter 2we presented the general theoretical framework of our approach, nearly all our com-putations for the homogeneous case (the other steps are given in Appendix A) andonly very selected steps of the elaborated computations for the inhomogeneous case,reporting the whole body in Appendix B.

The homogeneous case reduces essentially to a Fermi liquid approach. In Chapter 3we have tested its reliability and its drawbacks, in particular in the one-dimensionalcase where the exact solution of the Hubbard model is available (see Section § 3.2.1).

Then we have also explored the behaviour of the system in two and infinite di-mensions, with particular regard to the two-dimensional case because in closer contactto layered materials such as the cuprates; clearly we could exploit the fact that ourapproach works fairly well in two dimensions.

We have performed a systematic investigation of the charge instabilities of the ho-mogeneous system and we have found that under a sufficiently strong electron-phononcoupling it undergoes a transition towards a state with charge inhomogeneities: PeierlsCDW for small-intermediate strength of the Coulomb interaction and phase separationfor strong electronic correlations. As a by-product of this study, we have showed howthe Kohn anomalies are modified by strong correlations in one and two dimensions. Inone dimension we have also performed an approximate quasistatic expansion for thephonon damping. For a longer discussion of the results presented in Chapter 3, we


remind to the reader that we have already summarized our findings in Section § 3.5.

In Chapter 4 we have considered charge stripes within our GA+RPA approach.We have evaluated their charge response and showed the typical susceptibility peakassociated to their characteristic one-dimensional substructures. Then we have com-puted the renormalized optical spectrum of phonons coupled to electrons in the stripedphase. In this way we could reproduce quite satisfactorily the anomalous softening ofthe experimental dispersion; in particular, we have obtained a similar asymmetricaldip using a sizable electron-phonon coupling. For future work we could be interestedalso to include dynamical effects into the charge degrees of freedom. Indeed, we needto introduce the dynamical structure into the charge susceptibility to reproduce theexperimental phonon linewidths: the phonon damping is absent considering the staticform for the susceptibility. Clearly, working with dynamical charge stripes, a betteraccord with experimental results could be obtained; for this purpose, one could alsoextend the Hubbard interaction adding a longer range potential term in the model.

Now we would like to discuss possible extensions of our studies presented in Chap-ters 3 and 4. We want to make some extra considerations in order to clarify how ourresults are to be placed in the field of strongly correlated systems; in the same time wewill also determine possible routes for future work.

We want to comment Johannes and Mazin’s paper in Ref. [34], already mentionedin a footnote in Chapter 1, considering our results of Chapter 3. They contest that aPeierls mechanism properly takes place in many real classical CDW-materials, becausethe observed charge response peak is not always found at the wave vector predicted bythe peak position in the non-interacting susceptibility and expected from the nestingof the Fermi surface. They remark that the Peierls picture is quite naive for a mate-rial with a considerable electron-lattice coupling because it focuses essentially on theelectronic instabilities. In their opinion, one should give a more accurate account forthe structural phase transition of solids too. We think that another explanation forthe experimental results can be reasonably provided. According to our findings, themomentum dependence of the non-interacting susceptibility is qualitatively preservedfor small Coulomb repulsion and substantially modified for larger electronic correla-tions. In this Thesis we have showed that the typical wave vector of the charge-orderedstate can migrate from the nesting vector to q = 0 (phase separation) for sizable cor-relations. This is a general trend which can be found both for quasi one-dimensionalmaterials and quasi two-dimensional ones. If we add long-range Coloumb interactions,the phase separation could be removed and a new instability of the homogeneous statewill take place at a finite momentum other than the expected Peierls one (see forexample Refs. [18, 19, 208]). It is also conceivable that more realistic forms of theelectronic interactions (also at short distances) can change the position of the dom-inant instability. See for example the susceptibility curves displayed in Fig. 3.11, in


particular the one in solid line: one can conjecture that the peak at small momentacan become the maximum one instead of the Peierls peak or the peak at q = 0, uponincreasing a non-local e-e interaction. In principle, this behaviour could be explainedby our Dyson’s equation, given in Eq. (2.99) for any band-fillings. This mechanismpresumably happens in real materials and answers Mazin’s objections, that the Peierlsinstability has not true analogs in the mechanisms of real materials.

The inclusion of long-range Coulombic interactions will also allow us for a richerdescription of the physics of charge fluctuations as done in Ref. [124]. For example,we could give a more reliable representation of the phonon softening. And more ingeneral it is related to the possibility to represent correctly the physics for small wavevectors, to study properties associated to low-energy and low-momentum electronicmodes, such as zero sound excitations, plasmons ...

One may also wonder what will happen performing all the studies we have presentedin this Thesis for the case of transitive electron-phonon coupling, i.e. for Su-Schrieffer-Heeger interactions between two neighbour atoms. This study has already started inRef. [220] which is dedicated to one-dimensional systems but surely more attention willbe attributed to two-dimensional ones.

In the future we could also be interested to reproduce other evidences of the anoma-lous phonon softening. In fact, this is quite a recurring anomaly in “unconventional”materials: in our Thesis we have considered LSCO and LBCO systems with particularcare, but anomalies emerge also in YBCO and in graphene [221].

More in general, we could also consider charge excitations for various examples ofcharge orderings, always with the goal to be in touch with experimental results forthese other materials.

We have reported our results and suggested for their possible extensions.However, for the future we also plan to consider other new effects to be discov-

ered in strongly correlated oxydes, in novel superconductors such as the iron-basedoxypnictides, and in graphene.

Appendix B - The GA+RPAapproach to broken-symmetrystates

Preliminary notes

Here we provide the full formal derivation corresponding to Section § 2.6, presenting theGA+RPA computations performed in our work to obtain the results given in Chapter 4.

We develop a general formalism to study RPA fluctuations on top of a GA inho-mogeneous ground state, i.e. with charge and spin superstructures. In particular, weare interested at d× 2-cell magnetic charge stripes, d being the interstripe distance.

We consider a two-dimensional Bravais superlattice whose positions are given byR = n1a1 + n2a2, with ni integers. In reciprocal space we can associate a magneticreduced Brillouin zone (RBZ) to the superlattice in real space. The RBZ is spanned byvectors of the form K = h1b1 + h2b2, with hi integers and which fulfils the conditionai · bj = 2πδij.

Real Space - We work in the basis of the electronic operators ciασ. In particularwe consider scattering processes from a state |i, α〉 to another state |j, β〉, being i, j cellindexes and α, β intracell indexes. Then, given a N -site lattice with Nc cells of sizeNa = d × 2, one deals with i, j = 1, ..., Nc and α, β = 1, ..., 2d. Each site is univocallydetermined by the position riα = Ri + δα. Considering the hopping parameter tijαβ,if i 6= j the hopping involves two different cells, else it is an intracell hopping; α = βdenotes equivalent sites.

Momentum Space - Notice that in the scattering between an electron with mo-mentum k and another one with momentum k′ (both momenta belonging to the RBZ),clearly also the transferred momentum q = k′ − k belongs to the RBZ.


Energy expansion - The interaction kernel Wq

Real space energy expansion

We expand the GA energy in real space with a second order expansion in the density ρand in the double occupancy D. The quantities with the label “0” are to be evaluatedfor the saddle-point solution, i.e. a d× 2-cell stripe.

Ee =∑

ijαβσ

tijαβziασzjβσ〈c†iασcjβσ〉 +∑

iα

UDiα = Ee0 + δE(1)e + δE(2)

e (4.16)

The single particle density matrix is ρjiβασ = 〈c†iασcjβσ〉0 ≡ 〈Sd|c†iασcjβσ|Sd〉, where Sd

is a Slater determinant. We will determine the form of δE(2)e :

δE(2)e =

∑

ijαβσ

tijαβδ〈c†iασcjβσ〉 [ziασ

∂zjβσ

∂ρjjβσ

δρjjβσ + ziασ

∂zjβσ

∂ρjjβ−σ

δρjjβ−σ +

+∂ziασ

∂ρiiασ

δρiiασzjβσ +∂ziασ

∂ρiiα−σ

δρiiα−σzjβσ]

+ 〈c†iασcjβσ〉0[∂ziασ

∂ρiiασ

∂zjβσ

∂ρjjβσ

δρiiασδρjjβσ +∂ziασ

∂ρiiασ

∂zjβσ

∂ρjjβ−σ

δρiiασδρjjβ−σ +

+∂ziασ

∂ρiiα−σ

∂zjβσ

∂ρjjβσ

δρiiα−σδρjjβσ +∂ziασ

∂ρiiα−σ

∂zjβσ

∂ρjjβ−σ

δρiiα−σδρjjβ−σ]

+ 〈c†iασcjβσ〉0 [(∂ziασ

∂ρiiασ

δρiiασ +∂ziασ

∂ρiiα−σ

δρiiα−σ)∂zjβσ

∂Djβ

δDjβ +

+∂ziασ

∂Diα

δDiα(∂zjβσ

∂ρjjβσ

δρjjβσ +∂zjβσ

∂ρjjβ−σ

δρjjβ−σ)]

+ δ〈c†iασcjβσ〉 [ziασ

∂zjβσ

∂Djβ

δDjβ + zjβσ

∂ziασ

∂Diα

δDiα]

+ 〈c†iασcjβσ〉0[∂ziασ

∂Diα

∂zjβσ

∂Djβ

δDiαδDjβ]

+1

2〈c†iασcjβσ〉0[ziασ

∂2zjβσ

∂ρ2jjβσ

(δρjjβσ)2 + ziασ

∂2zjβσ

∂ρ2jjβ−σ

(δρjjβ−σ)2 + 2ziασ

∂2zjβσ

∂ρjjβσ∂ρjjβ−σ

δρjjβσδρjjβ−σ

+ zjβσ

∂2ziασ

∂ρ2iiασ

(δρiiασ)2 + zjβσ

∂2ziασ

∂ρ2iiα−σ

(δρiiα−σ)2 + 2zjβσ

∂2ziασ

∂ρiiασ∂ρiiα−σ

δρiiασδρiiα−σ]

+ 〈c†iασcjβσ〉0[ziασ

∂2zjβσ

∂ρjjβσ∂Djβ

δρjjβσδDjβ + ziασ

∂2zjβσ

∂ρjjβ−σ∂Djβ

δρjjβ−σδDjβ +

+ zjβσ

∂2ziασ

∂ρiiασ∂Diα

δρiiασδDiα + zjβσ

∂2ziασ

∂ρiiα−σ∂Diα

δρiiα−σδDiα]

+1

2〈c†iασcjβσ〉0[ziασ

∂2zjβσ

∂D2jβ

(δDjβ)2 + zjβσ

∂2ziασ

∂D2iα

(δDiα)2] (4.17)


We can cast Eq. (4.17) in a more compact form, using this symbolic notation forthe GA hopping factors:

zασ = ziασ ; zβσ = zjβσ

z′

ασσ′ =∂ziασ

∂ρiiασ′

, z′

βσσ′ =∂zjβσ

∂ρjjβσ′

; z′

ασD =∂ziασ

∂Diα

, z′

βσD =∂zjβσ

∂Djβ

z′′

ασσ′σ′′ =∂2ziασ

∂ρiiασ′∂ρiiασ′′

; z′′

βσσ′σ′′ =∂2zjβσ

∂ρjjβσ′∂ρjjβσ′′

z′′

ασσ′D =∂2ziασ

∂ρiiασ′∂Diα

, z′′

βσσ′D =∂2zjβσ

∂ρjjβσ′∂Djβ

; z′′

ασD =∂2ziασ

∂D2iα

, z′′

βσD =∂2zjβσ

∂D2jβ

All the former quantities are evaluated on the saddle-point state.

In order to manipulate the form of Eq. (4.17), we also introduce the quantities δTiασ

and T 0ijαβσ which are

δTiασ =∑

jβ

tijαβzβσ(δ〈c†iασcjβσ〉 + δ〈c†jβσciασ〉)

T 0ijαβσ = tijαβ(〈c†iασcjβσ〉0 + 〈c†jβσciασ〉0)


Then Eq. (4.17) will take the following form:

δE(2)e =

∑

iαβσσ′

Xiiαβσσ′δρiiασ′δTiασ (A)

+∑

ijαβσ′σ′′

1

2V 1

ijαβσ′σ′′δρiiασ′δρjjβσ′′ (B)

+∑

iασ′σ′′

1

2Yiiαασ′σ′′δρiiασ′δρiiασ′′ (B2)

+∑

ijαβσ′

L1ijαβσ′δρiiασ′δDjβ (C)

+∑

iασ′

Miiαασ′δρiiασ′δDiα (C2)

+∑

iαβσ

ZiiαβσδDiαδTiασ (D)

+∑

ijαβ

1

2U1

ijαβδDiαδDjβ (E)

+∑

iα

1

2Kiiαα(δDiα)2 (E2) (4.18)

with

V 1ijαβσ′σ′′ =

∑

σ

T 0ijαβσz

′ασσ′z′βσσ′′

Yiiαασ′σ′′ =∑

jβ

∑

σ

T 0ijαβσzβσz

′′ασσ′σ′′

L1ijαβσ′ =

∑

σ

T 0ijαβσz

′βσDz

′ασσ′

Miiαασ′ =∑

jβ

∑

σ

T 0ijαβσzβσz

′′ασσ′D

U1ijαβ =

∑

σ

T 0ijαβσz

′ασDz

′βσD

Kiiαα =∑

jβ

∑

σ

T 0ijαβσzβσz

′′ασD

Xiiαβσσ′ = z′ασσ′δαβ

Ziiαβσ = z′ασDδαβ


Compact form for δE(2)e in Real Space:

δE(2)e =

∑

iαβσσ′

Xiiαβσσ′δρiiασ′δTiασ (A)

+∑

ijαβσ′σ′′

1

2Vijαβσ′σ′′δρiiασ′δρjjβσ′′ (BB)

+∑

ijαβσ′

Lijαβσ′δρiiασ′δDjβ (CC)

+∑

iαβσ


+∑

ijαβ

1

2UijαβδDiαδDjβ (EE)

We rename the spin labels as dummy indexes:

δE(2)e =

∑

iαβσσ′

Xiiαβσσ′δρiiασδTiασ′ (A)

+∑

ijαβσσ′

1

2Vijαβσσ′δρiiασδρjjβσ′ (BB)

+∑

ijαβσ

LijαβσδρiiασδDjβ (CC)

+∑

iαβσ


+∑

ijαβ

1

2UijαβδDiαδDjβ (EE) (4.19)

with

Vijαβσσ′ = V 1ijαβσσ′ + Yiiαασσ′ ∗ δiα,jβ

Lijαβσ = L1ijαβσ +Miiαασ ∗ δiα,jβ

Uijαβ = U1ijαβ +Kiiαα ∗ δiα,jβ

δiα,jβ indicates diagonal elements of these interaction matrices.


Momentum space energy expansion

Here we define how to Fourier Transform (FT) the quantities introduced in real space.

A lattice site is given by riα = Ri + δα where Ri = n1a1 + n2a2 belongs to theBravais superlattice and δα specifies the α = 1, ... 2d intracell positions.

A point of the reciprocal lattice is given by k + K where k belongs to the first Re-duced magnetic Brillouin Zone (RBZ) and K = m1b1 +m2b2 belongs to the reciprocalsuperlattice. Thanks to the combination k + K, we recover all the Bloch states |k〉present in the Extended Brillouin Zone (EBZ). Clearly Ri ·Ki = 2πm′ (ni, mi and m′

are all integers) and so eiRi·Ki = 1.Given that the Bravais superlattice has a cell with Na atoms, the reciprocal super-

lattice will be given by a set of Na vectors Q and one will consider N=NaNc momentafor the FT over all the momenta of the EBZ.

In the following computations we will perform a partial FT, i.e. restricted to the1st RBZ, and thus the momenta to be considered are Nc.

c†iασ =1√Nc

∑

k

eik·(Ri+δα)c†kασ

δρiiασ =1

Nc

∑

kq

ei(k+q)·(Ri+δα)e−ik·(Ri+δα)δ〈c†k+q,ασckασ〉

=1

Nc

∑

q

eiq·(Ri+δα)∑

k

δρk+q,ασ;kασ

=1

Nc

∑

q

eiq·(Ri+δα)δρqασ

In summary, we shall work with the quantities:

δρiiασ =1

Nc

∑

q

eiq·(Ri+δα)δρqασ

δTiασ =1

Nc

∑

q

eiq·(Ri+δα)δTqασ

δDiα =1

Nc

∑

q

eiq·(Ri+δα)δDqα

Then we report the FT’s of all the terms of Eqs. (4.18) and (4.19).


∑

iαβσσ′

Xiiαβσσ′δρiiασδTiασ′ =∑

qαβσσ′

[Xqαβσσ′ ] δρqασδT−qβσ′ (A)

with

Xqαβσσ′ = Xooαβσσ′δαβ = z′ασ′σδαβ

∑

ijαβσσ′

1

2V 1

ijαβσσ′δρiiασδρjjβσ′ =1

2

∑

qαβσσ′

[V 1

qαβσσ′

]δρqασδρ−qβσ′ (B)

with

V 1qαβσσ′ =

∑

j

[∑

σ′′

T 0ojαβσ′′z′ασ′′σz

′βσ′′σ′

]eiq·(Ro−Rj)eiq·(δα−δβ)

=∑

j

V 1ojαβσσ′eiq·(Ro−Rj)eiq·(δα−δβ)

∑

iασσ′

1

2Yiiαασσ′δρiiασδρiiασ′ =

1

2

∑

qαβσσ′

[Yqαβσσ′ ] δρqασδρ−qβσ′ (B2)

with

Yqαβσσ′ =

[∑

jβσ′′

T 0ojαβσ′′zβσ′′z′′ασ′′σσ′

]δαβ

= Yooαασσ′δαβ

∑

ijαβσ

L1ijαβσδρiiασδDjβ =

∑

qαβσ

[L1

qαβσ

]δρqασδD−qβ (C)

with

L1qαβσ =

∑

j

[∑

σ′

T 0ojαβσ′z′βσ′Dz

′ασ′σ


=∑

j

L1ojαβσe

iq·(Ro−Rj)eiq·(δα−δβ)


∑

iασ

MiiαασδρiiασδDiα =∑

qαβσ

[Mqαβσ] δρqασδD−qβ (C2)

with

Mqαβσ =

[∑

jβσ′

T 0ojαβσ′zβσ′z′′ασ′σD

]δαβ

= Mooαασδαβ

∑

iασ

ZiiαβσδDiαδTiασ =∑

qαβσ

[Zqαβσ] δDqαδT−qβσ (D)

with

Zqαβσ = Zooαβσδαβ = z′ασDδαβ

∑

ijαβ

1

2U1

ijαβδDiαδDjβ =1

2

∑

qαβ

[U1

qαβ

]δDqαδD−qβ (E)

with

U1qαβ =

∑

j

[∑

σ′

T 0ojαβσ′z′ασ′Dz

′βσ′D


=∑

j

U1ojαβe


∑

iα

1

2Kiiαα(δDiα)2 =

1

2

∑

qαβ

[Kqαβ] δDqαδD−qβ (E2)

with

Kqαβ =

[∑

jβσ

T 0ojαβσzβσz

′′ασD

]δαβ

= Kooααδαβ


Compact form for δE(2)e in Momentum Space - Representation (I)

δE(2)e =

∑

qαβσσ′

[Xqαβσσ′ ] δρqασδT−qβσ′ (A)

+1

2

∑

qαβσσ′

[Vqαβσσ′ ] δρqασδρ−qβσ′ (BB)

+∑

qαβσ

[Lqαβσ] δρqασδD−qβ (CC)

+∑

qαβσ

[Zqαβσ] δDqαδT−qβσ (D)

+1

2

∑

qαβ

[Uqαβ] δDqαδD−qβ (EE) (4.20)

with

Vqαβσσ′ = V 1qαβσσ′ + Yqαβσσ′

Lqαβσ = L1qαβσ +Mqαβσ

Uqαβ = U1qαβ +Kqαβ

This is the representation (I), in which the energy is expanded using the deviationsδρσ and δTσ.

Xqαβσσ′ = Xooαασσ′δαβ

V 1qαβσσ′ =

∑

j

V 1ojαβσσ′eiq·(Ro−Rj)eiq·(δα−δβ)

Yqαβσσ′ = Yooαασσ′δαβ

L1qαβσ =

∑

j

L1ojαβσe


Mqαβσ = Mooαασδαβ

Zqαβσ = Zooαασδαβ

U1qαβ =

∑

j

U1ojαβe


Kqαβ = Kooααδαβ


Transformation from the Representation (I) to the Representation (II)

In the representation (I), the energy is expanded using the deviations δρσ and δTσ.Now we intend to move to the representation (II), where we expand the energy

using the deviations δρη and δTη (η = ±); for example δρ+ gives the charge deviationsand δρ− the spin deviations.

We give the transformation laws for the density deviation δρ, being the ones for thedeviation δT totally analogous:

δρqα,+ =∑

σ

δρqασ

δρqα,− =∑

σ

σδρqασ

δρqασ =1

2(δρqα,+ + σδρqα,−) (4.21)

where σ = ±1.Thus the terms in Eq. (4.20) will undergo the following transformations:

∑

qαβσσ′

[Xqαβσσ′ ] δρqασδT−qβσ′ =

∑

qαβσσ′

[Xqαβσσ′ ]1

4(δρqα,+ + σδρqα,−)(δT−qβ,+ + σ′δT−qβ,−) =

∑

qαβηη′

[Xqαβηη′ ] δρqα,ηδT−qβ,η′ (A)

with

Xqαβ++ =1

4

∑

σσ′

Xqαβσσ′ Xqαβ+− =1

4

∑

σσ′

Xqαβσσ′σ′

Xqαβ−+ =1

4

∑

σσ′

Xqαβσσ′σ Xqαβ−− =1

4

∑

σσ′

Xqαβσσ′σσ′

1

2

∑

qαβσσ′

[Vqαβσσ′ ] δρqασδρ−qβσ′ =

1

2

∑

qαβσσ′

[Vqαβσσ′ ]1

4(δρqα,+ + σδρqα,−)(δρ−qβ,+ + σ′δρ−qβ,−) =

1

2

∑

qαβηη′

[Vqαβηη′ ] δρqα,ηδρ−qβ,η′ (BB)


with

Vqαβ++ =1

4

∑

σσ′

Vqαβσσ′ Vqαβ+− =1

4

∑

σσ′

Vqαβσσ′σ′

Vqαβ−+ =1

4

∑

σσ′

Vqαβσσ′σ Vqαβ−− =1

4

∑

σσ′

Vqαβσσ′σσ′

∑

qαβσ

[Lqαβσ] δρqασδD−qβ =

∑

qαβσ

[Lqαβσ]1

2(δρqα,+ + σδρqα,−)δD−qβ =

∑

qαβη

[Lqαβη] δρqα,ηδD−qβ (CC)

with

Lqαβ+ =1

2

∑

σ

Lqαβσ Lqαβ− =1

2

∑

σ

Lqαβσσ

∑

qαβσ

[Zqαβσ] δDqαδT−qβ,σ =

∑

qαβσ

[Zqαβσ] δDqα

1

2(δT−qβ,+ + σδT−qβ,−) =

∑

qαβη

[Zqαβη] δDqαδT−qβ,η (D)

with

Zqαβ+ =1

2

∑

σ

Zqαβσ Zqαβ− =1

2

∑

σ

Zqαβσσ

Obviously, the quantity (EE) in Eq. (4.20) is invariant under the transformationillustrated in Eq. (4.21).


Compact form for δE(2)e in Momentum Space - Representation (II)

δE(2)e =

∑

qαβηη′

[Xqαβηη′ ] δρqαηδT−qβη′ (A)

+1

2

∑

qαβηη′

[Vqαβηη′ ] δρqαηδρ−qβη′ (BB)

+∑

qαβη

[Lqαβη] δρqαηδD−qβ (CC)

+∑

qαβη

[Zqαβη] δDqαδT−qβη (D)

+1

2

∑

qαβ

[Uqαβ ] δDqαδD−qβ (EE) (4.22)

Antiadiabatic Condition

∂δE(2)e

∂Dqα

=∑

βη

Lqαβηδρ−qβη +∑

βη

ZqαβηδT−qβη +∑

β

UqαβδD−qβ = 0

We can present this derivation in a more compact matricial formalism. In principle,one could signal a vectorial quantity with a single underline (i.e., δρ) and a matricialquantity with a double underline (i.e., Uq). In the following derivation we usually prefer

to omit this supplement of notation, if there is no risk of ambiguity.

We will exploit the property U tq = U−q which essentially follows from the observa-

tion that Uijαβ = Ujiβα:

U−q → U−qαβ =∑

j

Uojαβe−iq·(Ro−Rj)e−iq·(δα−δβ)

U tq → Uqβα =

∑

j

Ujoβαeiq·(Rj−Ro)eiq·(δβ−δα)

=∑

j

Uojαβeiq·(Rj−Ro)eiq·(δβ−δα)


δE(2)e =

∑

qηη′

δρtqη [Xqηη′ ] δT−qη′ (A)

+1

2

∑

qηη′

δρtqη [Vqηη′ ] δρ−qη′ (BB)

+∑

qη

δρtqη [Lqη] δD−q (CC)

+∑

qη

δDtq [Zqη] δT−qη (D)

+1

2

∑

q

δDtq [Uq] δD−q (EE) (4.23)

So the antiadiabatic condition yields:

∂δE(2)e

∂D−q

=∑

η

[δρt

qηLtqη + δT t

qηZtqη

]+ δDt

qU−q = 0

∂δE(2)e

∂Dtq

=∑

η

[Lqηδρ−qη + ZqηδT−qη] + UqδD−q = 0

and then

δDtq = −

∑

η

[δρt

qηLtqηU

−1−q + δT t

qηZtqηU

−1−q

]

δD−q = −∑

η

[U−1

q Lqηδρ−qη + U−1q ZqηδT−qη

](4.24)

Replacing Eq. (4.24) into Eq. (4.23), one eliminates the double occupancy deviations

δD and obtains δE(2)e in terms of charge and spin deviations δρ and δT only.

δE(2)e =

1

2

∑

qηη′

δρtqη

[Vqηη′ − Lt

qηU−1−qLqη′

]δρ−qη′

+1

2

∑

qηη′

δρtqη

[Xqηη′ − Lt

qηU−1−qZqη′

]δT−qη′

+1

2

∑

qηη′

δT tqη

[X t

qηη′ − ZtqηU

−1−qLqη′

]δρ−qη′

+1

2

∑

qηη′

δT tqη

[−Zt

qηU−1−qZqη′

]δT−qη′ (4.25)


Eq. (4.25) exhibits this general structure:

δE(2)e =

1

2

∑

qαβηη′

(δρqαη

δTqαη

)(Aqαβηη′ Bqαβηη′

Bqαβηη′ Cqαβηη′

)(δρ−qβη′

δT−qβη′

)

=1

2

∑

q

δRqWqδR−q (4.26)

where the interaction kernel Wq is

Wqαβηη′ ≡(Aqαβηη′ Bqαβηη′

Bqαβηη′ Cqαβηη′

)(4.27)

and in a more explicit form:

Wqαβηη′ =

Aqαβ++ Bqαβ++ Aqαβ+− Bqαβ+−

Bqαβ++ Cqαβ++ Bqαβ+− Cqαβ+−

Aqαβ−+ Bqαβ−+ Aqαβ−− Bqαβ−−

Bqαβ−+ Cqαβ−+ Bqαβ−− Cqαβ−−

(4.28)

When we consider the expansion from a charge-ordered state without spin or-der or in the case of a collinear spin solution, Wqαβηη′ presents zero off-diagonalblocks due to the decoupling of the charge and the spin sector: Aqαβηη′=Aqαβηηδηη′ ;Bqαβηη′=Bqαβηηδηη′ ; Cqαβηη′=Cqαβηηδηη′ . We then have a complete separation of thecharge and the spin sector within Wq: we can extract a purely charge interaction kernel

(Wqαβ++) and a purely spin one (Wqαβ−−), being the off-diagonal blocks Wqαβ+− =Wqαβ−+ = 0. Thus in these cases Wq is decoupled in a charge block and a spin block,

both along the diagonal with size 2Na × 2Na.In more general cases Wqαβηη′ and also the susceptibility χ0

qαβηη′ and χqαβηη′ arenon-sparse matrices with size 4Na×4Na and thereby 8d×8d in the case of d×2 stripes.


Transformation to the Momentum Space II

Within the momentum space we will definitely evaluate the quantities in the basisof the quasiparticle operators akλσ which diagonalize simultaneously the Gutzwillerhamiltonian h and the density matrix ρ. We use the transformation:3

a†kλσ =∑

α

φ∗λα(kσ)c†kασ

akλσ =∑

α

φλα(kσ)ckασ

c†kασ =∑

λ

φλα(kσ)a†kλσ

ckασ =∑

λ

φ∗λα(kσ)akλσ (4.29)

φ is a function of the electron momentum k in the RBZ; its matrix indexes are givenby the Na quasiparticle band indexes λ and by the Na supercell indexes α. Usingthe second couple of laws given in Eq. (4.29), we can formulate the quantities earliercalculated in the basis of the c-operators using the expectation values in the basis ofthe a-operators, where we know the eigenvalues ǫkλσ and the eigenstates |kλσ〉.

Thus the explicit transformation of δρqα,η will be

δρqα,+ =∑

kσ

δρk+q,ασ;kασ =∑

kσ

δ〈c†k+q,ασckασ〉

=∑

kλµσ

φλα(k + q, σ)φ∗µα(kσ)δ〈a†k+q,λσakµσ〉 (4.30)

δρqα,− =∑

kλµσ

σφλα(k + q, σ)φ∗µα(kσ)δ〈a†k+q,λσakµσ〉

δρ−qβ,+ =∑

kλµσ

φλβ(k − q, σ)φ∗µβ(kσ)δ〈a†k−q,λσakµσ〉

In the next Sections we will present the elements of the Lindhard non-interactingmatrix χ0

q associated to the superlattice, starting from

• the evaluation of the correlator 〈δρqα,ηδρ−qβ,η′〉0;

• a proper relation between δTqα,+ and δ〈c†k+q,ασckβσ〉.3In Eq. (4.29) one considers that the matrix of the canonical transformation φλα is hermitian, then

by definition φt = φ∗ → φαλ=φ∗

λα.


PARADIGMATIC EVALUATION OF THE CORRELATOR (χ0qαβ,++)ρρ

(χ0qαβ,++)ρρ(ω) = 〈δρqα,+δρ−qβ,+〉0

=1

Nc

1

i

∫dteiωt〈Ttρqα,+(t)ρ−qβ,+〉

=1

Nc

1

i

∫dt

∑

kk′λλ′µµ′σσ′

ei(ω+ǫk+q,λσ−ǫkµσ)tφλα(k + q, σ)φ∗µα(kσ)φλ′β(k′ − q, σ′)φ∗

µ′β(k′σ′) ×

×[e−δtθ(t)〈a†k+q,λσakµσa

†k′−q,λ′σ′ak′µ′σ′〉 + eδtθ(−t)〈a†k′−q,λ′σ′ak′µ′σ′a†k+q,λσakµσ〉

]

=1

Nc

1

i

∫dt

∑


ei(ω+ǫk+q,λσ−ǫkµσ)tφλα(k + q, σ)φ∗µα(kσ)φλ′β(k′ − q, σ′)φ∗

µ′β(k′σ′) ×

×δk′,k+qδλµ′δλ′µδσσ′

[e−δtθ(t)nk+q,λσ(1 − nkµσ) + eδtθ(−t)nkµσ(1 − nk+q,λσ)

]

= − 1

Nc

∑

kλµσ

φλα(k + q, σ)φ∗µα(kσ)φµβ(kσ)φ∗

λβ(k + q, σ) ×

×[

nk+q,λσ(1 − nkµσ)

ω + ǫk+q,λσ − ǫkµσ + iδ− nkµσ(1 − nk+q,λσ)

ω + ǫk+q,λσ − ǫkµσ − iδ

]

with δ = 0+.

Actually we will consider the static correlator (χ0qαβ,++)ρρ (ω, δ = 0, 0):

(χ0qαβ,++)ρρ = − 1

Nc

∑kλµσ φλα(k + q, σ)φ∗

µα(kσ)φµβ(kσ)φ∗λβ(k + q, σ) ×

× nk+q,λσ − nkµσ

ǫk+q,λσ − ǫkµσ

. (4.31)


RELATION δT vs. δρ

δρiiα,+ =∑

σ

δρiiασ ; δρiiα,− =∑

σ

σδρiiασ

δTiα,+ =∑

σ

δTiασ ; δTiα,− =∑

σ

σδTiασ

Homogeneous Case

For the homogeneous ground state one has found:

δTi,+ =∑

jσ

tijz0(δ〈c†iσcjσ〉 + δ〈c†jσciσ〉)

δTq,+ =∑

kσ

δTk+q,σ;kσ =∑

kσ

γk+q,kσδρk+q,σ;kσ

with γk+q,kσ = z0(ǫ0kσ + ǫ0k+q,σ) where ǫ0kσ is the bare electronic dispersion and z0 the

GA hopping factor.

Inhomogeneous Case

Here we give the derivation in the case of an inhomogeneous ground state.

δTqα,+ =∑

i

e−iq·(Ri+δα)δTiα,+ with

δTiα,+ =∑

jβσ

tijαβzβσ(δ〈c†iασcjβσ〉 + δ〈c†jβσciασ〉)


δTqα,+ =1

Nc

∑

ijβkq′σ

tijαβzβσe−iq·(Ri+δα) ×

×[ei(k+q′)·(Ri+δα)e−ik·(Rj+δβ)δ〈c†k+q′,ασckβσ〉 +

ei(k+q′)·(Rj+δβ)eiq′·Rie−iq′·Rie−ik·(Ri+δα)δ〈c†k+q′,βσckασ〉]

δTqα,+ =1

Nc

∑

ijβkq′σ

tijαβzβσ ×

×[ei(q′−q)·Riei(q′−q)·δαeik·(Ri−Rj)eik·(δα−δβ)δ〈c†k+q′,ασckβσ〉 +

ei(q′−q)·Rie−i(k+q′)·(Ri−Rj)e−ik·(δα−δβ)e−i(q·δα−q′·δβ)δ〈c†k+q′,βσckασ〉]

δTqα,+ =∑

kβσ

zβσ[tkαβeik·(δα−δβ)δ〈c†k+q,ασckβσ〉 +

t−(k+q),αβe−i(k+q)·(δα−δβ)δ〈c†k+q,βσckασ〉]

with tkαβ =∑

j tojαβeik·(Ro−Rj);

t−(k+q),αβ =∑

j tojαβe−i(k+q)·(Ro−Rj).

Thus we find

δTqα,+ =∑

kβσ

zβσ

[τkαβδ〈c†k+q,ασckβσ〉 + τ−(k+q),αβδ〈c†k+q,βσckασ〉

](4.32)

with

τkαβ = tkαβeik·(δα−δβ) =

∑

j

tojαβeik·(Ro−Rj)eik·(δα−δβ) ; (4.33)

τ−(k+q),αβ = t−(k+q),αβe−i(k+q)·(δα−δβ)

=∑

j

tojαβe−i(k+q)·(Ro−Rj)e−i(k+q)·(δα−δβ)


Therefore, considering Eqs. (4.31)-(4.33), one is able to evaluate all the correlatorswhich are in the non-interacting susceptibility χ0

qQQ′ηη′ .Using Eq. (4.29), one obtains the transformation laws for δT into the quasiparticle

basis, as it has been already done for δρ in Eq. (4.30):

δTqα,+ =∑

kβσ

zβσ

[τkαβδ〈c†k+q,ασckβσ〉 + τ−(k+q),αβδ〈c†k+q,βσckασ〉

]

=∑

kβλµσ

zβσ[τkαβφλα(k + q, σ)φ∗µβ(kσ)δ〈a†k+q,λσakµσ〉 +

τ−(k+q),αβφλβ(k + q, σ)φ∗µα(kσ)δ〈a†k+q,λσakµσ〉]

δTqα,+ =∑

kβλµσ

γk+q,k;λµαβσδ〈a†k+q,λσakµσ〉 (4.34)

δTqα,− =∑

kβλµσ

σγk+q,k;λµαβσδ〈a†k+q,λσakµσ〉

δT−qβ,+ =∑

kαλµσ

γk−q,k;λµβασδ〈a†k−q,λσakµσ〉

with

γk+q,k,λµαβσ = zβσ

[τkαβφλα(k + q, σ)φ∗

µβ(kσ) + τ−(k+q),αβφλβ(k + q, σ)φ∗µα(kσ)

]

γk−q,k;λµβασ = zασ

[τkβαφλβ(k − q, σ)φ∗

µα(kσ) + τ−(k−q),βαφλα(k − q, σ)φ∗µβ(kσ)

]

Recipe Notes

γk′−q,k′;λ′µ′βασ′ = zασ′ [τk′βαφλ′β(k′ − q, σ′)φ∗µ′α(k′σ′)

+ τ−(k′−q),βαφλ′α(k′ − q, σ′)φ∗µ′β(k′σ′)]δk′,k+qδλµ′δλ′µδσσ′ =

γk,k+q;µλβασ = zασ

[τk+q,βαφµβ(kσ)φ∗

λα(k + q, σ) + τ−kβαφµα(kσ)φ∗λβ(k + q, σ)

]

tkβα =∑

i

toiβαeik·(Ro−Ri) = tkαβ.


Other Correlators

In Eq. (4.31) we have presented (χ0qαβ,++)ρρ:

(χ0qαβ,++)ρρ = − 1

Nc

∑

kλµσ

φλα(k + q, σ)φ∗µα(kσ)φµβ(kσ)φ∗

λβ(k + q, σ)nk+q,λσ − nkµσ


We can obtain (χ0qαβ,++)Tρ and (χ0

qαβ,++)TT with a derivation similar to the onegiven for (χ0

qαβ,++)ρρ:

(χ0qαβ,++)Tρ(ω) = 〈δTqα,+δρ−qβ,+〉0

=1

Nc

1

i

∫dteiωt〈TtTqα,+(t)ρ−qβ,+〉

=1

Nc

1

i

∫dt

∑


ei(ω+ǫk+q,λσ−ǫkµσ)t ×∑

β′

γk+q,k;λµαβ′σφλ′β(k′ − q, σ′)φ∗µ′β(k′σ′) ×

δk′,k+qδλµ′δλ′µδσσ′

[e−δtθ(t)〈a†k+q,λσakµσa


]

(χ0qαβ,++)Tρ = − 1

Nc

∑

kλµσ

∑

β′

γk+q,k;λµαβ′σφµβ(kσ)φ∗λβ(k + q, σ)

nk+q,λσ − nkµσ


=

− 1

Nc

∑

kλµσ

∑

β′

zβ′σ

[τkαβ′φλα(k + q, σ)φ∗

µβ′(kσ) + τ−(k+q),αβ′φλβ′(k + q, σ)φ∗µα(kσ)

]×

×φµβ(kσ)φ∗λβ(k + q, σ)



(4.35)

(χ0qαβ,++)ρT = − 1

Nc

∑

kλµσ

φλα(k + q, σ)φ∗µα(kσ)

∑

α′

γk,k+q;µλβα′σ



=

− 1

Nc

∑

kλµσ

φλα(k + q, σ)φ∗µα(kσ) × nk+q,λσ − nkµσ


×∑

α′

zα′

[τk+q,βα′φµβ(kσ)φ∗

λα′(k + q, σ) + τ−kβα′φµα′(kσ)φ∗λβ(k + q, σ)

](4.36)


(χ0qαβ,++)TT (ω) = 〈δTqα,+δT−qβ,+〉0

=1

Nc

1

i

∫dteiωt〈TtTqα,+(t)T−qβ,+〉

=1

Nc

1

i

∫dt

∑


ei(ω+ǫk+q,λσ−ǫkµσ)t ×∑

β′

γk+q,k;λµαβ′σ

∑

α′

γk′−q,k′βα′λ′µ′σ′ × δk′,k+q

δλµ′δλ′µδσσ′

[e−δtθ(t)〈a†k+q,λσakµσa


]

(χ0qαβ,++)TT = − 1

Nc

∑

kλµσ

∑

β′

γk+q,k;λµαβ′σ

∑

α′

γk,k+q;µλβα′σ



=

− 1

Nc

∑

kλµσ

∑

β′

zβ′σ

[τkαβ′φλα(k + q, σ)φ∗

µβ′(kσ) + τ−(k+q),αβ′φλβ′(k + q, σ)φ∗µα(kσ)

]×

∑

α′

zα′σ

[τk+q,βα′φµβ(kσ)φ∗

λα′(k + q, σ) + τ−kβα′φµα′(kσ)φ∗λβ(k + q, σ)

]×



(4.37)


The Susceptibility χ0qαβηη′

χ0q =

∑

αβηη′

χ0qαβηη′ with χ0

qαβηη′ = (4.38)

〈δρqα,+δρ−qβ,+〉0 〈δρqα,+δT−qβ,+〉0 〈δρqα,+δρ−qβ,−〉0 〈δρqα,+δT−qβ,−〉0

〈δTqα,+δρ−qβ,+〉0 〈δTqα,+δT−qβ,+〉0 〈δTqα,+δρ−qβ,−〉0 〈δTqα,+δT−qβ,−〉0

〈δρqα,−δρ−qβ,+〉0 〈δρqα,−δT−qβ,+〉0 〈δρqα,−δρ−qβ,−〉0 〈δρqα,−δT−qβ,−〉0

〈δTqα,−δρ−qβ,+〉0 〈δTqα,−δT−qβ,+〉0 〈δTqα,−δρ−qβ,−〉0 〈δTqα,−δT−qβ,−〉0

This is the superlattice GA susceptibility χ0qαβηη′ .

The Susceptibility χqαβηη′

Using Eqs. (4.25) and (4.28) for the RPA interaction Wqαβηη′ and Eq. (4.38) for theGA susceptibility χ0

qαβηη′ , we can find the superlattice GA+RPA susceptibility χqαβηη′ :

χqαβηη′ = χ0qαβηη′ −

∑

β′β′′

χ0qαβ′ηη′Wqβ′β′′ηη′χqβ′′βηη′

χqηη′ = (I + χ0qηη′Wqηη′)−1χ0

qηη′ (4.39)


Determination of χqQQ′ηη′ by Fourier analysis

Insofar we have worked in the mixed real-momentum basis (A), generated by theoperators ckασ:

c†iασ =1√Nc

∑

k

eik·(Ri+δα)c†kασ

δRiiασ =1

Nc

∑

q

eiq·(Ri+δα)δRqασ

δDiα =1

Nc

∑

q

eiq·(Ri+δα)δDqα

Here we use δR to express the similar equations for δρ and δT .Now we prefer to present χqαβηη′ using a purely momentum basis (B) given by the

operators ck+K,σ:

c†iασ =1√N

∑

kK

ei(k+K)·δαeik·Ric†k+K,σ

δRiiασ =1

N

∑

qQ

ei(q+Q)·δαeiq·RiδRq+Q,σ

δDiα =1

N

∑

qQ

ei(q+Q)·δαeiq·RiδDq+Q

Actually, when we move

• from real space to the momentum space (A), we FT on the supercells (q ↔ i);

• from real space to the momentum space (B), we FT on the sites (q,Q ↔i, α);

• then from the momentum space (A) to the momentum space (B), we FT withinthe supercells (Q ↔ α).


Equalizing the r.h.s.’s of the equations previously given for c†iασ, δRiiασ, δDiα,we can relate the quantities expressed in momentum space (A) and in momentumspace (B):

c†kασ =1√Na

∑

K

eiK·δαc†k+K,σ

δRqασ =1

Na

∑

Q

eiQ·δαδRq+Q,σ

δDqα =1

Na

∑

Q

eiQ·δαδDq+Q

Then we can obtain the relation between χqαβηη′ and χqQQ′ηη′ :

χqαβηη′ =1

N2a

∑

QQ′

ei(Q·δα−Q′·δβ)χqQQ′ηη′

and

χqQQ′ηη′ =∑

αβ

e−i(Q·δα−Q′·δβ)χqαβηη′ . (4.40)

List of Publications and Pre-prints

1-A. Di Ciolo, M. Grilli, J. Lorenzana, and G. Seibold, “Charge inhomogeneitycoexisting with large Fermi surfaces”, Physica C 460, 1176, (2007).

2-A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, “Charge instabilities andelectron-phonon interaction in the Hubbard-Holstein model”, arxiv:cond-mat/0806.3385.

3-M. Grilli, G. Seibold, A. Di Ciolo, and J. Lorenzana, “Fermi surface dichotomyon systems with fluctuating ordering”, arxiv:cond-mat/0809.2197.

4-A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, “Kohn anomaly in stronglycorrelated systems”, pre-print.

5-E. von Oelsen, A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, “Phononrenormalization from local and transitive electron-lattice couplings in strongly corre-lated systems”, pre-print.

Bibliography

[1] A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu,T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S.Uchida, Z. Hussain, and Z. X. Shen, Nature (London) 412, 510 (2001).

[2] M. D’Astuto, P. K. Mang, P. Giura, A. Shukla, P. Ghigna, A. Mirone, M. Braden,M. Greven, M. Krisch, and F. Sette, Phys. Rev. Lett. 88, 167002 (2002).

[3] G.-H. Gweon, T. Sasagawa, S. Y. Zhou, J. Graf, H. Takagi, D.-H. Lee, and A.Lanzara, Nature (London) 430, 187 (2004).

[4] D. Reznik, L. Pintschovius, M. Ito, S. Iikubo, M. Sato, H. Goka, M. Fujita,K. Yamada, G. D. Gu, and J. M. Tranquada, Nature (London) 440, 1170 (2006).

[5] D. Reznik, L. Pintschovius, M. Fujita, K. Yamada, G. D. Gu, and J. M. Tran-quada, Journal of Low Temp. Phys. 147, 353 (2007).

[6] D. Reznik, T. Fukuda, D. Lamago, A. Q. R. Baron, S. Tsutsui, M. Fujita, andK. Yamada, arxiv:cond-mat/0710.4782.

[7] J. Lee, K. Fujita, K. McElroy, J. A. Slezak, M. Wang, Y. Aiura, H. Bando,M. Ishikado, T. Masui, J.-X. Zhu, A. V. Balatsky, H. Eisaki, S. Uchida, andJ. C. Davis, Nature (London) 442, 546 (2006).

[8] J. P. Falck, A. Levy, M. A. Kastner, and R. J. Birgeneau, Phys. Rev. Lett. 69,1109 (1992).

[9] P. Calvani, M. Capizzi, S. Lupi, P. Maselli, A. Paolone, and P. Roy, Phys. Rev.B 53, 2756 (1996).

[10] P. Calvani, M. Capizzi, S. Lupi, and G. Balestrino, Europhys. Lett. 31, 473(1995).

[11] K. M. Shen, F. Ronning, D. H. Lu, W. S. Lee, et al., Phys. Rev. Lett. 93, 267002(2004).

[12] L. Boeri, O. V. Dolgov, and A. A. Golubov, Phys. Rev. Lett. 101, 026403 (2008).

142 BIBLIOGRAPHY

[13] D. Bhoi, P. Mandal, and P. Choudhury, Supercond. Sci. Technol. 21, 125021(2008).

[14] A. J. Millis, R. Mueller, and B. I. Shraiman, Phys. Rev. B 54, 5405 (1996).

[15] P. S. Cornaglia, H. Ness, and D. R. Grempel, Phys. Rev. Lett. 93, 147201 (2004).

[16] O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997).

[17] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti, Science 296, 2364 (2002).

[18] M. Grilli and C. Castellani, Phys. Rev. B 50, 16880 (1994).

[19] C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995).

[20] U. Low, V. J. Emery, K. Fabricius, and S. A. Kivelson, Phys. Rev. Lett. 72, 1918(1994).

[21] J. Lorenzana, C. Castellani, and C. Di Castro, Phys. Rev. B 64, 235127 (2001).

[22] J. Lorenzana, C. Castellani, and C. Di Castro, Phys. Rev. B 64, 235128 (2001).

[23] J. Lorenzana, C. Castellani, and C. Di Castro, Europhys. Lett. 57, 704 (2002).

[24] C. Ortix, J. Lorenzana, and C. Di Castro, Phys. Rev. B 73, 245117 (2006).

[25] C. Ortix, J. Lorenzana, M. Beccaria, and C. Di Castro, Phys. Rev. B 75, 195107(2007).

[26] C. Ortix, J. Lorenzana, and C. Di Castro, Phys. Rev. Lett. 100, 246402 (2008).

[27] G. Gruner, Density Waves in Solids, ed. Addison Wesley (1994).

[28] O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod.Phys. 79, 353 (2007).

[29] A. H. Castro Neto and C. Morais Smith, “Charge Inhomogeneities in StronglyCorrelated Systems”, in the volume “Strong Interactions in Low Dimensions”(Kluwer, 2004).

[30] H. Frolich, Proc. Roy. Soc. London Ser. A223 (1954).

[31] R. E. Peierls, Quantum Theory of Solids (Oxford: Clarendon, 1955).

[32] A. W. Overhauser, Phys. Rev. Lett. 4, 462 (1960).

[33] A. W. Overhauser, Phys. Rev. 128, 1437 (1962).

BIBLIOGRAPHY 143

[34] M. D. Johannes and I. I. Mazin, Phys. Rev. B 77, 165135 (2008).

[35] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Brooks/Cole - ThomsomLearning Inc, Cornell University (1976).

[36] C. Schlenker, ”Low dimensional properties of Molybdenum bronzes and oxides”,(Kluwer Ed. , 1989).

[37] M. Greenblatt, Int. Mod. Phys. B 7, 4045 (1993).

[38] C. Schlenker, J. Dumas, M. Greenblatt, and S. van Smaalen, ”Physics and Chem-istry of Low Dimensional Inorganic Conductors”, NATO-ASI Series B, Vol. 354,(Plenum, 1996).

[39] J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature375, 561 (1995).

[40] T. Hanaguri, C.Lupien, Y. Kohsaka, D.-H. Lee, et al., Nature 430, 1001 (2004).

[41] C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993).

[42] J. M. Tranquada, J. E. Lorenzo, D. J. Buttrey, and V. Sachan, Phys. Rev. B 52,3581 (1995).

[43] C. H. Chen, S.-W. Cheong, and H. Y. Hwang, J. Appl. Phys. 81, 4326.

[44] S. Mori, C. H. Chen, and S.-W. Cheong, Nature 392, 473.

[45] A. Rusydi, P. Abbamonte, H. Eisaki, Y. Fujimaki, G. Blumberg, S. Uchida, andG. A. Sawatzky Phys. Rev. Lett. 97, 016403 (2006).

[46] S. Cox, J. C. Lashley, E. Rosten, J. Singleton, A. J. Williams, and P. B. Little-wood, J. Phys.: Condens. Matter 19, 192201 (2007).

[47] T. Vogt, P. M. Woodward, P. Karen, B. A. Hunter, P. Henning, and A. R.Moodenbaugh, Phys. Rev. Lett. 84, 2969 (2000).

[48] J. E. Hoffmann, E. W. Hudson, K. M. Lang, V. Madhavan, et al. Science 295,466 (2002).

[49] J. E. Hoffmann, K. McElroy, D.-H. Lee, K. M. Lang, et al. Science 297, 1148(2002).

[50] C. Howald, H. Eisaki, N. Kaneko, M. Greven, and A. Kapitulnik, Phys. Rev. B67, 014533 (2003).

144 BIBLIOGRAPHY

[51] M. Vershinin, A. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science 303,1995 (2004).

[52] K. McElroy, R. W. Simmonds, J. E. Hoffmann, D.-H. Lee, et al., Nature 422422, 592 (2003).

[53] K. McElroy, D.-H. Lee, J. E. Hoffmann, K. M. Lang, et al., Phys. Rev. Lett. 94,197005 (2005).

[54] W. D. Wise, M. C. Boyer, K. Chatterjee, T. Kondo, et al., Nature Phys. 4, 696(2008).

[55] R. L. Withers and J. A. Wilson, J. Phys. C 19, 4809 (1986).

[56] R. M. Fleming, D. E. Moncton, D. B. McWhan, and F. J. Di Salvo, Phys. Rev.Lett. 45, 576 (1980).

[57] S. V. Borisenko, A. A. Kordyuk, A. N. Yaresko, V. B. Zabolotny, et al., Phys.Rev. Lett. 100, 196402 (2008).

[58] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).

[59] P. Dai, B. C. Chakoumakos, G. F. Sun, K. W. Wong, Y. Xin, and D. F. Lu,Physica C 243, 201 (1995).

[60] J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, andB. Nachumi, Phys. Rev. B 54, 7489 (1996).

[61] J. M. Tranquada, J. D. Axe, N. Ichikawa, A. R. Moodenbaugh, Y. Nakamura,and S. Uchida, Phys. Rev. Lett. 78, 338 (1997).

[62] K. Yamada, C. H. Lee, K. Kurahashi, J. Wada et al., Phys. Rev. B 57, 6165(1998).

[63] M. Fujita, H. Goka, K. Yamada, and M. Matsuda, Phys. Rev. Lett. 88, 167008(2002).

[64] J. Lorenzana and G. Seibold, Phys. Rev. Lett. 89, 136401 (2002).

[65] P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A. Sawatzky, and D. L. Feng,Nature Phys. 1, 155 (2005).

[66] Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri,M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science315, 1380 (2007).

BIBLIOGRAPHY 145

[67] P. V. Bogdanov, A. Lanzara, S. A. Kellar, X. J. Zhou et al., Phys. Rev. Lett. 85,2581 (2000).

[68] X. J. Zhou, T. Yoshida, A. Lanzara, P. V. Bogdanov et al., Nature (London)423, 398 (2003).

[69] G. Sangiovanni, Ph-D Thesis.

[70] J. Hubbard, Proc. R. Soc. London, A 276, 238 (1963).

[71] F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 (1988).

[72] P. W. Anderson, Science 235, 1196 (1987).

[73] J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7391 (1989).

[74] A. R. Moodenbaugh, Y. Xu, M. Suenaga, T. J. Folkerts, and R. N. Shelton, Phys.Rev. B 38, 4596 (1988).

[75] X. J. Zhou,P. V. Bogdanov, S. A. Kellar, T. Noda, H. Eisaki, S. Uchida, Z. Hus-sain, and Z.-X. Shen, Science 286, 268 (1999).

[76] X. J. Zhou, T. Yoshida, S. A. Kellar, P. V. Bogdanov et al., Phys. Rev. Lett. 86,5578 (2001).

[77] S. Wakimoto, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, et al., Phys. Rev. B 61,3699 (2000).

[78] Y. Kohsaka, C. Taylor, P. Wahl, A. Schmidt, et al., Nature 454, 1072 (2008).

[79] B. Batlogg, R. J. Cava, A. Jayaraman, R. B. van Dover et al., Phys. Rev. Lett.58, 2333 (1987).

[80] P. Franck, J. Jung, M. A.-K. Mohamed, S. Gygax et al., Phys. Rev. B 44, 5318(1991).

[81] T. Schneider and H. Keller, Phys. Rev. Lett. 69, 3374 (1992).

[82] Guo-Meng Zhao, M. B. Hunt, H. Keller, and K. A. Muller, Nature (London) 385,236 (1997).

[83] J. Hofer, K. Conder, T. Sasagawa, Guo-Meng Zhao et al., Phys. Rev. Lett. 84,4192 (2000).

[84] R. Khasanov, A. Shengelaya, K. Conder, E. Morenzoni et al., J. Phys.: Condens.Matter 15, L17 (2003).

146 BIBLIOGRAPHY

[85] D. R. Temprano, J. Mesot, S. Janssen, K. Conder et al., Phys. Rev. Lett. 84,1990 (2000).

[86] S. Andergassen, S. Caprara, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 87,056401 (2001).

[87] B. Batlogg, H. Takagi, H. L. Kao, and J. Kwo, Electronic Properties of High-TcSuperconductors. The Normal State and the Superconducting State, (Springer-Verlag-New York: 1992).

[88] O. Gunnarson and M. Calandra, Phys. Rev. B 66, 205105-1 (2002).

[89] T. Cuk, F. Baumberger, D. H. Lu, N. Ingle et al. Phys. Rev. Lett. 93, 117003(2004).

[90] J. Hwang, T. Timusk, and G. D. Gu. Nature (London) 427, 714 (2004).

[91] T. P. Devereaux, T. Cuk, Z.-X. Shen, and N. Nagaosa, Phys. Rev. Lett. 93,117004 (2004).

[92] M. Eschrig and M. R. Norman, Phys. Rev. Lett. 85, 3261 (2000).

[93] A. Kaminski, M. Randeria, J. C. Campuzano, M. R. Norman et al., Phys. Rev.Lett. 86, 1070 (2001).

[94] G. Seibold and M. Grilli, Phys. Rev. B 72, 104519 (2005).

[95] F. Giustino, M. L. Cohen, and S. G. Louie, Nature 452, 975 (2008).

[96] S. Y. Savrasov and O. K. Andersen, Phys. Rev. Lett. 77, 4430 (1996).

[97] K. P. Bohnen, R. Heid, and M. Krauss, Europhys. Lett. 64, 104 (2003).

[98] R. Heid, K. P. Bohnen, R. Zeyher, and D. Manske, Phys. Rev. Lett. 100, 137001(2008).

[99] D. Reznik, G. Sangiovanni, O. Gunnarson, and T. P. Deveraux, Nature 455, E6(2008).

[100] J. Graf, G.-H. Gweon, K. McElroy, S. Y. Zhou et al., Phys. Rev. Lett. 98, 067004(2007).

[101] B. P. Xie, K. Yang, D. W. Shen, J. F. Zhao et al., Phys. Rev. Lett. 98, 147001(2007).

[102] T. Valla, T. E. Kidd, W.-G. Yin, G. D. Gu et al., Phys. Rev. Lett. 98, 167003(2007).

BIBLIOGRAPHY 147

[103] D. S. Inosov, J. Fink, A. A. Kordyuk, S. V. Borisenko et al., Phys. Rev. Lett. 99,237002 (2007).

[104] J. Chang, S. Pailhes, M. Shi, M. Mansson et al., Phys. Rev. B 75, 224508 (2007).

[105] W. Zhang, G. Liu, L. Zhao, H. Liu et al. Phys. Rev. Lett. 100, 107002 (2008).

[106] D. S. Inosov, R. Schuster, A. A. Kordyuk, J. Fink et al., Phys. Rev. B 77, 212504(2008).

[107] D. S. Inosov, Ph-D Thesis (2008).

[108] W. Cochran, The Dynamics of Atoms in Crystals, Edward Arnold Publishers(London, 1973).

[109] G. Shirane, Rev. Mod. Phys. 46, 437 (1974).

[110] Methods of Experimental Physics: Neutron Scattering, edited by K. Skold andD. L. Price, Academic Press Inc. (London , 1986).

[111] S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, OxfordScience Publications - Clarendon Press, (Oxford, 1987).

[112] L. Pintschovius and W. Reichardt, in Neutron Scattering in Layered Copper-Oxide Superconductors, edited by A. Furrer, volume 20, page 165 (Kluwer Aca-demic, Dordrecht, 1998).

[113] L. Pintschovius, N. Pyka, W. Reichardt, et al., Physica C 185, 156.

[114] S. L. Chaplot, W. Reichardt, L. Pintschovius, and N. Pyka, Phys. Rev. B 52,7230 (1995).

[115] R. J. McQueeney, Y. Petrov, T. Egami, et al., Phys. Rev. Lett. 82, 628 (1999).

[116] L. Pintschovius and M. Braden, Phys. Rev. B 60, 15039 (1999).

[117] J. Graf, M. D’Astuto, P. Giura, A. Shukla, et al., Phys. Rev. B 76, 172507 (2007).

[118] M. D’Astuto, G. Dhalenne, J. Graf, M. Hoesch, et al., Phys. Rev. B 78, 140511(R)(2008).

[119] L. Pintschovius, D. Reznik, W. Reichardt, Y. Endoh et al., Phys. Rev. B 69,214506 (2004).

[120] O. Rosch and O. Gunnarson, Phys. Rev. Lett. 93, 237001 (2004).

[121] M. Vojta, T. Vojta, and R. K. Kaul, Phys. Rev. Lett. 97, 097001 (2007).

148 BIBLIOGRAPHY

[122] V. Hinkov, P. Bourges, S. Pailhs, Y. Sidis, et al., Nature Phys. 3, 780 (2007).

[123] S. I. Mukhin, A. Mesaros, J. Zaanen, and F. V. Kusmartsev, Phys. Rev. B 76,174521 (2007).

[124] R. Citro, S. Cojocaru, and M. Marinaro, Eur. Phys. J. B 63, 179 (2008).

[125] G. Khaliullin and P. Horsch, Phys. Rev. B 54, R9600 (1996).

[126] R. Peierls, Proc. Phys. Soc. London, A 49, 72 (1937).

[127] N. F. Mott, Proc. Phys. Soc. London, A 62, 416 (1949).

[128] P. Fulde, Electron Correlations in Molecules and Solids, Springer (Springer Seriesin Solid State Sciences 100), New York (1995).

[129] J. Hubbard, Proc. R. Soc. London, A 281, 401 (1964).

[130] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).

[131] J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).

[132] E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).

[133] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68,13 (1996).

[134] W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970).

[135] M. C. Gutzwiller, Phys. Rev. 134, 923 (1964).

[136] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1965).

[137] T. Holstein, Ann. Phys 8, 325 (1959).

[138] T. Holstein, Ann. Phys. 8, 343 (1959).

[139] C. Castellani, G. Kotliar, R. Raimondi, M. Grilli, Z. Wang, and M. Rozenberg,Phys. Rev. Lett.69, 2009 (1992).

[140] R. Raimondi and C. Castellani, Phys. Rev. B (R) 48, 11453 (1993).

[141] Z. B. Huang, W. Hanke, and E. Arrigoni, Phys. Rev. B 68, 220507 (2003).

[142] J. Keller, C. E. Leal, and F. Forsthofer, Physica B 206 & 207, 739 (1995).

[143] E. Koch and R. Zeyher, Phys. Rev. B 70, 094510 (2004).

BIBLIOGRAPHY 149

[144] E. Cappelluti, B. Cerruti, and L. Pietronero, Phys. Rev. B 69, 161101 (2004).

[145] R. Citro and M. Marinaro, Eur. Phys. J. B 20, 343 (2001).

[146] R. Citro, S. Cojocaru, and M. Marinaro, Phys. Rev. B 72, 115108 (2005).

[147] H. Yokoyama and H. Shiba, J. Phys. Soc. Jpn. 56, 3582 (1987).

[148] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 59, 121 (1987).

[149] W. Metzner and D. Vollhardt, Phys. Rev. B 37, 7382 (1988).

[150] G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001).

[151] G. Seibold, F. Becca, and J. Lorenzana, Phys. Rev. B 67, 085108 (2003).

[152] G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986).

[153] J W. Rasul and T. Li, J. Phys. C 21, 5119 (1988).

[154] M. Lavagna, Phys. Rev. B 41, 142 (1990).

[155] E. Arrigoni and G. C. Strinati, Phys. Rev. Lett. 71, 3178 (1993).

[156] E. Arrigoni and G. C. Strinati, Phys. Rev. B 52, 2428 (1995).

[157] P. Ring and P. Schuck, The nuclear many-body problem (Springer-Verlag, NewYork, 1980).

[158] J. P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (The MIT Press,Cambridge, Massachusetts, 1986).

[159] J. Lorenzana and G. Seibold, private communications.

[160] P. Nozieres and D. Pines, The Theory of Quantum Liquids (Addison-WesleyPublishing Company, Inc, ADDRESS, 1990).

[161] D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).

[162] F. Gebhard, Phys. Rev. B 41, 9452 (1990).

[163] J. Lorenzana and G. Seibold, Phys. Rev. Lett. 90, 066404 (2003).

[164] G. Seibold, F. Becca, P. Rubin, and J. Lorenzana, Phys. Rev. B 69, 155113(2004).

[165] G. Seibold and J. Lorenzana, Phys. Rev. Lett. 94, 107006 (2005).

150 BIBLIOGRAPHY

[166] G. Seibold, F. Becca, and J. Lorenzana, Phys. Rev. Lett. 100, 016405 (2008).

[167] A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, arxiv:cond-mat/0806.3385.

[168] A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, unpublished.

[169] J. E. Hirsch, Phys. Rev. Lett. 51, 296 (1983).

[170] J. E. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 50, 1168 (1983).

[171] J. E. Hirsch and D. J. Scalapino, Phys. Rev. B 29, 5554 (1984).

[172] J. E. Hirsch, Phys. Rev. B 31, 6022 (1985).

[173] E. Berger, P. Valasek, and W. von der Linden, Phys. Rev. B 52, 4806 (1995).

[174] A. Dobry, A. Greco, J. Lorenzana, and J. Riera, Phys. Rev. B 49, 505 (1994).

[175] A. Dobry, A. Greco, J. Lorenzana, J. Riera, and H. T. Diep, Europhys. Lett. 27,617 (1994).

[176] J. Lorenzana and A. Dobry, Phys. Rev. B 50, 16094 (1994).

[177] M. Capone, M. Grilli, and W. Stephan, Eur. Phys. J. B 11, 551 (1999).

[178] J. K. Freericks and M. Jarrell, Phys. Rev. Lett. 75, 2570 (1995).

[179] M. Capone, G. Sangiovanni, C. Castellani, C. Di Castro, and M. Grilli, Phys.Rev. Lett. 92, 106401 (2004).

[180] W. Koller, D. Meyer, Y. Ono, and A. C. Hewson, Europhys. Lett. 66, 559 (2004).

[181] W. Koller, D. Meyer, and A. C. Hewson, Phys. Rev. B 70, 155103 (2004).

[182] G. S. Jeon, T.-H. Park, J. H. Han, H. C. Lee, and H.-Y. Choi, Phys. Rev. B 70,125114 (2004).

[183] G. Sangiovanni, M. Capone, C. Castellani, and M. Grilli, Phys. Rev. Lett. 94,026401 (2005).

[184] G. Sangiovanni, M. Capone, and C. Castellani, Phys. Rev. B 73, 165123 (2006).

[185] M. L. Kulic and R. Zeyher, Phys. Rev. B 49, 4395 (1994).

[186] R. Zeyher and M. L. Kulic, Phys. Rev. B 53, 2850 (1996).

[187] P. Barone, R. Raimondi, M. Capone, C. Castellani, and M. Fabrizio, Phys. Rev.B, 73, 085120 (2006).

BIBLIOGRAPHY 151

[188] F. Gunther, G. Seibold, and J. Lorenzana, Phys. Rev. Lett. 98, 176404 (2007).

[189] G. Sangiovanni, O. Gunnarson, E. Koch, C. Castellani, and M. Capone, Phys.Rev. Lett. 97, 046404 (2006).

[190] E. Muller-Hartmann, Z. Phys. B 74, 507 (1989).

[191] Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun(Dover Publications, New York, 1965).

[192] E. Koch, Phys. Rev. B 64, 165113 (2001).

[193] G. Seibold, F. Becca, F. Bucci, C. Castellani, C. Di Castro, and M. Grilli, Eur.Phys. J. B 13, 87 (2000).

[194] H. Shiba, Phys. Rev. B 6, 930 (1972).

[195] H. J. Schulz, Phys. Rev. Lett. 64, 2831 (1990).

[196] P. B. Littlewood and V. Heine, J. Phys. C: Solid State Phys. 14, 2943 (1981).

[197] M. Vekic, R. M. Noack, and S. R. White, Phys. Rev. B 46, 271 (1992).

[198] M. Fabrizio, A. Parola, and E. Tosatti, Phys. Rev. B 44, 1033 (1991).

[199] G. Sangiovanni, O. Gunnarsson, E. Koch, C. Castellani, and M. Capone, Phys.Rev. Lett. 97, 046404 (2006).

[200] W. Kohn, Phys. Rev. Lett. 2, 393 (1959).

[201] E. J. Woll Jr. and W. Kohn, Phys. Rev. 126, 1693 (1962).

[202] Earlier experimental evidence in B. N. Brockhouse, K. R. Rao, and A. D. B.Woods, Phys. Rev. Lett. 7, 93 (1961).

[203] B. N. Brockhouse, T. Arase, G. Caglioti, A. D. B. Woods, Phys. Rev. 128, 1099(1962).

[204] Y. Nakagawa, A. D. B. Woods, Phys. Rev. Lett. 11, 271 (1963).

[205] R. Stedman, L. Almqvist, G. Nilsson, and G. Raunio, Phys. Rev. 162, 545,(1967).

[206] R. Stedman, L. Almqvist, and G. Nilsson, Phys. Rev. 162, 549 (1967).

[207] See e.g. G. D. Mahan, Many-Particle Physics, Kluwer Academic Plenum Pub-lishers, New York (2000).

152 BIBLIOGRAPHY

[208] F. Becca, M. Tarquini, M. Grilli, and C. Di Castro, Phys. Rev. B 54, 12443(1996).

[209] G. Seibold and J. Lorenzana, Phys. Rev. B 69, 134513 (2004).

[210] G. Seibold and J. Lorenzana, Phys. Rev. B 73, 144515 (2006).

[211] V. I. Anisimov, M. A. Korotin, A. S. Mylnikova, A. V. Kozhevnikov, D. M.Korotin, and J. Lorenzana, Phys. Rev. B 70, 172501 (2004).

[212] J. Lorenzana and G. Seibold, Physica C 460, 271 (2007).

[213] G. Seibold and J. Lorenzana, J. Supercond. Nov. Magn. 20, 619 (2007).

[214] R. Coldea, S. M. Hayden, G. Aeppli, et al., Phys. Rev. Lett. 86, 5377 (2001).

[215] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen,Phys. Rev. Lett. 87, 047003-1 (2001).

[216] J. Lorenzana, G. Seibold, and R. Coldea, Phys. Rev. B 72, 224511 (2005).

[217] P. Sengupta, R. T. Scalettar, and R. R. P. Singh, Phys. Rev. B 66, 144420 (2002).

[218] T. Tohyama, Y. Inoue, K. Tsutsui, and S. Maekawa, Phys. Rev. B 72, 045113(2005).

[219] S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B43, 7942 (1991).

[220] E. von Oelsen, A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, pre-print.

[221] J. Yan, E. A. Henriksen, P. Kim, and A. Pinczuk, Phys. Rev. Lett. 101, 136804(2008).

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