andreas martin lauchli- quantum magnetism and strongly correlated electrons in low dimensions

8/3/2019 Andreas Martin Lauchli- Quantum Magnetism and Strongly Correlated Electrons in Low Dimensions

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Diss. ETH No. 14908

Quantum Magnetism and

Strongly Correlated Electronsin Low Dimensions

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

Doctor of Natural Sciences

presented by

Andreas Martin Lauchli

Dipl. Phys. ETH

born February 8th, 1972

citizen of Remigen (AG)

accepted on the recommendation of

Prof. Dr. T. M. Rice, examiner

Prof. Dr. M. Troyer, co-examiner

Prof. Dr. F. Mila, co-examiner

2002


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Abstract

In this thesis low dimensional strongly correlated electron systems and frustrated

quantum magnets are investigated employing large scale numerical simulations.

The difference between doping of lithium and zinc in undoped two leg spin ladders

is discussed. While zinc dopants induce local moments and suppress the spin gap,

weak Lithium doping is predicted to maintain a stable spin gap. The relevance of

our results to recent Zn and Li impurity experiments in the underdoped cuprates

is discussed. New experiments are proposed.

A novel numerical approach to the understanding of the strong coupling fixed points

of perturbative Renormalization Group (RG) treatments is introduced and its ap-

plicability to different low dimensional models is demonstrated. The method is

a combination of a standard perturbative RG treatment followed by a numerical

analysis of the flow to strong coupling by exact diagonalization methods. For sys-

tems such as the one dimensional g1g2 model or the two leg Hubbard ladder athalf filling, good agreement with existing analytical predictions is found. Future

applications to the two dimensional Hubbard model are outlined.

The phase diagram of a spin ladder with cyclic four spin exchange has been in-

vestigated. The phase diagram is surprisingly rich. In addition to conventional

phases such as the rung singlet phase, the ferromagnetic phase and the dimerized

phase, two more exotic phases with strong chiral correlations are found. One shows

long range order in the staggered scalar chirality, while the other has short range

order in the staggered vector chirality. First results for the square lattice indicate

that a phase with long range order in the staggered vector chirality is stabilized for

strong cyclic four spin exchange. The influence of four spin exchange on the magnon

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dispersion is determined and compared to recent experiments on La2CuO4.

Finally the phase diagram of a generalized Shastry-Sutherland model is reported.

We find two different Neel ordered phases, two short range ordered resonating va-

lence bond phases (with strong dimers or strong plaquette singlets) and along the

standard Shastry-Sutherland line a valence bond crystal phase with long range order

in plaquette singlet correlations, thereby breaking a discrete lattice symmetry.

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Zusammenfassung

In der vorliegenden Doktorarbeit werden stark korrelierte Elektronensysteme und

frustrierte Quantenmagnete in niedrigen Dimensionen mittels Computersimulatio-

nen untersucht.

Der Unterschied zwischen Zink und Lithium-Dotierung in undotierten Spinleitern

wird erlautert. Zink Storstellen induzieren lokale magnetische Momente und un-

terdrucken dadurch die Spinanregungslucke. Im Gegensatz dazu lasst die Dotierung

mittels Lithium die Anregungslucke intakt. Die Auswirkungen unserer Resultate

und die Interpretation von Experimenten mit Zink- und Lithium-Dotierung in den

unterdotierten Kupraten werden diskutiert.

Wir stellen eine neue numerische Methode zur Analyse von Renormierungsgrup-

penflussen hin zu starker Kopplung vor. Wir zeigen die Anwendbarkeit der Methode

auf verschiedene niedrigdimensionale Modelle. Der Zugang besteht aus der Anwen-

dung einer storungstheoretischen Renormierungsgruppe auf das System, welche in

einem zweiten Schritt durch die numerische Diagonalisierung des asymptotischen

Flusses erganzt wird. Fur Systeme wie das eindimensionale g1g2 Modell oder diezwei-Bein Hubbard Leiter bei halber Fullung finden wir sehr gute Ubereinstimmung

mit analytischen Berechnungen. Zukunftige Anwendungen auf das zweidimension-

ale Hubbardmodell werden skizziert.

Das erstaunlich reichhaltige Phasendiagramm einer Spinleiter mit Ringaustausch

wurde bestimmt. Zusatzlich zu konventionellen Phasen wie der Sprossen-Singlet

Phase, der ferromagnetischen Phase und einer dimerisierten Phase finden wir zwei

Regionen mit starken chiralen Korrelationen. Eine davon ist besitzt langreichweit-

ige Ordnung in der alternierenden skalaren Chiralitat. Die andere tragt kurzreich-

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weitige vektor-chirale Korrelationen. Erste Resultate fur das Quadratgitter weisen

darauf hin, dass fur grossen Ringaustausch die letztgenannte Phase ordnet. Wir

berechnen den Einfluss des Ringaustauschtermes auf die Dispersion der Magnonen

im Antiferromagneten auf dem Quadratgitter und vergleichen die Resultate mit

La2CuO4 Experimenten.

Zuletzt behandeln wir das Phasendiagram eines verallgemeinerten Shastry-Sutherland

Modells. Wir charakterisieren zwei Neel geordnete Phasen, zwei Resonating Valence

Bond Phasen mit starken Dimer- oder Plaketten-Singlets und einen Valence Bond

Kristall mit langreichweitiger Ordnung in den Plakett-Singlet Korrelationen fur das

normale Shastry-Sutherland Modell.

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Acknowledgements

First of all I would like to thank my advisor Prof. Maurice Rice for the opportunity

to work with him on a number of exciting and challenging physical questions. I

learned a lot about condensed matter physics and the way to look at strongly

correlated electrons.

My thanks also go to Prof. Matthias Troyer for teaching me all the fine details

about powerful algorithms in strongly correlated systems, about C++, generic pro-

gramming and supercomputers.

I am very grateful to Prof. Frederic Mila for accepting to be one of my coreferees

and for many stimulating discussions throughout my PhD time.

Some of the projects have been done in collaboration with other people. I found

it very interesting to work in a collaboration where different approaches to the

same problem meet. I would therefore like to thank Andreas Honecker, Carsten

Honerkamp, Didier Poilblanc, Guido Schmid, Manfred Sigrist, Stefan Wessel and

Steven White for their valuable contributions.

What would life be at the institute without all my colleagues: Malek, Hanspeter,

Jerome, Prakash, Stefan, Mathias, Igor, Samuel, Guido, Arno, Simon, Paolo, Mar-

tin, Fabien and Synge. I thank you all for many stimulating coffee breaks including

discussions about physics and the rest of life, and not to forget the outstanding

Toggeli1 games.

Im especially grateful to my parents. They always provided me strong support

throughout my Studienjahre. Thank you very much.

Finally I would like to thank Johanna. She knows why.

1swiss german: tabletop soccer

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VI


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Contents

1 Introduction 1

1.1 The cuprate high Tc superconductors . . . . . . . . . . . . . . . . . 1

1.2 Frustrated quantum magnets . . . . . . . . . . . . . . . . . . . . . . 4

2 Lithium induced charge and spin excitations in a spin ladder 7

2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Binding energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Magnon-Lithium bound state . . . . . . . . . . . . . . . . . . . . . 14

2.5 Local density of states . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Numerical analysis of Renormalization Group flows 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Mesh in k-space . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 The coupling function . . . . . . . . . . . . . . . . . . . . . 25

3.2.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Test case : a one-dimensional problem . . . . . . . . . . . . . . . . 28

3.4 The two-leg Hubbard ladder at half filling . . . . . . . . . . . . . . 31

3.4.1 Repulsive U - The D-Mott phase . . . . . . . . . . . . . . . 35

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3.4.2 Zoo of insulating phases . . . . . . . . . . . . . . . . . . . . 36

3.5 The Two-patch model . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Quantum magnets with cyclic four spin exchange 45

4.1 Phase diagram of a two leg ladder with cyclic four-spin interactions 46

4.1.1 Rung singlet phase . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.2 Staggered dimer phase . . . . . . . . . . . . . . . . . . . . . 50

4.1.3 Scalar chirality phase . . . . . . . . . . . . . . . . . . . . . . 51

4.1.4 Dominant vector chirality region . . . . . . . . . . . . . . . . 52

4.1.5 Dominant collinear spin region . . . . . . . . . . . . . . . . . 54

4.1.6 Ferromagnetic phase . . . . . . . . . . . . . . . . . . . . . . 54

4.1.7 Phase transitions and universality classes . . . . . . . . . . . 54

4.1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Square lattice with cyclic four spin exchange . . . . . . . . . . . . . 57

4.3 Magnon dispersion of La2CuO4 . . . . . . . . . . . . . . . . . . . . 66

5 Phase diagram of the quadrumerized Shastry-Sutherland model 71

5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Boson operator approach . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1 Dimer-boson approach . . . . . . . . . . . . . . . . . . . . . 74

5.2.2 Quadrumer-boson approach . . . . . . . . . . . . . . . . . . 75

5.3 Exact Diagonalization studies . . . . . . . . . . . . . . . . . . . . . 78

5.4 Shastry-Sutherland model . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Numerical techniques 89

6.1 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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6.1.2 Basis construction, Symmetries . . . . . . . . . . . . . . . . 90

6.1.3 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1.4 Lanczos Algorithm . . . . . . . . . . . . . . . . . . . . . . . 95

6.1.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1.6 Implementation Details . . . . . . . . . . . . . . . . . . . . . 99

6.2 Density Matrix Renormalization Group . . . . . . . . . . . . . . . . 101

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Chapter 1

Introduction

1.1 The cuprate high Tc superconductors

The study of strongly correlated electron systems is one of the most active fields in

condensed matter physics. Since the seminal discovery of high Tc superconductivity

in the cuprates in 1986 [1] steady progress has been made in the understanding of

these strongly interacting systems. But still a consistent theory is lacking. It is

generally believed that the strong Coulomb repulsion inside the two dimensional

CuO2 planes plays an important role.

d-wave

Superconductivity

Fermi Liquid

Non-Fermi Liquid

Pseudogap

NeelOrder

TemperatureT

hole concentration x

Figure 1.1: Schematic phase diagram of the hole doped cuprate high Tc superconductors.

The phase diagram of the hole doped cuprates has the schematic form shown in

Fig. 1.1. The undoped system is a good example of a Mott insulator (i.e. an

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insulating state induced by correlation effects, not by band structure) and exhibits

antiferromagnetic long range order. This Neel order is rapidly destroyed by the

doping of holes and a strange metallic state emerges which goes superconducting

for low enough temperatures. Upon further doping the critical temperature Tc

of the superconducting phase raises as high as 133 K for certain mercury based

compounds. Doping beyond optimal doping reduces Tc again and the material

turns into a conventional metal (Fermi liquid). Superconductivity is not the only

unconventional phenomenon in this phase diagram. Another puzzle is the presence

of a pseudogap in the single particle spectral function within the underdoped region

of the phase diagram. An early high Tc paradigm stated that the key to the solution

of the high Tc puzzle lies in the understanding of the strange normal state properties

of the cuprates. This seems to be true still today.

The broad range of phenomena present in the phase diagram illustrate the difficulty

of developing a consistent theory of the cuprate materials. Parts of the phase

diagram are well understood in their respective framework, but these cease to be

valid for other regions.

Soon after the discovery of the high Tc superconductors it was realized that analyt-

ical treatments alone will not immediately solve the puzzle. The reason for this are

the strong correlations and fluctuations present in the CuO2 planes which render

the usual mean field approaches unreliable. Also perturbative schemes are difficult

to put to work as there is no well defined limit about which to expand. Therefore

numerical simulations became very important tools in this field and algorithms such

as Quantum Monte Carlo, Exact Diagonalization, Density Matrix renormalization

Group and series expansions helped improve our understanding that the basic mod-

els such as the single band Hubbard model or its descendant, the tJ model capturethe essential physics of the cuprates. In this thesis we will mainly use Exact Diago-

nalization (ED) and the density matrix renormalization group (DMRG) algorithm.

These algorithms are presented in chapter 6.

In recent years it has been realized that the presence of impurities in these strongly

correlated materials can actually provide new insights in the properties of the host

system. Beautiful scanning tunnelling microscope (STM) experiments [3] on super-

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conducting BSCCO1 samples were able to measure the local density of states around

individual Zn impurites on an atomic scale. The local density of states exhibited

a d-wave like pattern. A different series of experiments using Nuclear Magnetic

Resonance (NMR) techniques analyzed the effect of Zinc and Lithium impurities

in YBCO [19] in the metallic and the superconducting state. These nonmagnetic

impurities both induce a magnetic moment close to the impurity site. This mag-

netic moment is finally Kondo screened at low temperatures. Our work presented

in chapter 2 was motivated by these fascinating results and we discuss the behavior

of Zinc and Lithium impurities in an undoped Resonating Valence Bond (RVB)

system, the two leg spin ladder.

180 K

95 K

120 K

cT =85 K

Figure 1.2: Evolution of the Fermi surface as a function of temperature in an underdoped

sample. Schematized ARPES results (taken from [7])

As pointed out before the nature of the pseudogap phase is one of the hotly debated

topics today. In this region of the phase diagram the single particle spectral function

develops a gap in certain regions of the Fermi surface despite the fact that the

system is not yet superconducting. Very nice photoemission experiments [6, 4, 5]

revealed a successive destruction of the Fermi surface in underdoped BSCCO as the

temperature is lowered. This is illustrated in Fig. 1.2. For high temperatures the

Fermi surface is intact. As the temperature is lowered extending regions close to

1BSCCO stands for the compound Bi2Sr2CaCu2O8x, YBCO stands for YBa2Cu3O6+x

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to the (, 0) and (0, ) points develop a gap. Finally below the superconducting

transition at 85 K only the d-wave nodal quasiparticles are left. Various theoretical

approaches to this phenomenon have emerged in the meantime. Here we briefly

focus on the results of weak coupling Renormalization Group (RG) calculations for

the twodimensional t

t Hubbard model [38, 37, 40]. These studies suggest a weak

coupling instability towards a state which develops a gap in the aforementioned

regions of the Brillouin zone. Interestingly this state shares much of the physics

of the two leg Hubbard ladder. In order to characterize these RG results more

precisely, we propose and discuss a novel numerical approach for the analysis of the

flow to strong coupling in chapter 3.

1.2 Frustrated quantum magnets

Frustrated (quantum) magnets form another class of strongly interacting electron

systems. In these systems all the elementary interactions between spins cannot

be satisfied simultaneously. Therefore they are called frustrated. The inherent

competition induces strong fluctuations. These in turn can induce unconventional

phases in which no simple magnetic structure such as ferromagnetism or Neel order

is stabilized.

An exciting example with classical spins is the so called spin ice [8]. For example

the Ho2Ti2O7 compound is a magnetic system on the pyrochlore structure. The

Ising anisotropy constrains the spins to point either in or out of the elementary

tetrahedron. The competition between exchange and dipolar interactions dictates a

local structure with two spins pointing in a two spins pointing out. This rule, which

is very similar to the ice rules proposed by Pauling, leads to a highly degenerate

groundstate with a finite entropy per spin at T = 0. This has been confirmed

experimentally.

When turning to fully quantum spins (S = 1/2) a few models with unconventional

behavior are known. For reviews see [9]. The antiferromagnetic Heisenberg model

on the Kagome lattice for example does not seem to order and might also have a

groundstate with extensive entropy. Numerical calculations report a finite triplet

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gap and a large number of singlets below the triplet gap. In the multiple spin

exchange models on the triangular lattice a variety of different phases has been

reported, including a long sought spin liquid without any local order parameter.

The discovery of significant cyclic four spin exchange in the La2CuO4 [54] compound

sparked our interest in the physics of higher order spin interactions. We determine

numerically the phase diagram of a spin ladder and of the square lattice Heisenberg

model with additional four spin interactions. Our results reveal several unexpected

phases and also underline the fact that the physics of multiple spin exchange models

on square geometries is rather different compared to the triangular lattice.

Finally we calculate the phase diagram of a generalized Shastry Sutherland model.

This model has attracted a lot of interest, especially because it is realized in the

magnetic structure of the SrCu2(BO3)2 compound. A open problem was the identi-

fication of the phase between the established Neel order in one limit and the exact

dimer product state in the other limit. We present evidence for a valence bond

crystal phase with plaquette singlets in an intermediate range of parameters.

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Chapter 2

Lithium induced charge and spin

excitations in a spin ladder

In this chapter we investigate properties of strongly correlated systems upon impu-

rity doping. In recent years beautiful experiments demonstrated that the doping of

impurities into strongly correlated systems leads to interesting phenomena which

reveal a lot about the properties of the host material itself. Important advances in

experimental techniques such as NMR and STM lead to these interesting experi-

mental results.

We concentrate on the impurity doping of the high-Tc cuprate superconductors

which is an effective tool to explore the low temperature physics of these strongly

correlated systems.

The similarities or differences observed upon doping non-magnetic zinc (Zn) and

lithium (Li) ions must find their explanations in the nature of the host and in

the peculiarities of each dopant. In the antiferromagnetic (AF) phase of La2CuO4,

Li [11] is far more effective at suppressing AF order than Zn [12], although both enter

the same planar Cu(2) site. Li introduces a hole (due to the difference in formal

valence of Cu2+ and Li+) which is tightly bound since the alloy series La2LixCu1xO4

remains insulating for all 0 < x < 0.5. The rapid destruction of AF order is

attributed to the effect of the bound hole1. On the other hand Zn (which has the

1The detailed mechanism is however still not understood. The skyrmion topological defect

scenario proposed in Phys. Rev. Lett. 77, 3021 (1996) has failed to be reproduced in our extensive

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same valence as Cu) does not destroy AF order up to the percolation threshold.

This has been evidenced in experimental [14] and numerical studies [15].

Strikingly, non-magnetic Zn and Li ions behave similarly in conducting (hole-doped)

YBa2Cu3O6+x (YBCO) by inducing local magnetic moments [16] which sit predom-

inantly on the four nearest neighbor (NN) Cu. They both exhibit static [17] and

dynamic [18] susceptibilities reminiscent of a Kondo-like behavior with a very low

effective temperature in the underdoped samples (TK 2.8 K) [19]. Previous cal-culations using a vacant site model for a Zn-dopant found that it acted as a strong

scattering center for holes with even a bound state (hole-Zn bound state) which

could be the source of an effective Kondo coupling [20].

Undoped spin ladders [21] offer an ideal system to investigate doping in a spin liq-

uid or resonating valence bond [22] (RVB) state with short range AF correlations

and a spin gap, which can help the understanding of their two dimensional (2D)

analogs. Zn doping into the (undoped) spin-1/2 Heisenberg two-leg ladder com-

pound SrCu2O3 leads to local moments which form an AF ordered state [23] at

low temperature. Local moments and a rapid suppression of the spin gap were ob-

tained theoretically in a Heisenberg ladder using the vacant-site model [24, 25] for

Zn doping (without additional holes). Further simulations [26] led to an effective

model with coupled local moments with an interaction which decays rapidly with

separation.

Doping Li into a two-leg spin ladder is an open problem both experimentally and

theoretically. Novel physics can be expected due to the additional (with respect to

Zn) hole when Li+ replaces Cu2+. In the following, we use a vacant-site model for

Li+ and show that Li+, unlike Zn2+, does not introduce low-energy spin excitations

but forms a dopant-magnon bound state (BS) just below the spin gap of the undoped

ladder. It follows that, unlike Zn-doped ladders, Li-doped ladders will keep a robust

spin liquid character at low temperature.

calculations.

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(a)

(b)

5

4 4221

53 3

Figure 2.1: Schematic representations of a Li-doped spin ladder. The cross denotes the dopant

site (Li+ ion). The circle stands for the injected mobile hole and the thick lines sketch the attractive

hole potential.(a) Pictorial representation of the groundstate where spins are paired in singlets

(shown shaded). Sites are labelled for convenience. (b) Sketch of the lowest triplet excitation:

dopant-magnon bound state (discussion in the text).

2.1 The model

For dilute concentrations, a single dopant as in Fig. 2.1 suffices. We model a Li+

dopant by an vacant site with a hard-core repulsion for holes and an attractivenearest neighbor (NN) Coulomb potential due to its negative charge with respect

to Cu2+. The Hamiltonian reads:

H = J

(Si Sj 14

ninj) (2.1)

t

,

(ci,cj, + h.c.) V

lI

(1 nlI ) ,

using standard notations and the primed sum is restricted to the NN bonds < ij >notconnected to the dopant. The sum over lI runs over dopant NN sites. Note, for

simplicity, we restrict ourselves to the case of a magnetically isotropic ladder i.e.

with equal rung and leg couplings, J. We use Exact Diagonalisations (ED) of small

periodic ladders (up to 2 12) supplemented by Density Matrix RenormalisationGroup (DMRG) calculations on larger open systems (up to 2 128). In open

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ladders, the dopant is placed in the center.

2.2 Binding energies

First, we investigate the localization of the injected hole versus the strength of the

Coulomb potential. Following Ref. [20], we define the hole-dopant binding energy

as,

S=01 h,1 do p = E0(1h, 1i) + E0(0h, 0i) (2.2)

(E0(1h, 0i) + E0(0h, 1i)) ,

where E0(nh, mi) is the groundstate (GS) energy with n = 0 or 1 (m = 0 or 1)

holes (dopants). It is negative when the hole and Li

+

-ion form a stable BS. SinceLi-doping removes two spins we expect a magnetically inert groundstate i.e. a

singlet (S = 0). As seen in Fig. 2.2(a) a stable bound state is found for almost

all couplings, even when V = 0, but the binding strength increases considerably

with V. Note, the magnitude of the binding energy is slightly larger than in a

2D planar geometry [20]. Fig. 2.1(a) shows schematically how the absence of local

moment can be understood from the RVB nature of the host (all remaining spins

are paired in singlets). Although a single Li-dopant binds the injected hole, caution

is required at finite concentration and the possibility of other decay channels has

to be considered, e.g. 2 holes from 2 dopants recombine into an itinerant hole pair.

This is ruled out since the dopant-hole binding energy is always larger in absolute

value than half of the hole-pair binding energy (see Fig. 2.2(a)) even when V = 0.

Note that the other decay channel consisting of a single dopant trapping two holes

can also be rejected on energetic grounds since the two hole-dopant binding energy

S=02 h, 2 dop

defined as E0(2h, 1i) + E0(0h, 1i)

2E0(1h, 1i) was always found to be

positive. At low concentrations we therefore find decoupled dopant bound states

each with one hole. Since the spatial extent of an isolated BS is small ( = 2 to 4

rungs even when V = 0), the system remains insulating up to large doping.

The binding energies related to the two decay processes in the triplet channel that

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0 0.5 1

J/t1

.2

0.8

0.4

0.0

Binding

en

ergies/t

V/t=0

V/t=0.5

V/t=1

2holes

/2

0 0.5 1

J/t0

.8

0.6

0.4

0.2

0.0

0 0.5 1

J/t0

.1

0.075

0.05

0.025

0

0.02

5

(a)(b) (c)

Figure 2.2: Various binding energies to the dopant (see text for definitions) vs J/t obtained

by ED on a 211 ladder. V denotes the attractive NN potential. (a) BE of the single hole inthe singlet GS. (for comparison, half of the hole-pair binding energy 2holes is also shown). (b)

Binding energy of the hole to the dopant-magnon complex (see equation 2.3) in the lowest triplet

state. (c) Binding energy of the magnon to the hole-dopant complex (see equation 2.3) in the

lowest triplet state.

are plotted in Figs. 2.2(b),(c) are defined as

S=11 h,1 do p = E1(1h, 1i) + E0(0h, 0i) (E0(1h, 0i) + E0(0h, 1i))and as

S 0S = E1(1h, 1i) + E0(0h, 0i) (E0(1h, 1i) + E1(0h, 0i)), (2.3)

where E0 (E1) is the lowest energy in the singlet (triplet) sector.

2.3 Magnetic properties

We now turn to the magnetic properties of the Li-doped spin ladder. We compute

the dynamical spin structure factor S(q, ) for an undoped ladder (Fig. 2.3(a)),

a Zn-doped ladder (Fig. 2.3(b)) and a Li-doped ladder described by Eq. (2.1)

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S(q,

)[a

.u.

]

0 1 2

/J0 1 2

/J0 1 2

/J

q=(0,)

q=(/5,)

q=(2/5,)

q=(3/5,)

q=(4/5,)

q=(,)

x 1/3

x 1/10x 1/10

x 1/3

x 1/20

x 1/6

x 1/6

x 1/3

x 1/3

x 1/3

(a) (b) (c)

Figure 2.3: Spin structure factors S(q, ) calculated on 2 10 ladders. The different curvescorrespond to decreasing q, from q = (, ) (bottom) to q = (0, ) (top). For clarity, reducing

scaling factors (as indicated) are applied on some curves. (a) Undoped periodic ladder; (b) Spin

ladder doped with a single Zn (full line) or two Zn dopants separated by the maximum distance

on the same leg (dashed blue line); The arrow marks the spectral weight which is generated inside

the spin gap. (c) Ladder doped with a single Li dopant with t = 2J and V /t = 0.5. The peak

originating from the bound state mentioned in text is marked with an arrow

(Fig. 2.3(c)). The dynamical structure factor is defined as:

S(q, ) =

n

|n|Szq|0|2 ( n), (2.4)

where these sum runs over all eigenstates n with energy n. The RVB picture, in

which spins are paired up into short range singlets, gives a qualitative understanding

of the magnetic properties. In the undoped ladder, a triplet excitation (magnon)

is well described by exciting a rung singlet into a triplet. Fig. 2.3(a) shows the

single magnon dispersion with a minimum at q = (, ) and = 0S, the spin

gap [27] of the undoped ladder. Introducing a Zn atom on a rung releases a free

spin-1/2 which leads to zero-energy spin fluctuations, predominantly at q = (, )

(Fig. 2.3(b)) and the undoped ladder magnon survives. Two Zn-dopants behave as

two S=1/2 moments with a weak effective exchange interaction, Jeff, which decays

rapidly with separation, in agreement with Ref. [24]. A small spectral weight at

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q = (, ) and low energy ( Jeff) appears below the undoped spin gap results.Li-doping (Fig. 2.3(c)) is drastically different with no weight at small energy. Since

a Li+ dopant has a bound hole, there is no free spin but a new type of excitation

appears just below the unperturbed spin gap - a bound state of a magnon with the

hole/Li+ as naively depicted in Fig. 2.1(b). Its binding energy defined as the energy

difference with respect to the free magnon energy 0S remains in general quite small

(in absolute value) as seen directly in Fig. 2.3(c) (and quantitatively in Fig. 2.2(c)).

Therefore a drastic reduction of the spin gap does not occur in this case. We checked

this conclusion by extending the DMRG calculations of Fig 2.4 to two Li dopants

separated by 64 sites for the case V /t = 1 (Fig. 2.5. In this case the lowest triplet

excitation is a magnon-hole BS strongly localized near one dopant quite different to

the case of two Zn dopants discussed above. The spin susceptibility should remain

activated with only a small reduction in the activation energy in the presence of

Li-dopants unlike the Curie term introduced by the Zn-dopants.

28 32 36

rung

0

0.2

0.4

0 16 32 48 64

rung

0

0.05

0.1

V/t=1.0

V/t=1.3

V/t=2.0

(a) (b)

Figure 2.4: Hole rung density (a) and Sz rung density (b) along the ladder direction in the

lowest energy triplet state calculated by DMRG for J/t = 0.5. The rung density is defined as the

algebraic sum of the densities (if any) on the two sites of a given rung. Different values of V /t (as

indicated) are shown.

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0 50 100

rung

0

0.05

0.1

0.15

Sz

(x)on

rung

0 50 1000

0.1

0.2

0.3

0.4

nh

(x)on

rung

J/t =0.5 , V/t=1

Spin Gap (two dopants) : 0.2412 tSpin Gap (one dopant): 0.2413 tSpin Gap (undoped): 0.2512 t

Figure 2.5: Hole rung density (upper panel) and Sz rung density (lower panel) along the ladder

direction in the lowest energy triplet state for a 2 128 ladder with two Li dopants, calculated byDMRG for J/t = 0.5 and V /t = 1. Each Li dopant confines one hole. The magnon is bound to

one of the Li-hole complexes. The values of the spin gap indicate that there is no drastic reduction

for two Li dopants contrary to the Zn case.

2.4 Magnon-Lithium bound state

The physical origin of this BS is of particular interest. It can be attributed to the

gain in hole kinetic energy associated with the spin triplet. DMRG calculations

give different spatial extents of the charge and spin perturbations. While the hole

is localized on the scale of a few rungs (Fig. 2.4(a)) the rung magnetization can

extend to large distances (Fig. 2.4(b)). This is consistent with our finding that the

binding energies of a hole to a dopant-magnon complex and that of a magnon to a

hole-dopant complex are quite different (Figs. 2.2(b) and 2.2(c)). In fact, increasing

the attraction V binds the hole more strongly and limits the ability of the hole toreduce its kinetic energy by moving in the spin polarized background of the magnon,

hence reducing the binding of the magnon to the dopant-hole complex [28]. Directly

from the binding energy (Fig. 2.2(c)), we can conclude that above a critical value

of V (typically VC/t 2 for J/t = 0.5.) the magnon escapes from the hole-dopantcomplex, as can be seen also in Fig. 2.4(b), where the rung Sz-density for V /t = 2

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indicates an unbound magnon.

2.5 Local density of states

-6

-4

-2

0

/t

-6

-4

-2

0

2

/t

impurity

site

12

2

3 3

4 4

5 5

Figure 2.6: Local DOS around a Li dopant obtained by ED of a 2 9 ladder with J/t = 0.5 andV/t = 0.5. Each panel corresponds to a site in the vicinity of the dopant (site labels correspond

to those of Fig. 2.1(a)). Occupied (empty) electronic states are shaded (left blank).

We calculate also the local density of state (LDOS) near the Li-dopant. The local

density of states is defined as follows:

Ni,i() =

n

|n|ci,|0|2 ( (n 0)), (2.5)

Results are shown in Fig. 2.6 for the spatially resolved DOS spectra. The LDOS can

be measured directly in scanning tunneling microscope (STM) experiments. The

> 0 ( < 0) spectra give the weights of the neutral (charged) target S = 1/2

states accessed by removing the hole (adding an extra hole) to the singlet GS. The

large peak at small positive energy on site # 1 (i.e. on the same rung as the dopant)

corresponds to a local moment (S=1/2)-dopant resonance. Other resonances are

seen at higher energies with larger spatial ranges i.e. 2 or 3 sites on the leg opposite

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to the dopant. Similarly, < 0 resonances are seen when adding an extra hole on

the same leg as the dopant, e.g on sites # 3 and 5. The lowest resonance energy

(in absolute value) is obtained when the second hole is added on sites # 3 next to

the dopant. This might indicate the possibility of a bound state of two holes close

to one impurity2. Note the local > 0 ( < 0) integrated weight provides directly

the local hole (electron) density in the GS. Hence the bound hole is located mainly

on the leg opposite to the dopant and extends roughly over three sites.

2.6 Conclusions

Our theory can be directly tested, if Li can be substituted for Cu in the ladder

compound, SrCu2O3. The extra bound hole around a Li dopant should ensure that

a free local S=1/2 moment is not created, unlike Zn doping. Hence SrCu2xLixO3

should not order antiferromagnetically at low temperature, unlike SrCu2xZnxO3

(Ref. [23]). Further the nature of the magnon-dopant bound state, the charge

distribution and local DOS could also be examined experimentally in these systems.

However, our analysis raises interesting questions regarding the close similarity be-

tween Li and Zn substitution in superconducting YBCO samples [19]. In particular

if we interpret the spin gap phase in underdoped YBa2Cu3O6.6 as a doped d-RVB

phase, then there should be a close similarity to the behavior of the doped lad-

der. However Bobroff et al. [16, 17] report a free S=1/2 moment (which is Kondo

screened only at very low temperatures) for both Zn and Li doping of the un-

derdoped samples. A possible way to reconcile this apparent contradiction is to

postulate that Li+ does not bind a hole in YBa2Cu3xLixO6.6, unlike the case of

La2Cu1xLixO4. This could occur if the mobile O2-ions in the chains were repelled

from the Li+-ions in the planes. A test of this hypothesis can be made by doping

Li+ and Zn2+ in YBa2Cu4O8 which as a stoichiometric compound has no mobile

O-ions. Our analysis then predicts free S=1/2 moments only for Zn-doping and not

for Li-doping in this case.

2In that case we would probably need to include the hole-hole repulsion into the problem as

well

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In conclusion, our analysis predicts a clear distinction between the magnetic prop-

erties of the two non-magnetic ions, Zn2+ and Li+ when doped into spin liquids due

to the binding of a hole in the latter case. Experiments to test these predictions

are proposed.

The content of this chapter has been published in

Phys. Rev. Lett. 88, 257201 (2002)

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Chapter 3

Numerical analysis of

Renormalization Group flows

This chapter is devoted to the presentation and analysis of a new numerical ap-

proach to correlated fermion systems. It is a combination of a weak-coupling renor-

malization group (RG) treatment of the fermionic system followed by a numerical

analysis of the asymptotic flow by an exact diagonalization (ED) scheme. It is

intended to give insights into the strong coupling state starting from the weak

coupling limit.

The outline of this chapter is as follows: In the first section we introduce the nu-

merical scheme in detail. The method is then illustrated with an application to a

one-dimensional system, the Luther-Emery liquid. Next we investigate the weak

coupling phase diagram of the two-leg Hubbard ladder at half-filling. Bosonization

studies revealed that the Hubbard ladder at half-filling can accommodate a large

variety of ordered and quantum disordered phases. All of them are insulating. We

show that the numerical scheme is able to characterize these phases in accord with

the Bosonization treatments. In particular we confirm the simultaneous enhance-

ment of several correlation functions. Finally results for the two-dimensional

two-patch model, a simplified model relevant for the 2D t t Hubbard model arepresented.

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3.1 Introduction

Renormalization group approaches to physical problems are powerful conceptual

and calculational tools. Initially developed for problems in particle physics, the

method was successfully extended to statistical and condensed matter physics. In

the context of strongly interacting electrons a milestone was set by the solution of

the Kondo problem by Wilson [33]. In the spirit of Wilsons ideas the RG method was

subsequently applied to one dimensional conductors. (see Refs. [29, 30, 31, 32, 34]

for reviews.)

In the framework of strongly correlated electrons the Renormalization Group is

often implemented in k-space. One starts with a theory at an initial cutoff1 0 with

bare (initial) couplings gi(0). In the next step one lowers the running cutoff

and integrates out the fermions in the narrow shell between 0 and . The mode

elimination leads to changes in the couplings and may also generate couplings which

were not present at the initial stage. The interest lies in the behavior of the various

coupling constants and susceptibilities as one lowers the cutoff to zero energy,

i.e. to the Fermi surface. This information is contained in the Renormalization

group equations. Depending on the physics of the system we can scale to zero

energy without encountering any singularity, which signifies that the bare system is

attracted to a weak coupling fixed point; or we find a divergency at a finite cutoff

c. This indicates an instability of the initial theory towards a strong coupling

fixed point. In one dimensional models both situations are known to occur (c.f.

section 3.3). For the t t Hubbard model in two dimensions however the couplingsgenerically flow to strong coupling.

In one dimension the perturbative RG approach is on solid grounds. In general

one encounters logarithmically divergent zero incoming momentum particle-particle

and q=2kF momentum transfer particle-hole diagrams. These can be treated consis-

tently in a one loop approach. For flows to weak coupling the RG approach remains

in its domain of validity. For flows to strong coupling the coupling constants leave

the perturbative regime and other methods are needed for a reliable analysis. Due

1The precise definition of the cutoff depends on the chosen approach. It could be a bandwidth

cutoff, a temperature cutoff, etc.

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to the special structure of the low energy excitations in one dimension we can resort

to the bosonization approach, where the fermionic operators are mapped to bosonic

ones. The resulting bosonic theory is weakly interacting and a semiclassical anal-

ysis yields an accurate description of the low energy physics. A different source of

understanding are the few one dimensional models which are exactly solvable. So

for example the standard Hubbard model for general filling and U/t and the tJmodel at J/t = 2, both solvable by Bethe-Ansatz; the Tomonaga-Luttinger liquid

and the Luther-Emery liquid, both solvable by Bosonization.

The discovery of the high-Tc superconductors in 1986 has sparked a tremendous

amount of research on two dimensional strongly correlated electron models, since

it is believed that superconductivity emerges mainly due to electron-electron inter-

actions, not due phonons as in standard BCS theory. Weak coupling approaches

played an important role from the start: By concentrating on the regions around the

van Hove singularities the two patch approach for the Hubbard model was derived

[49, 50]. In recent years improved RG schemes with a higher k-space resolution (N-

patch schemes) have been developed by several groups [35, 36, 37]. These studies

reported various instabilities but all agreed on the fact that the system scales to

strong coupling. In contrast to one dimension one can not apply the bosonization

mapping in two dimension to the cases of our interest. Exact solutions for non-

trivial models are also lacking. Therefore there is a need for an unbiased method

to complement the RG analysis.

3.2 The method

Our approach is built on a numerical investigation of the Hamiltonian which results

from the asymptotic couplings in the RG procedure. The scheme we developed

consists of three steps:

1. A weak coupling Renormalization Group scheme in kspace is implemented.We calculate the flow of the couplings g(k1, k2, k3) as a function of by

integrating the RG equations. The ratios of the couplings close to the crit-

ical scale are determined. In some cases analytical results for the ratios are

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available.

2. We map the coupling function gc(k1, k2, k3) to a Hamiltonian on a finite mesh

of k-points.

3. The discretized problem is diagonalized exactly with a numerical algorithm.

In practice a Lanczos type algorithm is implemented. This limits the maxi-

mum number of k-space orbitals to about 20. Energies and correlation func-

tions are calculated, enabling us to determine the energy gap structure and

different order parameter susceptibilities directly in a fermionic language.

In the following we discuss some details of the setup of the mesh in k-space and the

mapping of the asymptotic couplings to the k-space Hamiltonian.

3.2.1 Mesh in k-space

The first ingredient of an numerical implementation is the mesh of k-points in

reciprocal space. The mesh consists of a number of patches Np which corresponds to

the number of Fermi points in the case of a one dimensional system, or to a N-patch

approximation in 2D systems. In each of the patches we distribute Nppk k-points

with a fixed momentum assigned. This gives a total number ofk-points Nk=NpNppk .

The distribution of the k-points in a 1D setting with two Fermi points is illustrated

in Fig. 3.1. The k-points are chosen below a bandwidth cutoff , distributed in

a uniform way throughout the allowed region. The momenta of the individual k-

points should respect certain relations: when total incoming momentum zero pairing

instabilities could arise we should have pairs with momenta k and k present, andthe common 2kF instabilities should be respected by the presence of momenta k and

k2kF. These requirements are satisfied with the uniform spacing described above.

In the 1D situation illustrated in Fig. 3.1 the degeneracy of the noninteracting

system with Ne = Nk depends on the number of orbitals per patch Nppk . For N

ppk

even (odd) the noninteracting groundstate is not degenerate (is degenerate). This

will sometimes have an influence on the finite size behavior of the gaps for example.

Note that due to the initial RG procedure on the couplings and the bandwidth

cutoff, our approach is not a simple exact diagonalization of a Fourier transformed

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F-k

EF

kF

k

k

Figure 3.1: Schematic representation of the discretized k-space around the Fermi surface in a

1D geometry with two patches.

problem, but actually a simulation of a system of effective length L = 2/k

Nk.

Our calculations are carried out with Ne=Nk for the groundstate sector. The Fermi

energy EF then lies in the middle of the bandwidth . Gaps are calculated with

respect to that state. The actual filling of the parent systems is encoded in the

asymptotic interactions, e.g. by the presence or absence of umklapp processes,

differences in Fermi velocities, etc.

ky

=

E

-kF A

kF A

-kF B

kF B

FE

Fk

y= 0

Figure 3.2: Schematic representation of the discretized k-space around the four Fermi points in

a ladder geometry.

The mesh of k-points for the two leg Hubbard ladder is analogous to the 1D case,

we just have four patches instead of two (Fig. 3.2). Now umklapp processes are

included as well. This is ensured by the following relation on the longitudinal Fermi

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momenta: |kF,A + kF,B | = . The important wavevector at half filling therefore is(, ).

The situation is different for the 2D two patch case, as illustrated in Fig. 3.3. Due

to the restricted number of available orbitals in the numerical calculation together

with the physical requirements on the momenta very few discretization schemes are

possible. We have chosen the two arrangements in Fig. 3.3 for our calculations.

The noninteracting groundstate has a closed shell structure and all desired k-point

relations are satisfied. The mesh is however not a homogeneous refinement and

could therefore pose some difficulties in the finite size scaling process. Qualitative

results should nevertheless be possible.

Figure 3.3: Discretization of the k-space in the two patch model. The left panel is the choice for8 (resp. 12 with points on the Fermi arcs) k-points. The right panel for 16 (resp. 20) k-points. The

grey points show optional points on the Fermi arcs which could be included in future calculations.

The empty points represent folded ( = ) existing k-points.

In our program code we exploit the conservation ofSztot, the number of particles Ne

and the conservation of momentum to reduce the size of the Hilbert space before

the diagonalization. This gives us the additional advantage to resolve energies as

a function of total momentum. The reduction factor for a subsector with fixedmomentum can be important: e.g. for a two leg Hubbard ladder at half filling

with Nk = 16, Ne = 16, Sztot = 0 and total momentum (0, 0) we reduce the size

of the Hilbert space from (C168 )2 = 165636900 down to 3370670. In comparison

to standard real space exact diagonalizations where the number of different total

momenta is equal to the number of orbitals, our special Hilbert space structure

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allows for much more total momentum sectors. In the one dimensional situation

in Fig. 3.1 with Ne=18 our allowed momenta are clustered with a certain width

around specific momenta ranging from 18kF (all particles on the left branch) to+18kF (all particles on the right branch) in steps of 2kF. (transfer of one particlefrom the left to the right branch).

3.2.2 The coupling function

The Hamiltonian which acts on the mesh of k-points is generically of the following

SU(2) invariant form:

H =k,

(k) ck,ck,

+ 12

k1,k2,k3

V(k1, k2, k3),

ck3,ck4,ck2,ck1,. (3.1)

Where (k) denotes the kinetic energy, is a global coupling constant, the total

volume (usually Nk), V(k1, k2, k3) is the discretized coupling function and k4 =k1 + k2 k3 (modulo umklapp) by momentum conservation. The fact that it onlycontains four-fermion terms can be justified by RG arguments2.

The functional dependence of the coupling function V(k1, k2, k3) is basically deter-

mined by the RG couplings close to the critical scale:

V(k1, k2, k3) = g[c] (Patch(k1), Patch(k2), Patch(k3)) ; (3.2)

Here g[c] denotes the ratios of the diverging couplings and Patch(k) assigns a

patch index to every k-point. If we discretize the coupling function this way we will

generate O(N3k ) different scattering processes. As this number can become quite

large even for small Nk we would like to minimize the number of coupling processes3.

One possible way to reduce the number of couplings is to retain only those processeswhich exactly satisfy the momentum relations of the suspected instabilities. In a

certain sense this amounts to solving an effective reduced Hamiltonian similar to

2The processes with higher order fermionic interactions have a different scaling dimension and

are therefore irrelevant.3For a two leg Hubbard ladder at half filling with Nk = 16 we have 1200 different couplings

in the D-Mott phase.

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k1,

k2,

k ,

k ,4

3

Figure 3.4: The elementary scattering vertex V(k1, k2, k3). Two incoming particles with mo-menta and spin (k1, ) and (k2,

) are scattered to (k3, ) and (k4, ). Momentum is conserved

by k4 = k1 + k2 k3 (modulo umklapp).

mean-field theories, e.g. the reduced BCS Hamiltonian only scatters pairs with

total momentum exactly zero. In a 1D setting we would then only keep processes

with a) k2 = k1 b) k3 = k1 2kF and c) k4 = k1 2kF after the reduction. Thislimits the number of processes to O(N2k ). We have noted in the application of the

present scheme that the finite size behavior of gaps and structure factors depends tosome extent on the discretization of the couplings. For the one dimensional system

in section 3.3 the reduced number of couplings gave results in good agreement

with the theoretical expectations. For the two leg ladder case, where we have an

insulating, fully gapped state without quasi-long range order, we had to include

all the couplings in order to obtain stable, finite gaps. With the restricted set of

couplings the spin and the two particle gap would scale to zero. The single particle

gap however was stable even for the reduced set. These observations might reflectthe fact that the 1D system has a single dominant correlation function, whereas the

two leg ladder has a spin liquid groundstate with several equally dominant short

range correlations.

3.2.3 Observables

As in the well-known real space ED calculations we can measure finite size expec-

tation values of almost any observable. In our approach we are mainly interested

in energy gaps and structure factors related to several types of orders. The energy

gaps we determine are defined as follows:

Spin Gap:

(S=1)(Nk) = E0(Nk, Ne, 1) E0(Nk, Ne, 0) (3.3)

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Single particle gap:

1p(Nk) =1

2

E0(Nk, Ne + 1,

1

2) + E0(Nk, Ne 1, 1

2)

E0(Nk, Ne, 0) (3.4)

Two particle gap:2p(Nk) =

1

2(E0(Nk, Ne + 2, 0) + E0(Nk, Ne 2, 0))

E0(Nk, Ne, 0) (3.5)

where E0(Nk, Ne, Sz) denotes the groundstate energy of the discretized system for a

fixed number of k-points Nk, fixed number of particles Ne and total magnetization

Sz. The energies have to be measured in the appropriate total momentum sector,

but the momentum of the gap is model dependant.

The next observables are the structure factors associated to different order param-

eters. The particle-hole (p-h) instabilities with momentum q and form-factor fA(k)

are defined as follows:

Spin singlet channel :

OACDW =

1

Nk k, fA(k) ck,ck+q, (3.6)

Spin triplet channel (z-component) :

OASDW = 1Nk

k,

1

2fA(k) c

k,ck+q, (3.7)

whereas the particle-particle (p-p) instabilities are defined as:

OASC =

1

Nk k fA(k) c

k,

c

k,

(3.8)

In the singlet channel the particle-hole order parameters correspond to standard

charge density wave (CDW) instabilities for fA(k) 1 (s-wave). The triplet ana-logue corresponds to a spin density wave (SDW). The higher angular momentum

analogues will be discussed in more detail in section 3.4.

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In order to measure the structure factors in the ED approach one calculates the

groundstate |0 and subsequently applies the appropriate operator on the ground-state:

SX = 0|OXOX |0 = |OX |0|2 (3.9)It is also possible to calculate dynamical response functions (e.g. single particle

spectral functions or dynamical spin structure factors) with the continued fraction

method in the present scheme.

3.3 Test case : a one-dimensional problem

Let us illustrate our approach on a simple one-dimensional problem: the 1D Fermi-

gas with two Fermi points. The relevant couplings are labeled g1 to g4 (illustrated

in Fig. 3.5). The g1 processes denote backscattering processes. The g2 and g4

processes are of the forward scattering type. Finally the g3 processes are so called

umklapp processes which violate momentum conservation in general, but are allowed

at special fillings, e.g. at half filling. In the following we consider a system at a

g1

g3

g2

-k k FF

-k kF F-k k

k-k F F

FF

g4

Figure 3.5: The g-ology of the 1D spinful Fermi gas with two Fermi points. g1 denotes backscat-

tering processes, g2 forward scattering, g3 umklapp scattering and g4 chiral forward scattering

processes.

generic filling (away from half filling) and we therefore neglect the g3 coupling. For

simplicity we also discard the g4 processes. Their effect is more on a quantitative

level, they renormalize velocities but are not expected to change the overall phase

diagram.

The phase diagram of the so called g1g2 model has been the subject of many studiesin the 1970s. Reviews can be found in [29, 34]. The one-loop renormalization group

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g1=2g2

g2

g1

sSC CDW

SDWpSC

Figure 3.6: Phase diagram of the g1g2 model. The phases are characterized by the leadingalgebraic correlation function. The two phases for g1 > 0 are gapless. The two regions with g1 < 0

flow to strong coupling and develop a spin gap.

equations for the g1g2 model have been derived as follows:

g1 = 1

g21

g2 = 12

g21, (3.10)

with g = dg/dl and l = ln(/0) +. These equations have a solution in

closed form:

g1(l) =g1(0)

1 + g1(0)l

g1(l) 2g2(l) = const. (3.11)

The flow of the couplings thus depends on the sign of the initial coupling g1(0).

If g1(0) > 0 (repulsive backscattering) we scale to a weak coupling fixed point:

the Luttinger liquid [g1() = 0, g2() = g2(0) 12g1(0)]. If however g1(0) < 0

(attractive backscattering) the weak coupling fixed point is unstable and we flow

to strong coupling [g1, g2 ]. Using bosonization it has been shown that thestrong coupling fixed point develops a spin gap. The weak coupling fixed point

has no gaps. The leading correlation functions have been determined; the resulting

phase diagram is shown in Fig. 3.6.

As an example we calculate the various gaps in the lower left region of the phase

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0 0.5 1 1.5 2

0

1

2

3

4

5

6

G

ap/

spin gap

0 0.5 1 1.5 2

0

1

2

3

4

5

6

g1=-2, g

2=-2

two particle gap

nk=8

Nk=10

N=12

0 0.5 1 1.5 2

0

1

2

3

4

5

6

single particle gap

Figure 3.7: Gaps as a function of and system size in the Luther-Emery part (dominant

superconductivity correlations and a finite spin and one particle gap) of the g1g2 phase diagram.

diagram, i.e g1 < 0, g2 < g1/2. The dominant correlation function is s-wave sin-

glet pairing. For this state we expect a finite spin and single particle gap but no

two particle gap in analogy to a superconducting state. Our numerical results are

shown in Fig. 3.7. The different curves in each panel represent different numbers of

k-points. The horizontal axis denotes the interaction strength . This parameter

allows us to tune between the noninteracting limit and the fully interacting limit,

where kinetic energy plays almost no role anymore. The evolution of the gaps as a

function of indicates also where the finite size effects due to the discretization be-

come unimportant. Note that the Nk=10 system has zero gaps due to a degenerate

groundstate at =0. The finite size behavior of the gap curves strongly suggests a

finite spin and single particle gap, while the two particle gap scales to zero. At the

present stage we do not attempt to measure the gaps quantitatively, but we merely

determine the qualitative gap signature. We have also calculated the gaps in the

other regions of the phase diagram. It turned out that the results for g1 > 0 are

less regular than for g1 < 0. This could be related to the fact that for the latter

case a large spin gap develops.

We calculated also the structure factors corresponding to charge density wave

(CDW), spin density wave (SDW), singlet superconductivity (sSC) and triplet su-

perconductivity (pSC). The results in Fig. 3.8 display dominant correlation func-

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CDW SDW sSC pSC0

2

4

6

8

10

Structure

Factors

[a.u.]

g1=-2, g

2=-2

CDW SDW sSC pSC0

2

4

6

8

Structure

Factors

[a.u.]

g1=2, g2=-2

Nk=8

Nk=10

Nk=12

Nk=14

CDW SDW sSC pSC0

2

4

6

8

10g

1=-2, g

2=2

CDW SDW sSC pSC0

2

4

6

8g1=2, g2=2

Figure 3.8: Structure factors of different orderparameters for the g1g2 model at four differentpoints in the phase diagram. The phases in the two upper panels scale to weak coupling in the

RG process (g1 > 0), while the phases in the lower panels scale to strong coupling region (g1 < 0).

The dominant correlation function agrees with phase diagram in Fig. 3.6. The finite size behavior

is much more regular in the spin gapped strong coupling phases.

tions consistent with the phase diagram in Fig. 3.6. The finite size behavior is very

regular for the two phases with g1 < 0. For the phases with g1 > 0 we detect a

systematic difference between the systems with and without orbitals at the Fermi

energy.

We conclude from the application of our scheme to the 1D g1g2 model that theresults are consistent with the analytical results. Especially in the phases where

the RG flow diverges to strong coupling we get good agreement.

3.4 The two-leg Hubbard ladder at half filling

The two leg Hubbard ladder is an interesting system on the path from one-dimensional

to two-dimensional physics. At half filling the system is insulating (finite charge

gap, incompressible) for any U > 0. In the strong coupling limit (U t) the modelmaps to a two leg spin ladder (also an insulator) with antiferromagnetic exchange

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interaction (J t2/U). This system is a nice realization of a resonating valencebond (RVB) spin liquid state in the spirit of Andersons early proposal [22]. It

shows exponentially decaying spin correlations and an excitation gap to singlet and

triplet modes. The strong coupling limit is well understood, in particular in the

limit where the coupling on the rung is strongest. The groundstate is then well

approximated by a rung singlet product state. We call such a state with a charge

gap and short range spin correlations an Insulating Spin Liquid (ISL).

Our present interest in the two leg ladder is twofold. First we want to test our

numerical scheme on a different system where reliable results are available. As we

will see the variety of potential phases arising in the weak coupling phase diagram

is fascinating. The fact the we will encounter phases without power-law correlation

functions (only short range order) will mark a difference to the 1D model discussed

before. The second motivation comes from recent results of N-patch RG calcula-

tions of the 2D tt Hubbard model. [37]. The properties of the flow to strongcoupling in the region between antiferromagnetism and d-wave superconductivity

were reminiscent of the two leg Hubbard ladder and a scenario based on the forma-

tion of an ISL in parts of the Brillouin zone was put forward [37, 40]. Our aim is to

characterize the relevant phases of the ladder model within the present numerical

scheme in order to compare to the more complex 2D models later on.

In the following we focus again on the limit of weak coupling (U t). This limithas been discussed in detail in a series of publications [42, 40, 43, 44, 45]. Here

we just give a brief outline of the weak coupling RG results before we compare our

numerical results to the analytical predictions.

Let us first discuss the noninteracting starting point. The band structure of the

nearest neighbor tight binding model on the two leg ladder reads:

(kx, ky) = t cos(ky) 2t cos(kx), ky {0, } (3.12)

where t (t) denotes the hopping amplitude on the legs (rungs). The dispersion is

plotted in Fig 3.9. At half filling and for t/t < 2 both bands are partially filled.

There are four Fermi points present. The Fermi velocities on the two bands are

equal at half filling and the Fermi wavevectors kAF, kBF add up to .

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-1 -0.5 0 0.5 1

kx/

-4

-2

0

2

4

(k

x,

ky

)/t

ky=

ky=0

-kF

Ak

F

A

kF

B-k F

B

Figure 3.9: Noninteracting dispersion of a two leg Hubbard ladder with t/t = 1 at half filling.

kAF + kBF add up to . The Fermi velocities are equal on all four branches (valid for t/t < 2).

The relevant processes4 in the g-ology of the two leg Hubbard ladder at half filling

are sketched in Fig. 3.10 (notation according to [39, 40]). The g3 couplings denote

umklapp processes, i.e processes where the total incoming and the total outgoing

momenta differ by a reciprocal lattice vector. We discard the completely chiral g4

processes in our analysis because they are not important on a qualitative level, i.e.

their main effect is to renormalize velocities.

The one loop RG equations at half filling have the following form [42, 39]:

g1x = g21x + g2yg1p + g

23x g3xg3c

g2x =1

2

g21x + g

22y + g

21p g23c

g3x = 2g1xg3x g1xg3c g2xg3x g2eg3x g2yg3eg3c = 2g1eg3c + g1pg3e g1eg3x g2yg3e g2eg3c g2xg3cg2y = g1xg1p + g2yg2x g2yg2e g3eg3xg1p = 2g1eg1p + g1xg2y

g1eg2y + g1pg2x

g1pg2e + g3eg3c

g3eg3x

g1e = g21e + g

21p g2yg1p + g23c g3cg3x

g2e =1

2

g21e g23e g23x g22y

4Here we call a process relevant if is logarithmically divergent in second order perturbation

theory. The solution of the RG equation will show wether the process is also relevant in the RG

sense.

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g1x

g2x

g3e

g1e

g2e

g3c

g1p

g2y

g3x

Figure 3.10: Coupling constants labelling the relevant processes connecting the four Fermi points

at half filling. The g3s are umklapp processes. The couplings g1x, g2x, g1p and g2y denote cooper

processes, couplings g3e, g2e, g2y and g3x are SDW type processes and finally g3e, g1e, g3c and g1p

are CDW type processes

g3e = 2g1pg3c g1eg3e g2yg3c g1pg3x g2yg3x 2g2eg3e, (3.13)

where g = dg/d and we decrease from an initial cutoff 0.

For general initial couplings gi(0) these RG equations are too complicated for an

analytical solution and we need to perform a numerical integration. It is possible

however to obtain solutions in closed form for special values of the initial couplings.

The ansatz

gi() =gi(0)

log(/c)(3.14)

is a solution of 3.13, provided that all gis and gis are replaced by gi(0) and the

gi(0) solve the resulting algebraic system of equations. c denotes the critical scale

where all the couplings diverge. It depends on the initial couplings and scale. Note

that these solutions dictate the initial couplings at the scale 0. Surprisingly the

numerical integration generally yields divergencies which are well captured by this

ansatz. In particular the couplings diverge with fixed ratios. It is the task of the

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numerical integration to decide for which initial couplings we flow to which special

solution of the above form.

3.4.1 Repulsive U - The D-Mott phase

10-12

10 -9

10 -6

-20

2

4

6

g

/t

/ t

Figure 3.11: Flow of the nine coupling constants of the two leg Hubbard ladder at half filling for

purely repulsive initial couplings. The flow diverges at a critical scale c 0.5 1012 t. The dark,

dashed couplings are the d-wave pairing type, the dark, solid couplings the AF couplings and thelight, dashed couplings are d-density wave couplings diverging more slowly than the others. The

initial couplings at 0 = 0.5 t were 0.1 t. (Plot taken from [40])

In Fig. 3.11 we plot the flow of the couplings for purely repulsive initial couplings

[gi(0) = U] as would be the case for a simple repulsive Hubbard model. Seven out

of nine couplings diverge at a critical scale c. The remaining two couplings diverge

not as fast as the others. The ratios of the diverging couplings for this particular

flow are as follows:

g1x g1e g1p g2x g2e g2y g3e g3c g3x

-1 0 1 -1/2 1/2 1 1 0 1

The fixed ray was characterized using Bosonization in [42]. The resulting phase has

only short range order, no power-law correlations. Nevertheless some correlation

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functions are enhanced, while others are suppressed with respect to the noninteract-

ing limit. In the case at hand enhanced spin density wave correlations and d-wave

superconducting correlations are predicted. The staggered flux correlations (dCDW

or d-density wave) are expected to be enhanced as well. All gaps in this phase have

been predicted to be same, based on a dynamically generated SO(8) symmetry. The

phase was termed D-Mott in [42].

We have again diagonalized the corresponding fermionic Hamiltonian and deter-

mined the gap structure. Our results are show in the uppermost panel of Fig. 3.12.

Interestingly the spin and the two particle gap are identical, while the single particle

gap is slightly larger. On the level of our expected accuracy we consider them to

be the same. The spin and the two particle gap show small finite size corrections,

but are clearly consistent with a finite gap as Nk

.

We then calculated the s-wave and d-wave component of the charge and spin density

wave and the pairing correlations. The resulting structure factors are shown in the

upper left plot in Fig. 3.14. We indeed find enhanced sSDW, dCDW and dSC

short range correlations as expected. The size dependence of the structure factors

is significantly reduced compared to the g1g2 model where the correlations werequasi long ranged.

3.4.2 Zoo of insulating phases

The general weak coupling phase diagram of the two leg Hubbard ladder at half

filling has attracted much interest in recent years. Lin, Balents and Fisher [42]

reported new phases in addition to the D-Mott phase discussed before: an ordered

charge density wave state (CDW), an ordered state with staggered circulating or-

bital currents (dCDW) 5 and a s-wave analog to the D-Mott phase (S-Mott). The

dCDW phase is also known as a staggered flux phase or orbital antiferromag-

net and can be viewed as a charge density wave state with angular momentum

l = 2 (d-density wave). It is also intensively discussed as a candidate ground-

state in the pseudogap region of the underdoped cuprates. In two recent preprints

5This phase was misinterpreted in the original paper, but later identified correctly in [43]

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[44, 45] the zoo of insulating phases at half filling has been expanded by other den-

sity wave states with non-zero angular momentum [pCDW ( l = 1) and fCDW

( l = 3)] and two quantum disordered superconductors (S-Mott and D-Mott),

which presumably differ from the unprimed states in topology.

For the first four phases the asymptotic coupling ratios have been determined in

[42] and we recast them in our notation in table 3.1. It was noticed by Lin, Balents

and Fisher that for a large set of initial couplings the flow is attracted to a manifold

with enhanced dynamical symmetry. In particular the four fixed rays in table 3.1

correspond each to an integrable SO(8) Gross-Neveu model. This allows a detailed

characterization of the phases.

Phase g1x g1e g1p g2x g2e g2y g3e g3c g3x

D-Mott -1 0 1 -1/2 1/2 1 1 0 1S-Mott -1 0 -1 -1/2 1/2 -1 -1 0 1

sCDW 0 -1 -1 0 0 0 -1 -1 0

dCDW 0 -1 1 0 0 0 1 -1 0

Table 3.1: Ratios of the coupling constants for the four dominant fixed rays on the

SO(5) manifold.

We map the asymptotic couplings to a discrete lattice in k-space with 8, 12 and

16 k-points. This corresponds to 2,3 and 4 k-points per patch. The noninteracting

( = 0) groundstate is nondegenerate for the systems with 8 and 16 k-points. We

calculate the gap structure in each phase by plotting the evolution of the three

gaps as a function of . The parameter allows us to follow the gaps between

the noninteracting limit and the limit where kinetic energy is not so important

anymore. The results are shown in Fig. 3.12. A common feature is that for each

phase the triplet gap and the two particle gap are equal. The two disordered and

the two ordered states have exactly the same energy gaps respectively. The gaps in

the first two phases (D-Mott and S-Mott) are small, but remain finite for Nk .This can easily be inferred in our plots. According to the SO(8) symmetry of the

asymptotic Hamiltonian all three gaps in these two phases should be of equal size.

That is however not verified exactly in our calculations, as the single particle gap is

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slightly larger than the other gaps. The reason for this could be attributed to our

discretization scheme. It might not respect all of the the nonlocal SO(8) symmetry

elements of the Hamiltonian.

The ordered phases have very clear-cut gaps with small finite size corrections. The

spin gap and two particle charge gap are roughly twice as large as the single parti-

cle gap. In addition we find that the groundstate is twofold degenerate: the lowest

singlets at momentum (0, 0) and (, ) have exactly the same energy. Such de-

generacies are expected for a spontaneous translation symmetry breaking scenario

(Z2 symmetry breaking). In other situations the two states are very close in en-

ergy because of the finite tunneling amplitude in finite systems but are not exactly

degenerate like in the present case.

0 1 20

2

4

6

CDW

0 1 20

2

4

6

dCDW

0 1 20

1

2

3

S-Mott

0 1 20

1

2

3

D-Mott

Nk=8

Nk=12

Nk=16

Spin Gap /

0 1 2

Interaction Strength

0 1 2

0 1 2

0 1 2

Two Particle Gap /

0 1 2

0 1 2

0 1 2

0 1 2

Single Particle Gap /

Figure 3.12: Finite size scaling of the three relevant gaps in the four phases on the SO(5)

manifold discussed by Lin, Balents and Fisher. The finite size scaling of the gaps in the quantum

disordered phases (D-Mott and S-Mott) is consistent with small but finite gaps in the limit Nk . The two ordered phases display very clear gaps with only small finite size corrections. Contraryto the first two phases the ordered phases have a twofold degenerate groundstate, i.e. a singlet at

momentum (, ) is degenerate with the lowest state at (0, 0). This is another evidence for long

range order.

Let us now investigate the different correlation functions. We first discuss the

particle-hole instabilities with momentum q=(, ) and different formfactors fA(k).

The values of the formfactors fA(k) only depend on the patch index (see Fig. 3.13).

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The s-wave channel corresponds to the well-known charge density wave (CDW)

and spin density wave (SDW) order parameters. The higher angular momentum

analogues are also well known by now. The d-density wave or staggered flux

instability corresponds to the CDW operator with d-wave formfactor. The p-wave

CDW order is a spin-Peierls type instability with an alternation in the kinetic

energy on the leg bonds and a phase shift between the legs6. Finally the fCDW

state has been identified as a directed current phase with currents flowing across

the diagonals of a plaquette [44, 45]. Each of these channels also has a triplet

counterpart. D-wave SDW correlations for example are found in the dominant

vector chirality region on the two leg spin ladder with cyclic four spin exchange

(discussed in section 4.1 of this thesis). They can be understood as staggered spin

currents. In the particle-particle channel singlet pairing correlations are measured.

This is the case for s-wave and d-wave formfactors. The p-wave and f-wave cases

would correspond to triplet cooper pairs.

1 1

11

1 1

-1 -1 -1

-11

1

-1

-1

1

1

f-wavep-waved-waves-wave

Figure 3.13: Formfactors fA(k) for the two leg Hubbard ladder.

Our results are summarized in Fig. 3.14 for each phase. We measured the structure

factor: i.e diverging quantities ( Nk) signal true long range order, while saturatedbehavior indicates short range order. We discuss the results in each of the phases

in the following:

D-Mott - The D-Mott phase discussed before has enhanced SDW, dCDW(d-density wave) and dSC response.

S-Mott - The S-Mott phase was described as a disordered s-wave supercon-ductor in earlier work. Our results agree with this, the s-wave Cooper response

6This state is also known as bond order wave or bond charge density wave

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is clearly enhanced. We detect equally enhanced response in the sCDW and

the dSDW channels, which has not been reported in earlier studies.

dCDW - The true long range order in this phase is reflected in an almostperfect scaling of the staggered flux structure factor with system size Nk. No

other order parameter is enhanced.

CDW - The same statements as for the dCDW phase are valid. There is justan interchange in the formfactors from d-wave to s-wave.

sCDW dCDW sSDW dSDW sSC dSC0

5

10

15

20

S

tructure

Factors

[a.u.]

dCDW / Staggered Flux


1

2

3

4

Structure

Factors

[a.u.]

DMott

Nk

=8

Nk=12

Nk=16


5

10

15

20CDW


1

2

3

4SMott

Figure 3.14: Finite size scaling of the different structure factors in the four dominant phases

on the SO(5) manifold discussed by Lin, Balents and Fisher. The finite size scaling behavior in

the quantum disordered phases (D-Mott and S-Mott), where they are slowly approaching a finite

value, is clearly different compared to the true ordered phases [Staggered Flux (dCDW) and charge

density wave (CDW)], where they are proportional to the system size, thereby signalling long range

order. The structure factors are normalized to the values in the noninteracting groundstate. is

set to 2.

Additional phases at half filling

Recent work [44, 45] reanalyzed the weak coupling phase diagram of the Hubbard

ladder by bosonization and reported four additional phases as possible groundstates.

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Phase g1x g1e g1p g2x g2e g2y g3e g3c g3x

D-Mott -1 0 1 -1/2 1/2 1 -1 0 -1

S-Mott -1 0 -1 -1/2 1/2 -1 1 0 -1

pCDW 0 -1 -1 0 0 0 1 1 0

fCDW 0 -1 1 0 0 0 -1 1 0

Table 3.2: Ratios of the coupling constants for the four new phases discussed by

Wu et al.[44] and Tsuchiizu et al.[45].

The postulated phases encompass the pCDW and fCDW ordered phases and the

two disordered phases S-Mott and D-Mott. We have checked within our approach

that by using the asymptotic couplings in table 3.2 one obtains states with the

correct signals in the correlation function. The pCDW and the fCDW are long

range ordered phases as the sCDW and dCDW phases discussed before. The only

difference between these ordered phases is the formfactor. The gaps behave exactly

the same way. The primed Mott-states are slightly more subtle to characterize.

They are disordered phases and all correlation functions decay exponentiall

andreas martin lauchli- quantum magnetism and strongly correlated electrons in low dimensions

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