Theory of Quantum and Complex systems Polaron concept and its applicationsJ. T. DevreeseTheorie van Kwantumsystemen en Complexe Systemen (TQC),Universiteit Antwerpen, BelgiumOutlinePolaron conceptPolaron optical absorption: analytic approximationsMany-polaron optical absorption of doped strontium titanateMechanisms of the Fermi-liquid response of Nb-doped strontium titanateSuperconductivity in the LaAlO3-SrTiO3 heterostructure12:112Polaron concept312:113Polaron concept1 L. D. Landau, Phys. Z. Sowjetunion 3, 664 (1933)2 S. I. Pekar, Untersuchungen ber die Elektronentheorie der Kristalle, Berlin, Akademie, 19543 H. Frhlich, Adv. Phys. 3, 325 (1954)4 A. S. Alexandrov and N. Mott, Polarons and bipolarons, World Scientific, Singapore, 1996 5 J. T. L. Devreese, Moles agitat mentem. Ontwikkelingen in de fysika van de vaste stof. Rede uitgesproken bij de aanvaarding van het ambt van buitengewoon hoogleraar in de fysica van de vaste stof, in het bijzonder de theorie van de vaste stof, bij de afdeling der technische natuurkunde aan de Technische Hogeschool Eindhoven, March 9, 19796 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009)A conduction electron repels the negative ions and attracts the positive ionsA self-induced potential arises, which acts back on the electron and modifies its physical propertiesAn artists view of a polaron 5

A conduction electron (or hole) together with its self-induced polarization in a polar crystal forms a quasiparticle, which is called a polaron 1-3 Properties of polarons have attracted increasing attention due to their possible relevance to physics of high-Tc superconductors 4 12:1144Landau 1933, self-induced potential, localization, high-Tc superconductors, nanostructures

The polaron concept was introduced by Landau in 1933.

In the figure, I show an artists view of a polaron.

The physical properties of a polaron differ from those of a band-carrier. A polaron is characterized by its binding energy, by an effective mass and by its characteristic response to external electric and magnetic fields (e. g. dc mobility and optical absorption coefficient).

If the spatial extension of a polaron is large compared to the lattice parameter of the solid, the latter can be treated as a polarizable continuum. This is the case of a "large polaron".

An electron or a hole trapped by its self-induced atomic (ionic) displacement field in a region of linear dimension, which is of the order of the lattice constant, is called small polaron.

More recent extensions of the polaron concept have been invoked, e. g., to study the properties of conjugated polymers, colossal magnetoresistance perovskites, high-Tc superconductors, fullerenes, quasi-1D conductors, nanostructured materials - semiconductor quantum wells, superlattices, quantum dots, some exotic object such as multielectron bubbles in liquid helium, and impurity polarons in atomic Bose-condensates.Electron-phonon couplingThe large-polaron coupling constant was introduced by Frhlich 11 H. Frhlich, Adv. Phys. 3, 325 (1954)wLO is the long-wavelength frequency of a LO phonone and e0 are, respectively, the electronic and the static dielectric constant of the polar crystalmb is the electron (hole) band mass

12:115Frohlich in 1954 applied the second quantization form of the electron-lattice interaction.

The large, or continuum polaron is characterized by the electron-LO-phonon coupling constant, which was first introduced by Frhlich.

The coupling strength of the large polaron is directly related to the macroscopic polarizability of a medium expressed in terms of high-frequency and static dielectric constants. 5Strong- and weak-coupling polaron1 L. D. Landau and S. I. Pekar, Zh. Eksper. Teor. Fiz. 18, 419 (1948)2 S. I. Pekar, Untersuchungen ber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 19513 N. N. Bogolubov, Ukr. Matem. Zh. 2, 3 (1950)4 S. J. Miyake, J. Phys. Soc. Jpn. 38, 181 (1975)5 H. Frhlich, Adv. Phys. 3, 325 (1954)6 M. A. Smondyrev, Teor. Math. Fiz. 68, 29 (1986) [English translation: Theor. Math. Phys. 68, 653 (1986)]7 J. Rseler, Phys. Stat. Sol. (b) 25, 311 (1968)8 Wu Xiaoguang, F. M. Peeters, and J. T. Devreese, Phys. Rev. B 31, 3420 (1985)

The weak-coupling limit 5 is obtained from the leading terms for a 0. Weak-coupling results for the polaron parameters are

The 3D case 6,7:

The 2D case 8:where ws is the surface-optical-phonon frequency The first studies on polarons were devoted to the calculation of the self-energy and the effective mass of polarons in the limit of large a, or strong coupling 1-3 The ground-state energy and the effective mass of a strong-coupling polaron 4 are:

12:116The early analytic investigations of the polaron problem resulted in weak-coupling and strong-coupling series expansions of polaron parameters the ground state energy and the polaron effective mass.6Feynmans path-integral treatment12:117The polaron problem is formulated1 as an equivalent one-particle problem in which the interaction, non-local in time or "retarded", is between the electron and itself. 1 R. P. Feynman, Phys. Rev. 97, 660 (1955)2 R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M. Platzman, Phys. Rev. 127, 1004 (1962)3 K. K. Thornber and R. P. Feynman, Phys. Rev. B 1, 4099 (1970)The propagator resulting from the elimination of the phonon field

All-coupling theoryb = 1/(kBT)For many materials it is useful to have an adequate all-coupling polaron theory at one's disposal.

Such an all-coupling theory was developed by Feynman using his path-integral formalism.

He got the idea to formulate the polaron problem into the Lagrangian form of the quantum mechanics and then eliminate the field oscillators.

The resulting path integral for the partition function contains the action in which the interaction of an electron with a quantized field is replaced by the retarded interaction of an electron with itself.

This path integral (72) with (73) has a great intuitive appeal: it shows the polaron problem as an equivalent one-particle problem in which the interaction, non-local in time or retarded, occurs betweenthe electron and itself.

Feynman studied first the self-energy and the effective mass and later the mobility of polarons.

Subsequently the path-integral approach to the polaron problem was generalized and developed to become a powerful tool to study optical absorption, magnetophonon resonance, cyclotron resonance, and other phenomena.

7Polaron optical absorption:analytic approximations812:118Optical properties of polarons at weak coupling1,21 V. L. Gurevich, I. G. Lang, and Yu. A. Firsov, Sov. Phys. Solid State 4, 918 (1962)2 J. Devreese, W. Huybrechts, and L. Lemmens, Phys. Stat. Sol. (b) 48, 77 (1971)High densities, , where z is the Fermi energy:Low densities, :At T = 0, the optical ab-sorption coefficient can be expressed in terms of elementary functions in two limiting cases

Elementary polaron scattering process

12:119At zero temperature and in the weak-coupling limit, the optical absorption is due to the elementary polaron scattering process.

At nonzero temperature, the absorption of a photon can be accompanied not only by emission, but also by absorption of one or more phonons. In the weak-coupling limit (1) the polaron absorption coefficient was first obtained by Gurevich, Lang and Firsov.

At zero temperature, the absorption coefficient for absorption of light with frequency can be expressed in terms of elementary functions in two limiting cases: in the region of comparatively low frequencies and in the high-frequency region.

A simple derivation in Ref. 2 using a LLP canonical transformation method gives the absorption coefficient of free polarons, which coincides with the perturbation result obtained by Gurevich, Lang and Firsov.

9Internal excitations of a polaron at strong coupling

12:1110The opposite limit of the strong coupling polaron optical conductivity was also considered in 60ths. The scheme of polaron internal excitations was developed which subsequently was intensely discussed.

These excitations are manifested in the polaron optical absorption spectra.

1) The polaron ground state.

2) If the lattice polarization is allowed to relax or adapt to the electronic distribution of the excited electron (which itself then adapts its wave function to the new potential etc. leading to a self-consistent final state), the so-called relaxed excited state (RES) results.

3) The (unstable) polaron state, in which the lattice polarization corresponds to the electron ground state, while the electron is excited, is referred to as a Franck-Condon (FC) state of the polaron.

4) If one or more real phonons are excited, the polaron is in a scattering state.

10Dynamic response of continuum polarons:the path-integral treatment12:11111 R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M. Platzman, Phys. Rev. 127, 1004 (1962)2 K. K. Thornber and R. P. Feynman, Phys. Rev. B 1, 4099 (1970)3 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)4 R. P. Feynman, Phys. Rev. 97, 660 (1955)

The path-integral treatment 1-3 of the polaron response is based on the Feynman polaron model 4 FHIP approximation 1

Absorption coefficient is related to impedance asThe expansion of the impedance leads to the resonant structure

Starting from ~1970, great efforts are devoted to develop an analytic all-coupling theory of the polaron optical response.

The most important results in this direction are obtained using the Feynman path-integral formalizm. The path-integral treatment of the polaron response is based on the Feynman polaron model.

FHIP approximation: the double path integral with the exact polaron action is replaced by the expansion in powers of the difference betrween exact and trial actions.

In 1972, Devreese, De Sitter and Goovaerts obtained the polaron optical conductivity exploiting both imaginary and real parts of the polarizability derived by FHIP within the all-coupling Feynman path-integral approach.

They obtained the resonant structure of the optical conductivity using the expansion of the impedance rather than the expansion of the optical conductivity itself.11All-coupling path-integral Feynman approach Memory-function formalism 1,2The polaron optical conductivity within the memory-function formalism

Memory function1 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)2 F. M. Peeters and J. T. Devreese, Phys. Rev. B 28, 6051 (1983)

The two-point density-density Greens functionv and w are the variational frequency parameters of the Feynman polaron model.

12:1112The response-treatment for the polaron performed in 1983 by Devreese and Peeters uses the Mori-Zwanzig memory function technique, which projects the exact state space of the electron-phonon system to the state space of the trial Feynman system.

In that work the fact that the Mori-formalism allows to treat response properties in terms of the characteristics of the groundstate is exploited.

The polaron optical conductivity calculated in this memory function approach appears to be equivalent to the optical conductivity calculated within the Feynman-path integral treatment in 1972.

***The approach to the problem of the polaron polaron optical conductivity developed in 1972 PRB paper by DSG is based on the Feynman path integral technique, where the optical conductivitty is calculated starting from the Feynman variational model for the polaron and using the path integral response formalism.

Subsequently the path integral approach was rewritten in terms of the memory function formalism in 1983.

Within the memory-function formalism the interaction of the charge carriers with the free phonon oscillations can be expressed in terms of the electron density-density correlation function. It is calculated using the Feynman model, where the electron is coupled via a harmonic force to a fictitious particle that simulates phonons.

The frequency parameters of the model v and w are determined variationally using the Jensen-Feynman inequality.

12Optical absorption of polarons at arbitrary coupling12:1113 The memory function c(w) contains the dynamics of the polaronOptical-absorption spectrum of a single large polaron 1

At T = 0, a d-like central peak is at the origin For larger a, peaks attributed to transitions to RES are more pronounced

1 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)Here we show several polaron optical conductivity spectra obtained within the theory developed by Devreese with co-workers in 1972.

The optical absorption spectra calculated according the all-coupling theory are shown in the figure for values of the coupling constant in the range from 1 to 6.

The absorption spectrum for =1 consists of a "one-phonon line", similar to the weak-coupling result.

It is remarkable that within the Feynman model, the three different kinds of polaron excitations appear in the calculated optical absorption spectra for polarons at 5:

a) scattering states where e. g. one real phonon is excited (the structure starting at =LO); b) peak attributed to the relaxed excited state (RES); c) Franck--Condon (FC) states.

Furthermore, at zero temperature, the optical absorption spectrum for one polaron contains a "central peak" [~ ()] at the origin. For non-zero temperature, this "central peak" smears out and the optical absorption spectrum consists of both a broad envelope and an "anomalous" Drude-type low-frequency component.

This peak ensures a fulfillment of the sum rule for the polaron optical conductivity.

131 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)2 A.S. Mishchenko, N. Nagaosa, N. V. Prokofev, A. Sakamoto, B. V. Svistunov, Phys. Rev. Lett. 91, 236401 (2003) 3 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009)

Second-order perturbation theory

DSG approach1

Diagrammatic Quantum Monte-Carlo method 2Polaron optical conductivity: comparison between DSG and DQMC12:1114In the years 2000 to 2003, the DQMC results by Mishchenko et al. appeared in the polaron physics. There were intense and fruitful discussions on these results between M. and D. in Antwerpen. They shed light to some challenging questions of the polaron theory and led to a deeper understanding of the nature of internal polaron states. In particular, the concept of RES has been substantially extended.

Here, the polaron optical conductivity spectra calculated within the path-integral approach of DSG are compared with the optical conductivity calculated using the diagrammatic Monte Carlo method.

Calculations of the optical conductivity for the Frhlich polaron performed within the diagrammatic Quantum Monte Carlo method fully confirm the results of the DSG approach at 3.

In the intermediate coupling regime 37 the line widths of the peaks obtained using the FHIP treatment with the expansion of the impedance are unreliable.

Nevertheless, the position of the RES peak for >7 is meaningful for comparison with strong-coupling treatments of the RES energy for all considered values alpha.14Comparison of DQMC with the DSG model of the polaron optical conductivityThe positions of the main peak of the polaron optical-conductivity band1, obtained within DSG2, are in a remarkable agreement with the results of DQMC 3. The peak width within DSG is smaller than that in DQMC, especially at strong coupling.The origin of the peak width at strong coupling is not yet understood.

DQMC12:11151 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009)2 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)3 A. S. Mishchenko, N. Nagaosa, N. V. Prokofev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. Lett. 91, 236401 (2003)When the DQMC results for the polaron optical conductivity became available, the comparison of the dominant peak positions of the theory by Devreese with those given by DQMC was performed.

In this slide, the energy of the dominant peak in the optical conductivity spectra calculated within the DSG approach is plotted together with that given by the Diagrammatic Monte Carlo method.

We see the remarkable agreement between those positions for all considered alpha.

As seen from the figure, the main-peak positions, obtained within DSG, are in good agreement with the results of DMC for all considered values of including the strong-coupling region. The difference between the DSG and DMC results is relatively larger at = 8 and for = 9.5, but even there the agreement still seems quite satisfactory.

The main peak within the DSG model was interpreted in terms of the relaxed excited states of a polaron.

The remarkable agreement between the peak positions obtained within DSG and DMC is a strong argument in favor of the concept of the relaxed excited states of a polaron.

This coincidence of the peak positions allows us to suggest that in fact, the key fingerprint of multi-phonon processes the peak position - is really catched in the DSG approach. However, the point to be clarified is the line width of the optical absorption peaks.

15Arbitrary-coupling polaron optical conductivity: new analytical approximations12:1116Extended memory function formalism with phonon broadened levelsStrong coupling expansion based on the Franck-Condon principleThis deeper understanding was resulted in the recent paper by Prof. Devreese with collaborators in PRL (2006).

In this work the numerical data for the optical absorption of the Frhlich polaron model, obtained by a numerically exact Diagrammatic Monte Carlo method, are compared with two new approximate approaches, which consider the lattice relaxation effects in different ways.

These approaches are:

1. An extended memory function formalism with phonon broadened levels reproduces the optical response for small and intermediate coupling strengths, alpha