**VIII Training Course in the Physics of**

**Correlated Electron Systems and High-Tc
Superconductors**

*Vietri sul Mare (Salerno) Italy*

*6
- 17 October 2003*

*Lecture Topics and
Background References*

**
Prof. ****Antoine
Georges**

*Laboratoire
de Physique Theorique, Ecole Normale Superieure, Paris*

Correlated electron systems:

Materials, Models and Dynamical Mean-Field Theory

**I.
Introduction to the theory of strongly correlated
electron materials.**

In this lecture, I will explain why some solids display strong electron correlations. These are generally associated with open d- and f- shells, leading to quite localised orbitals. Some basic notions of the electronic structure of such materials will be presented. Physical aspects associated with strong correlations (e.g in transport, spectroscopy,...) will be briefly reviewed.

In the afternoon training session, simple concepts and approximation schemes will be discussed: Models (Hubbard, Anderson lattice, and extensions), Hubbard bands and Hubbard approximation(s), slave- boson schemes (the Brinkman-Rice heavy Fermi liquid), superexchange, the mean-field theory of spin-density wave ordering etc...

*Useful
reading related to lecture 1:* Refs. (24),(10).

**II.
Basic principle sof dynamical mean-field theory (DMFT).**

I will introduce in this lecture the basic principles of DMFT. This approach focuses on the local Green’s function (or local spectral density) of a lattice model (or a real solid). I will emphasize that an exact functional of this quantity can be constructed, along similar lines to density functional theory. The local Green’s function can be represented as the solution of a quantum impurity model. DMFT corresponds to the simplest local approximation on this functional, leading to a self-consistency condition relating the local Green’s function to the effective bath of the impurity model.

Extensions of the formalism to longer-range interactions will also be briefly considered.

In the afternoon training session, we shall practice on writing DMFT equations for various models, the connection with the limit of infinite dimensions, the description of ordered phases and the calculation of response functions.

*Useful
reading related to lecture 2:* Review articles on DMFT (6; 23). Some
original articles (5; 17). On the use of functionals:
Refs (2–4; 21).

**III.
Technical aspects: algorithms for solving quantum
impurity models and the DMFT equations.**

Algorithms for solving quantum impurity models will be discussed in this lecture, such as Quantum Monte Carlo, exact diagonalisation, etc... Some approximation schemes will also be presented. Some basic notions on the physics of the Anderson impurity model will be introduced.

In the afternoon session, some explicit examples will be presented on the computer. Time permitting, some applications of quantum Anderson impurity models to other field will be discussed, such as transport and the Kondo effect in quantum dots.

*Useful
reading related to lecture 3:* Ref. (6; 9).

**IV.
The Mott transition: DMFT confronts recent experiments.**

In this lecture, I will describe the detailed theory of the Mott transition which emerged from DMFT and has been one of its early success. Very recent experiments have confirmed some of the predictions of this theory. Photoemission on V2O3 and Ca(Sr)VO3 have revealed the quasi-particle peak at the Fermi level. Large transfers of spectral weight upon changing temperature are seen in optics. The liquid-gas critical behaviour at the Mott critical endpoint has been demonstrated in transport experiments, as well as the various transport regimes.

In the afternoon session, I will describe some issues of current interest concerning the Mott transition beyond a local (DMFT) description, e.g the role of superexchange, the possibility of hot/cold spots formation, and anisotropic transport in low-dimensional systems.

*Useful
reading related to lecture 4:* On the Mott transition within DMFT: Ref. (1;
6; 7; 12; 13; 19; 20). For recent experiments, see Refs.
(18; 22) (photoemission) and Refs. (11; 15; 16) (transport).

**V.
New methods of electronic structure calculations for
correlated materials**

Over the past few years, great progress has been achieved by combining DMFT ideas and methods with those of density functional theory. This allows to overcome some of the limitations of the local density approximation (LDA) for strongly correlated materials. I will describe the principles and concrete implementation of these new methods of electronic structure calculations. This will be illustrated by several examples on transition- metal oxides and f-electron materials. Some open issues in this rapidly evolving field will be discussed.

*Useful
reading related to lecture 5:* Refs. (8; 14)

See also the viewgraphs of recent talks at:

http://online.itp.ucsb.edu/online/cem02

http://online.itp.ucsb.edu/online/cem02/si-conf-schedule.html

http://www.ictp.trieste.it/ smr1512/contributionspage.html

**References**

[1] R. Bulla, T. A. Costi, and D. Vollhardt. Finite-temperature numerical renormalization group study of the mott transition. Phys. Rev. B , 64:45103, 2001.

[2] R. Chitra and G. Kotliar. Dynamical mean-field theory and electronic structure calculations. Phys. Rev. B, 62:12715, 2000.

[3] R. Chitra and G. Kotliar. Effective action approach to strongly correlated fermion systems. Phys. Rev. B, 63:115110, 2001.

[4] A. Georges. Exact functionals, e®ective actions and dynamical mean-field theories, in: Proceedings of the NATO ASI "Field Theory of Strongly Correlated Fermions and Bosons in Low Dimensional Disordered Systems". Kluwer Acad., 2002.

[5] A. Georges and G. Kotliar. Hubbard model in infinite dimensions. Phys. Rev. B, 45:6479, 1992.

[6] A. Georges, G. Kotliar, M. Rozenberg, and W. Krauth. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys., 68:13, 1996.

[7] A. Georges and W. Krauth. Phys. Rev. B , 48:7167, 1993.

[8] K. Held, I. A. Nekrasov, N. Bl¨umer, V. I. Anisimov, and D. Vollhardt. Realistic modeling of strongly correlated electron systems: an introduction to the lda+dmft approach. Int. J. Mod. Phys. B, 15:2611, 2001. cond-mat/0010395.

[9] J. E. Hirsch and R. M. Fye. Phys. Rev. Lett., 25:2521, 1986.

[10] M. Imada, A. Fujimori, and Y. Tokura. Metal-insulator transitions. Rev. Mod. Phys. , 70:1039, 1998.

[11] F. Kagawa, T. Itou, K. Miyagawa, and K. Kanoda. First-order mott transition and its critical endpoint in a quasi-two dimensional organic conductor ·-(bedt-ttf)2cu[n(cn)2]cl. preprint cond-mat/0307304.

[12] G. Kotliar. Landau theory of the mott transition in the fully frustrated hubbard model in infinite dimensions. Eur. J.Phys. B, 27:11, 1999.

[13] G. Kotliar, E. Lange, and M. J. Rozenberg. Landau theory of the finite temperature mott transition. Phys. Rev. Lett., 84:5180, 2000.

[14] G. Kotliar and S. Y. Savrasov. Dynamical Mean Field Theory, Model Hamiltonians and First Principles Electronic Structure Calculations. In "New Theoretical Approaches to Strongly Correlated Systems, A.M. Tsvelik Ed., Kluwer

Academic Publishers, 2001. Proc. of the Nato Advanced Study Institute on New Tehoretical Approaches to Strongly Correlated Systems, Cambridge, UK, 1999; preprint cond-mat/0208241.

[15] P. Limelette, A. Georges, D. J´erome, , P. Wzietek, P. Metcalf, and J.M. Honig. Universality and critical behavior at the mott transition. To be published in Science, Sep. 2003.

[16] P. Limelette, P. Wzietek, S. Florens, A. Georges, T. Costi, , C. Pasquier, D. Jerome, C. Meziere, and P. Batail. Mott transition and transport crossovers in the organic compound ·-(bedt-ttf)2cu[n(cn)2]cl. prl, 2003.

[17] W. Metzner and D. Vollhardt. Correlated lattice fermions in d = TRUE dimensions. Phys. Rev. Lett., 62:324, 1989.

[18] S. K. Mo, J. D. Denlinger, H. D. Kim, J. H. Park, J. W. Allen, A. Sekiyama, A. Yamasaki, K. Kadono, S. Suga, Y. Saitoh, T. Muro, P. Metcalf, G. Keller, K. Held, V. Eyert, V. I. Anisimov, and D. Vollhardt. Prominent quasi-particle peak in the photoemission spectrum of the metallic phase of v2o3. preprint cond-mat/0212110.

[19] M. J. Rozenberg, R. Chitra, and G. Kotliar. Finite temperature mott transition in the hubbard model in infinite dimensions. Phys. Rev. Lett., 83:3498, 1999.

[20] M. J. Rozenberg, G. Kotliar, and X. Y. Zhang. Phys. Rev. B , 49:10181, 1994.

[21] S. Y. Savrasov and G. Kotliar. Spectral density functional for electronic structure calculations. preprint cond-mat/0308053.

[22] A. Sekiyama, H. Fujiwara, S. Imada, H. Eisaki, S. I. Uchida, K. Takegahara, H. Harima, Y. Saitoh, and S. Suga. Genuine electronic states insensitive to the distortion in perovskite vanadium oxides revealed by high-energy photoemission. preprint cond-mat/0206471.

[23] T.Pruschke, M.Jarrell, and J.Freericks. Adv. Phys., 42:187, 1995.

[24] C. M. Varma and T. Giamarchi. Model for oxide metals and superconductors. Elsevier, 1991. Les Houches Summer School.

**
Prof. ****Masatoshi
Imada**

Institute for Solid State Physics, University of Tokyo

New Theoretical Tools for Correlated Insulators and Metals

**I.
Historical Overview for Physics of Strongly Correlated** **Electron Systems**

In the first lecture, I will overview the background of the problem by focusing on the properties of metals, insulators and correlated phases. Typical properties of transition metal oxides and organic compounds are picked up from material to material and their underlying common concept will be introduced.

Reference: [1], [2]Chap.I, Chap. IV

Training course: Connection of the experimental systems and theoretical models are discussed. Basic principles of the density functional theory are also summarized.

Reference: [2]Chap. IIA-D

II. Several Numerical Methods

I will tell basic methodology of several numerical methods such as quantum Monte Carlo methods, path-integral renormalization-group method together with some other numerical algorithms.

Before this treatments, I will discuss the relationship of the numerical methods to various approximations.

Using the Stratonovich-Hubbard transformation, the mean-.eld analysis and the numerical methods are formulated on a unified basis. Several outcome obtained by their applications are also reviewed.

Training session: I will take practicing time for more detailed description of algorithms of the numerical methods. For example we discuss how to calculate physical properties such as correlation functions. The updating procedure of obtaining low energy states will also be elucidated.

Reference : [2]Chap.IID-E, For Monte Carlo methods, [3] and [4, 5]. For path-integral renormalization-group method, [27, 8]. For some other methods, [6].

III. Operator Projection Method

I will start from the operator projection theory and tell about applications to correlted electron systems. The method is also reformulated as a method to improve the dynamical mean-field theory.

Training sessions: Some applications and details of this method are discussed. Particularly, formalism as an extension of the dynamical mean .eld theory is considered in detail.

References: [9], [10], [13], [14], [15, 16, 17],

**IV.
Properties of Correlated Electrons Revealed by** **Theories and Comparison
to Experiments**

In this lecture, I will discuss the diversity of the physical properties of correlated metals and insulators in the region near the Mott insulators in the light of the outcome from the numerical and analytical methods I explained in the previous lectures.

In the afternoon session, some detailed comparisons with the outcome of new tools and the experimental results will be made. In particular, the signi.cance of this region is explained as the multi-furcating physics in examples of two-dimensional systems. This includes the inhomogeneity and structural formations in real and momentum spaces.

Reference : [2]Chap.IID-E,IV, [18, 19, 20], [22, 23, 21], [27, 28, 29], For, Recent experiments,

see [32, 31, 33, 34, 35, 36, 37]

V. Present Status and Future Perspective

The comparison between theories and experiments will further be made. Then future of this .eld will also be discussed. I will talk on a formalism towards realistic theories of correlated electron systems.

References

[1]
N. F. Mott, *Metal-Insulator Transitions*, (Taylor and Francis, London/Philadelphia,
1990).

[2]
M. Imada, A. Fujimori and Y. Tokura:Rev. Mod. Phys. **70 **(1998) 1039.

[3] R. Blankenbecler, D.J. Scalapino, and R.L. Sugar: Phys. Rev. D 24, (1981) 2278.

[4]
M.Imada and Y.Hatsugai: J. Phys. Soc. Jpn. **58 **(1989) 2571.

[5] N. Furukawa and M. Imada: J. Phys. Soc. Jpn. 61 (1992) 3331.

[6] E. Dagotto: Rev. Mod. Phys. 66 (1994) 763.

[7]
T. Kashima and M. Imada: J. Phys. Soc. Jpn. **70 **(2001) 2287.

[8]
M. Imada and T. Kashima: J. Phys. Soc. Jpn. **69 **(2000) 2723.

[9]
S. Nakajima, Prog. Theor. Phys. **20**, 948 (1958); R. Zwanzig, *Lectures
in Theoretical* *Physics*, Vol. 3, (Interscience, New York, 1961); H.
Mori, Prog. Theor. Phys. **33**, 423 (1965); Prog. Theor. Phys. **34**,
399 (1965).

[10]
G. Baym, Phys. Rev. **127**, 1391 (1962); L. P. Kadano. and G. Baym, *Quantum
Statistical* *Mechanics*, (Benjamin, Menlo Park, 1962).

[11]
L. M. Roth, Phys. Rev. **184**, 451 (1969).

[12]
H. Matsumoto and F. Mancini, Phys. Rev. B **55**, 2095 (1997).

[13] S. Onoda and M. Imada: J. Phys. Soc. Jpn. 70 (2001) 632;J. Phys. Soc. Jpn. 70 (2001) 3398.

[14]
A. Georges, G. Kotliar, W. Krauth and M. Rozenberg: Rev. Mod. Phys. **68**,
13 (1996).

[15]
S. Onoda and M. Imada: Phys. Rev. B **67 **(2003) 161102.

[16]
Th. Maier *et al.*, Eur. Phys. J. B **13**, 613 (2000); M. Jarrell *et
al.*, Phys. Rev. B **64**, 195130 (2001); Th. Maier *et al.*,
cond-mat/0111368.

[17]
G. Kotliar *et al.*, Phys. Rev. Lett. **87**, 186401 (2001).

[18]
F.F. Assaad, and M. Imada, Phys. Rev. Lett.**76**, 3176 (1996).

[19]
H. Tsunetsugu and M.Imada, J. Phys. Soc. Jpn. **67**, 1864 (1998).

[20]
M. Kohno, Phys Rev. B **55**, 1435 (1997).

[21]
F. F. Assaad and M. Imada, Eur. Phys. J. B **10**, 595 (1999).

[22]
M. Imada, J. Phys. Soc. Jpn. **64 **(1995) 2954.

[23]
M. Imada and S. Onoda, *"Open Problems in Strongly Correlated Electron
Systems" *eds. J. Bonca et al. (Kluwer Academic Publishers 2001)
p.69-80.

[24]
M. Cyrot, J. Phys. (France) **33**, 125 (1972).

[25]
C. Castellani *et al.*, Phys. Rev. Lett. **43**, 1957 (1979).

[26]
M. J. Rozenberg *et al.*, Phys. Rev. Lett. **83**, 3498 (1999).

[27]
T. Kashima and M. Imada: J. Phys. Soc. Jpn. **70 **(2001) 3052.

[28]
H. Morita, S. Watanabe and M.Imada, J. Phys. Soc. Jpn. **71 **(2002) 2109.

[29] M. Imada, T. Mizusaki and S. Watanabe: cond-mat/0307022.

[30]
G. Kotliar, S. Murthy and M.J. Rosenberg, Phys. Rev. Lett. **89 **(2002)
046401.

[31]
Z. Wang et al., Phys. Rev. B **65**, 064509 (2002).

[32]
N. Harima, A. Fujimori, T. Sugaya, I. Terasaki: Phys. Rev. B **67 **(2003)
172501.

[33]
A. Casey, H. Patel, J. Nyeki, B.P. Cowan, J. Saunders, Phys. Rev. Lett.**90**,
(2003)115301.

[34]
K. Ishida, M. Morishita, K. Yawata and H. Fukuyama, Phys. Rev. Lett.**79 **(1997)
3451.

[35]
D. Fournier, M. Poirier, M. Castonguay, et al., Phys. Rev. Lett.**90 **(2003)
127002.

[36] Y. Shimizu, M. Maeda, G. Saito, K. Miyagawa and K. Kanoda, unpublished.

[37]
M. Tamura and R. Kato, J. Phys. Cond. Matt. **14 **(2002) L729.

**Prof.
Miodrag L. Kulic**

*University
of Augsburg, Theory II, 86135 Augsburg*

Electron-Phonon Interaction and Strong Correlations

in High-Tc Superconductors

**I.
Theory of Electron-Phonon Interaction (EPI) [1], [2]**

- self-energy effects in the Migdal theory

- effects of long-range Coulomb interaction and anharmonicity

- bare and renormalized coupling constant

**II.
Strongly Correlated Systems [3]**

- Hubbard operators vs slave bosons

**III.
1/N Expansion in Terms of Hubbard Operators [4], [5]**

- self-energy

- charge- and spin-vertex functions

**IV.
Renormalization of EPI by Strong Correlations in HTSC [4], [5]**

- Madelung coupling

- covalent coupling

- forward scattering peak in EPI

**V.
Eliashberg Equations for Strongly Correlated Superconductors [5], [6]**

- self-energy due to EPI and Coulomb interaction

- nonadiabatic effects and T

**Training
Sessions**

1. Functional derivative technique for EPI - details of calculations [1], [2]

2. Algebra of Hubbard operators, effective t-J model - details [3], [4], [5]

3. 1/N expansion for self-energy and vertex functions - details [4], [5]

4. What is pairing mechanism in HTSC? - pro and contra arguments for various

pairing potentials [5], [7], [8]

5. ARPES experiments and self-energy effects in HTSC - brief introduction in

ARPES; EPI and Coulomb self-energy effects in ARPES of HTSC [9], [10]

**References**

[1] E. G. Maksimov, in "High Temperature Superconductivity", by V. L. Ginzburg, D. A. Kirzhnits, Ch. III (Consultants Bureau, New York, 1982)

[2] G. Baym, Ann. of Phys. 14, 1 (1961)

[3] Yu. A. Izyumov, Physics-Uspekhi 40, 445 (1997)

[4] M. L. Kulic, R. Zeyher, Phys. Rev. B 49, 4395 (1994); R. Zeyher, M. L. Kulic, Phys. Rev. B 53, 2850 (1996); M. L. Kulic, R. Zeyher, Mod. Phys. Lett. B 11, 333 (1997)

[5] M. L. Kulic, Physics Reports 338, p.1-264 (2000)

[6] C. Grimaldi, L. Pietronero, S. StrÄassler, Phys. Rev. B 52, 10 516 (1995); ibid. Phys. Rev. B 52, 10 530 (1995)

[7] E. G. Maksimov, Uspekhi Fiz. Nauk, 170, 1033-1061 (2000)

[8] D. J. Scalapino, Physics Reports 250, 329 (1995)

[9] A. Damascelli, Z.-X. Shen, Z. Hussein, Rev. Mod. Phys., 75, 473 (2003)

[10] M. L. Kulic, O. V. Dolgov, cond-mat/ 03 08 597 (28 Aug. 2003)

**
Prof. A. Muramatsu**

*Institut
für Theoretische Physik III, Universität Stuttgart, Stuttgart*

Monte Carlo simulations of quantum systems with global updates

**I.**
**Quantum spin-systems I: world-lines and the loop-algorithm**

In
the first lecture we start with the simplest model for a strongly correlated system, namely an antiferromagnetic quantum *S*-1*/*2 chain. We will
review methods for its simulation starting with the world-line algorithm [1] and
then introducing the loop algorithm [2, 3].

Training session: Discussion of algorithmic details. General background on Monte Carlo simulations. [4, 5, 6].

**II.**
**Quantum spin-systems II: further developments**

The second lecture will be dedicated to further developments of the loop-algorithm, where issues like the measurement of correlation functions, improved estimators, and the algorithm without discretization time errors will be discussed [7, 3]. Also methods like maximum entropy [8] for spectral functions will be introduced.

Training session: Implementation of the algorithms.

**III.**
**Fermionic systems with strong correlations: the t-J model**

In this lecture we discuss models for doped antiferromagnets, with emphasis on the strong correlation limit that is central for high temperature superconductors and related materials [9]. Also an exact canonical transformation that leads to a formulation with separated charge and spin degrees of freedom [10, 11] will be discussed. Results for a single hole in antiferromagnetic chains [12], ladders [13], and planes [14] will be summarized, giving an initial picture of charge dynamics in a quantum antiferromagnet.

Training session: Background discussion on quasiparticles and charge-spin separation.

**IV.**
**The hybrid-loop algorithm**

We will discuss here a new algorithm for the simulation of finite doping in a quantum antiferromagnet. The elements given in the first 3 lectures will be put together, and an additional algorithmic element will be introduced, namely the determinantal method [15], resulting in a hybrid algorithm that combines the efficiency for spins of the loop-algorithm with the determinantal one for fermions.

Training session: Discussion of algorithmic details with emphasis on manipulation of Slater determinants.

**V.**
**Spinon, holons and antiholons in one dimension**

The
last lecture will be dedicated to a first application of the hybrid-loop
algorithm. We
discuss the spectral function of the t-J model in one dimension with finite doping. There we can see how charge spin separation shows up in the one-particle
spectral
function, and a detailed description can be achieved by comparison with an analitically
soluble model, namely the t-J model with 1*/r*^{2} interaction
[16, 17, 18]. The
results show that in addition to spinons and holons expected in one-dimensional
metals, antiholons with charge *Q *= 2*e*, spin *S *= FALSE, and
twice the mass of the holons are necessary to describe the inverse photoemission
spectra at finite doping [19].

Training session: Further discussion on charge-spin separation and the issues treated in the morning lecture.

References

[1]
J. E. Hirsch, R. L. Sugar, D. J. Scalapino, and R. Blankenbecler, Phys. Rev. B **26**,
5033 (1982).

[2]
H. G. Evertz, M. Marcu, and G. Lana, Phys. Rev. Lett **70**, 875 (1993).

[3]
H. G. Evertz, Adv. Phys. **52**, 1 (2003).

[4]
J. W. Negele and H. Orland, *Quantum many-particle sytems*, *Frontiers in
* *Physics
*(Addison-Wesley Publishing Company, Redwood City, 1988).

[5]
A. Muramatsu, in *Quantum Monte Carlo Methods in Physics and Chemistry*, edited
by M. P. Nightingale and C. J. Umrigar (Kluwer Academic Press, Dordrecht, 1999).

[6]
F. Assaad, in *Quantum Simulations of Complex Many-Body Systems: from* *Theory
to Algorithms*, edited by J. Grotendorst, D. Marx, and A. Muramatsu (NIC
Series, Vol. 10, FZ-Julich, 2002).

[7]
B. B. Beard and U.-J. Wiese, Phy. Rev. Lett. **77**, 5130 (1996).

[8]
M. Jarrell and J. Gubernatis, Phys. Rep. **269**, 133 (1996).

[9]
M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. **70**, 1039 (1998).

[10]
G. Khaliullin, JETP Lett. **52**, 389 (1990).

[11]
A. Angelucci, Phys. Rev. B **51**, 11580 (1995).

[12]
M. Brunner, F. F. Assaad, and A. Muramatsu, Eur. Phys. J. B **16**, 209
(2000).

[13]
M. Brunner, S. Capponi, F. F. Assaad, and A. Muramatsu, Phys. Rev. B **63**, 180511(R)
(2001).

[14]
M. Brunner, F. F. Assaad, and A. Muramatsu, Phys. Rev. B **62**, 15480
(2000).

[15]
R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Phys. Rev. D **24**, 2278
(1981).

[16]
Y. Kuramoto and H. Yokoyama, Phys. Rev. Lett. **67**, 1338 (1991).

[17]
Z. N. C. Ha and F. D. M. Haldane, Phys. Rev. Lett **73**, 2887 (1994).

[18]
M. Arikawa, Y. Saiga, and Y. Kuramoto, Phys. Rev. Lett **86**, 3096 (2001).

[19]
C. Lavalle *et al.*, Phys. Rev. Lett. **90**, 216401 (2003).