**X Training Course in the Physics of**

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

*Vietri sul Mare (Salerno) Italy*

*3 - 14 October 2005*

*Lecture Topics and
Background References*

*Département
de Physique de la Matière Condensée, University of Geneva*

Strong correlations in low dimensional systems

I. Peculiarities of one dimensional systems - crash course on Fermi liquids - Instabilities of the Fermi liquid - Differences in one dimension - Special treatement of interactions

II. Bosonization technique - Nature of excitations - Bosonization transformation - Physical properties of 1D fermions - Extension to other systems (bosons, 2D classical systems)

III. Spin chains and Ladders - Properties of 1D spin 1/2 chains - Extensions: frustrations and spin-phonons couplind (spin-Peierls transition) - S=1 chains

IV. Fermions on a Lattice - Hubbard and t-J models - Umklapps and Mott transition - Coupled Fermionic chains

V. Bosonic systems - Lieb-Lininger and Bose-Hubbard models - Mott transition for bosonic systems - Effects of disorder

Tutorials:

1) Dzialoshinskii-Larkin model

2) Application of bosonization for Coulomb interactions

3) Spin Ladder systems

4) Effect of boundaries

5) Higher dimensional systems. Conventional versus unconventional superconductivity.

**
References**

A. Useful to know before the course:

Basics of Many-Body physics: G. D. Mahan "Many particle physics" (Plenum)

Path Integral: J. W. Negele and H. Orland "Quantum many particle systems" (Addison-Wesley)

Renormalization group: J. Cardy "Scaling and Renormalization in Statistical Physics" (Cambridge University Press)

B. Material covered during the course:

The
lectures will follow closely: T. Giamarchi "Quantum physics in one
dimension" (Oxford University Press)

Max Planck Institute for Solid State Research, Stuttgart

Functional renormalization group approach to correlated electron systems

I. Intro: Correlated electrons and RG

Scale problem in Fermi systems, example HTSC; Hubbard model (2D,1D), simple properties; Perturbation theory, infrared divergences, partial summations (Baym-Kadanoff); basic idea of Wilson-type (momentum shell) RG

II. Functional RG: Fundamentals

Generating functionals from functional integrals; Derivation of exact flow equations; Truncations; Response functions

III. Simple flows:

phi^4-theory and Wilson-Fisher fixed point; Cooper instability; BCS theory (spontaneous symmetry breaking)

IV. Application: 2D Hubbard model

Instabilities from 1-loop vertex flow, magnetism and d-wave superconductivity, renormalization of single-particle excitations (self-energy)

V. Application: Impurities in Luttinger liquids

Truncated flow equations and results for density of states and conductance, single impurity and resonant tunneling through double barrier

Tutorials: Questions/discussion/exercises. Functional integrals (crash course), 1-loop flow for Luttinger model and 1D Hubbard model as exercise

**References**

Many-body formalism including functional integrals: J.W. Negele and H. Orland, "Quantum Many-Particle Systems" (Addison-Wesley, Reading, MA, 1987).

Introduction to Wilsonian RG for Fermi sytems: R. Shankar, "Renormalization group approach to interacting fermions", Rev. Mod. Phys. 66, 129 (1994).

Short review of functional RG for Fermi systems, including application to 2D Hubbard model and impurities in Luttinger liquids: W. Metzner, "Functional renormalization group computation of interacting Fermi systems", Proc. of Yukawa Symposium 2004, to appear in Prog. Theor. Phys. (preprint).

Derivation of functional RG flow equations: T. Enss, "Renormalization, conservation laws and transport in correlated electron systems", Ph.D. thesis (Stuttgart 2005), cond-mat/0504703, chapter 2.

*Laboratoire
de Physique des Solides, Université Paris-Sud *

Strongly Correlated Electron Behaviors and Heavy Fermions in Anomalous Rare-earth and actinide Systems

I. Introduction to anomalous rare-earth systems : experimental situation, « intermediate valence », example of the phase diagram of Cerium, the Anderson Hamiltonian.

II. The Kondo effect for a single impurity : perturbation calculation above the Kondo temperature Tk, exact single-impurity solution below Tk and Fermi Liquid behavior, heavy fermion behavior, the Schrieffer-Wolff transformation.

III. The Kondo effect for Ce, Yb and other anomalous rare-earth impurities : the « Coqblin-Schrieffer » Hamiltonian, the Kondo effect with crystalline field effect and application to transport properties, low temperature exact solution. The case of multichannel Kondo effect.

IV. The Kondo lattice : the Doniach diagram, the different mean-field approximations, the competition between the Kondo effect and the antiferromagnetic or ferromagnetic order, the spin glass-Kondo competition, the Periodic Anderson Hamiltonian.

V. The series of actinide metals, the undercreened model applied to Uranium compounds. Experimental review on the superconductivity occurring in Ce, U or even Pu systems. Final prospective analysis on future works on strongly correlated electron systems.

*Institute
of Theoretical and Computational Physics, Graz University of Technology *

Numerical Approaches to coupled Quantum Systems

A) Introduction into the Monte-Carlo Simulations and Exact Diagonalization, from the basics to tips and tricks. The students will have the opportunity to ecxperience the inner life of these techniques by practicle courses and analytic exercises. At the end they will have a working knowlege in these numerical techniques.

B) Brief introduction into the physics of Manganites, as example of coupled quantum degrees of freedom: What is special and how can it be modelled. The Kondo-Lattice Model will be one ingedient. What is the present physical understanding.

C) Application of QMC-Simulations and ED to the Kondo-lattice model to study the double exchange mechanism, the polaron formation, magnetic phases and phonon physics. In the afternoon courses there will be the oportunity also to compare with approximate analytic results.

1+2) Quantum-Monte-Carlo Simulations: SWOT analysis strengths, weaknesses, opportunities, threats. Status Quo. The students will learn how to do Monte-Carlo-Simulations and how to do it right.

3) Exact diagonalization

4) Manganites: Models

5) Manganites: Numerical results and open questions.