XI Training Course in the Physics of Strongly Correlated Systems


Vietri sul Mare (Salerno) Italy

2 - 13 October 2006


Lecture Topics and Background References



Prof. Michele Fabrizio

International School for Advanced Scientific Studies, Trieste (Italy)

Clusters of Anderson impurities and beyond: How Kondo effect dies and what can we learn about the Mott transition

I. The Mott transition and the energy scales that accompany it. - Mapping a lattice model onto an Anderson impurity model: Brief introduction to Dynamical Mean Field Theory. - How a realistic Mott transition translates into impurity models: the death of the Kondo effect. 

II. The Anderson impurity model and the Kondo effect: (a) introduction; (b) poor man's scaling and Wilson's Numerical Renormalization Group. 

III. The Anderson impurity model and the Kondo effect (continued): (c) the local Fermi-liquid description; (d) brief introduction to Conformal Field Theory applied to impurity models.

V. The two-impurity model: (a) introduction; (b) the phase diagram; (c) Fermi-liquid description; (d) dynamical behavior. 

V. Impurity clusters: (a) the trimer of impurities; (b) the plaquette of impurities. - Concluding remarks. 



T1 The scattering theory and perturbative results in the Anderson impurity model. 

T2 Basis and applications of the local Fermi-liquid theory. 

T3 Simple applications of Conformal Field Theory to impurity models. 

T4 Examples of non-perturbative approaches: the Gutzwiller variational wave-function.

T5 Simple lattice counterparts of the two-impurity models. 



Basic knowledge of Many Body Theory and Renormalization Group.





Prof. Didier Poilblanc

Laboratoire de Physique Theorique C.N.R.S. & Universite de Toulouse (France) 

Modelling and simulating strongly correlated fermions 



I.-II. Microscopic models and low-dimensional lattices - Exact diagonalisation & Lanczos methods - Comparison with other numerical methods (DMRG, QMC, CORE, etc...) 

III. Dimerized and/or frustrated chains - Application to copper-germanate - Spin-Peierls model - Adiabatic phonons - Low energy soliton excitations - Impurities imbedded in Spin-Peierls chains 

IV. Spin & doped ladders: a route towards understanding Hi-Tc ? Strong coupling limit & spin gap behavior - Magnon excitations - Superconducting properties 

V. Some application to frustrated quantum magnetism 2D and 3D frustrated lattices - Classical degeneracy - Spin liquids & Valence Bond Crystals - Effect of doping.



T1 Power and Lanczos methods: getting started - Some analytic derivations - Getting used to simple codes for 1D spin systems (GS energies, scaling, etc...) 

T2 Metal-insulator transition and Luttinger liquid exponents in a chain of correlated fermions 

T3 Full diagonalisations: how to calculate thermodynamic properties like specific heat, susceptibility, etc... 

T4 Simulating a dimerized (and/or frustrated) chain and/or a 2-leg spin ladder - Physical properties, spin gap, magnon dispersion,... - Spinon (de)confinement



* Basic textbooks: 

- "Interacting electrons and quantum magnetism", A. Auerbach (Spinger-Verlag, 1994).

- "Lecture Notes on Electron Correlation and Magnetism", P. Fazekas, Series in Modern Condensed Matter Physics-Vol.5 (World Scientific, 1999).

* More specialized textbooks: 

- "Quantum Magnetism", Eds. Schollwoeck et al., Lecture Notes in Physics 645 (Springer, Berlin, 2004). 

- "Effective models for low-dimensional strongly correlated systems", Eds. G.G. Batrouni & D. Poilblanc, AIP Conference Proceedings No 816 (AIP, Melville, New York, 2006). 

- "Diagonalization & Density Matrix Renormalization Group-based techniques...", R. Noack & S. Manmana, p.93 in "Lectures on the Physics of Highly Correlated Electron Systems X", Eds. A. Avella & F. Mancini (AIP 2005). 

- "Strong correlations in low dimensional systems", T. Giamarchi, p.94 in "Lectures on the Physics of Highly Correlated Electron Systems IX", Eds. A. Avella & F. Mancini (AIP 2004).




Prof. Richard Scalettar

Department of Physics University of California, Davis (USA)

Numerical Studies of Disordered Tight-Binding Hamiltonians



I. Introduction & The Translationally Invariant Hubbard model 

    A. Weak and Strong Coupling Approaches 

        i. Noninteracting Limit (U=0) 

        ii. Extreme Strong Coupling Limit (t=0) 

        iii. Greens Functions 

    B. Mean Field Theory 

    C. Exact Diagonalization 

        i. Two sites 

        ii. Pseudocode for many sites 

II. Disorder in the Absence of Interactions 

    A. Anderson Transition 

        i. Mobility Edge 

        ii. Participation ratio 

    B. Digression into Non-Hermiticity: The Hatano-Nelson Model 

        i. Connection between complex eigenvalues and delocalization 

        ii. Gauge Transformations/Boundary Conditions 

    C. Pseudospectra 

III. Practical Prescription for Determinant Quantum Monte Carlo 

    A. Basic Formalism 

        i. Pseudocode for Simplest code 

        ii. Refinements/Complexities 

    B. Results: Local Moment, Spin Correlations, and Specific Heat 

IV. Randomness and Interactions Together 

    A. Interaction driven Anderson Insulator to Metal Transition 

    B. Zeeman Field 

    C. The Role of Particle-Hole Symmetry 

    D. Interaction driven Band Insulator to Metal Transition 

V. Conclusions. 



T1 Creation and Destruction Operators and the Hubbard Hamiltonian 

T2 Review of Classical Monte Carlo 

T3 Derivation of Determinant Quantum Monte Carlo 

T4 Supplementary Material



- P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985). 

- D. Belitz and T.R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994). 

- S.V. Kravchenko, G.V. Kravchenko, J.E. Furneaux, V.M. Pudalov, and M. D'Iorio, Phys. Rev. B 50, 8039 (1994), 

- V. Dobrosavljevic, E. Abrahams, E. Miranda, and S. Chakravarty, Phys. Rev. Lett. 79, 455 (1997). 

- "Conducting phase in the two-dimensional disordered Hubbard model", P.J.H. Denteneer, R.T. Scalettar, and N. Trivedi, Phys. Rev. Lett. 83, 4610 (1999).




Prof. Dirk van der Marel

Département de Physique de la Matière Condensée Université de Genève (Switzerland) 

Optical probes of electron correlations in solids



I. Extracting the optical conductivity from optical experiments: Reflectivity, transmission, ellipsometry, grazing incidence reflection, time-domain experiments, Kramers-Kronig relations 

II. Drude weight, interband transitions, optical sum-rules. 

III. Collective modes, and optical experiments to probe them: Phonons, plasmons, excitons, spin fluctuations. 

IV. Quasi-particle excitations, (generalized) Drude model, dynamical scattering rate and mass renormalization, coupling to phonons, plasmons and spinfluctuations. 

V. Optical spectra of superconductors, correlation functions, kinetic energy, plasmons.



T1 Case studies of Fresnel’s equations, multilayers, anisotropic materials 

T2 Case studies of optical sumrules in molecules, the Heitler-London model. 

T3 Case studies of collective modes, multi-band superconductors. 

T4 Case studies of memory functions, polarons, heavy fermions 

T5 Case studies of restricted sumrules, spin-(dis)order, superconductors.



* Useful to know before the course: 

- Optical properties of solids, Frederick Wooten, Academic Press, 1972 

- Electrodynamics of Solids, Optical Properties of Electrons in Matter, M. Dressel and G. Gruner, Cambridge University Press, 2002 

* Material covered during the course: 

- D. van der Marel, cond-mat/0301506; "Optical signatures of electron correlations in the cuprates" in "Strong interactions in low dimensions", series: Physics and Chemistry of Materials with Low-Dimensional Structures, Vol. 25, Dionys Baeriswyl (Editor), L. Degiorgi (Editor), Kluwer (2005), VI, ISBN: 1-4020-1798-7. 

- D. van der Marel, cond-mat/0410473; "Optical spectroscopy of plasmons and excitons in cuprate superconductors ", Journal of Superconductivity 17, 559-575 (2004).


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