**XV
Training Course in the Physics of Strongly Correlated Systems**

*Vietri sul Mare (Salerno) Italy*

*October 4 -15, 2010*

*Lecture Topics and
Background References*

**Professor Ole K. Andersen**

Max-Planck Institute for Solid-State
Research

Stuttgart, Germany

Understanding the electronic structure of 3d-electron materials of recent interest, starting from LDA Wannier functions.

**Lectures:**

1. The single-particle picture. Periodic system of elements. Tight-binding
description of the band structure of solids [1]. L_{2}O_{3}
described in the dynamical-mean field approximation (DMFT) [3,4].

3. Transition-metal perovskites. Mott transition in a 3d(t_{2g})^{1}
series; cation control [5]. Coulomb-enhanced spin-orbit coupling in the 4d(t_{2g})^{5}
oxide Sr_{2}RhO_{4} [6]. Pressure-induced metal-insulator
transition in the 3d(t_{2g})(e_{g})^{1} oxide LaMnO_{4}
[7].

4. Trends in band structures of HTSC d^{9-h} cuprates [8].
Superconductors from heterostructures of d^{7} nickelates? [9].

5. Band structure and itinerant magnetism of the new iron-pnictide and
chalcogenide superconductors [10].

References (most of these
papers may be downloaded from
http://www.fkf.mpg.de/andersen/):

[1] Richard M. Martin, , t, tperp(k), and Jperp". J. Phys.Chem. Solids, 56,
(1995), 1573;

E. Pavarini et al. "Band-structure trend in cuprates and correlation with T_{cmax}".
Phys. Rev. Lett. 87 (2001), 047003;

P.R.C. Kent et al. "Combined density-functional and dynamical cluster quantum
Monte Carlo calculations for three-band Hubbard models for hole-doped cuprate
superconductors". Phys. Rev. B 78 (2008), 035132

[9] P. Hansmann et al. "Turning a Nickelate Fermi Surface into a Cupratelike One
through Heterostructuring". Phys. Rev. Lett. 103 (2009), 016401

[10] A. Yaresko et al. "Interplay between magnetic properties and Fermi surface
nesting in iron pnictides". Phys. Rev. B 79 (2009), 144421

Training:
Students will perform simple analytical calculations demonstrating e.g.
downfolding, and understanding covalency and itinerant magnetism via
tight-binding.

**Professor Adrian E. Feiguin**

Department of Physics and Astronomy

University of Wyoming

Laramie WY, USA

The density matrix renormalization group (DMRG) method and its time-dependent variants.

**Lectures:**

1. Exact diagonalization. Numerical Renormalization Group. Disentangling quantum
many body states: the Schmidt decomposition and the density matrix
transformation. The density matrix renormalization group method. Measuring
observables. Targeting multiple states. Calculating gaps. Extension to higher
dimensions. Quantifying entanglement

2. The wave-function transformation. Time evolution using DMRG. The
Suzuki-Trotter decomposition. Adaptive tDMRG. Time-targeting methods.
Time-evolution using the Krylov basis.

3. Applications of the tDMRG method. Calculating time-dependent correlation
functions. Quenches and entanglement growth. Thermo-field formalism. Quantum
purification. Evolution in imaginary time. Thermodynamics.

4. Matrix Product States and DMRG. MPS as a variational ansatz. The AKLT state.
Projected Entangled Pair States (PEPS) Infinite Time Evolving Block Decimation
method (iTEBD). Infinite size algorithms.

5. ALPS libraries. ALPS DMRG.

**References:
**1) S. R. White, Density matrix formulation for quantum renormalization
groups. PRL 69,2863 (1992).

2) S. R. White, Density-matrix algorithms for quantum renormalization groups. PRB 48,10345 (1993)

3) U. SchollwThe density-matrix renormalization group. RMP 77, 259 (2005)

4) K. Hallberg, Density Matrix Renormalization: A Review of the Method and its Applications. arXiv:cond-mat/0303557

5) R. Noack and S. Manmana, Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems. arXiv:cond-mat/0510321

6) Steven R. White and Adrian E. Feiguin, Real-Time Evolution Using the Density Matrix Renormalization Group

Phys. Rev. Lett. 93, 076401 (2004)

7) A. J. Daley, C. Kollath, U. Schollwoeck, G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces. J. Stat. Mech.: Theor. Exp. (2004) P04005

8) G. Vidal, Efficient Simulation of One-Dimensional Quantum Many-Body Systems. Phys. Rev. Lett. 93, 040502 (2004)

9) G. Vidal, Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension. Phys. Rev. Lett. 98, 070201 (2007)

10) Adrian E. Feiguin and Steven R. White, Time-step targeting methods for real-time dynamics using the density matrix renormalization group. Phys. Rev. B 72, 020404 (2005)

11) Adrian E. Feiguin and Steven R. White, Finite-temperature density matrix renormalization using an enlarged Hilbert space. Phys. Rev. B 72, 220401 (2005)

12) Peter Schmitteckert, Nonequilibrium electron transport using the density matrix renormalization group method. Phys. Rev. B 70, 121302 (2004)

13) F. Heidrich-Meisner, A. E. Feiguin, and E. Dagotto, Real-time simulations of nonequilibrium transport in the single-impurity Anderson model. Phys. Rev. B 79, 235336 (2009)

14) Luis G. G. V. Dias da Silva, F. Heidrich-Meisner, A. E. Feiguin, C. A. Bins, E. V. Anda, and E. Dagotto, Transport properties and Kondo correlations in nanostructures: Time-dependent DMRG method applied to quantum dots coupled to Wilson chains. Phys. Rev. B 78, 195317 (2008)

15) K. A. Al-Hassanieh, A. E. Feiguin, J. A. Riera, C. A. BAdaptive time-dependent density-matrix renormalization-group technique for calculating the conductance of strongly correlated nanostructures, Phys. Rev. B 73, 195304 (2006)

16) U. Schollwoeck, S. R. White, Methods for Time Dependence in DMRG. arXiv:cond-mat/0606018

17) U. Schollwoeck, The density matrix renormalization group in the age of matrix product states. arXiv:cond-mat/1008.3477

18) F. Verstraete, D. Porras, J. I. Cirac, DMRG and periodic boundary conditions: a quantum information perspective. Phys. Rev. Lett. 93, 227205 (2004)

19) F. Verstraete, J.I. Cirac, Matrix product states represent ground states faithfully. Phys. Rev. B 73, 094423 (2006)

20) F. Verstraete, M. M. Wolf, D. Perez-Garcia, J. I. Cirac, Criticality, the area law, and the computational power of PEPS. Phys. Rev. Lett. 96, 220601 (2006).

21) D. Perez-Garcia, F. Verstraete, M.M. Wolf, J.I. Cirac, Matrix Product State Representations. Quantum Inf. Comput. 7, 401 (2007)

22) F. Verstraete, J.I. Cirac, V. Murg, Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems. Adv. Phys. 57,143 (2008)

**Professor Hans R. Ott**

ETH Zurich - Department of Physics

Laboratory for Solid State Physics

Zurich, Switzerland

Experimental approaches to strong correlations.

**Lectures topics:**

Fermi liquid, normal liquid 3He, Kondo (dilute and lattice), heavy-electron
metals, non Fermi-liquid aspects

Superconductivity (key properties and related experiments)

Unconventional Superconductivity (examples and experiments)

Physics of low-dimensional systems (examples and experimental approaches)

**Professor Michael Potthoff**

Institute for Theoretical Physics

University of Hamburg

Hamburg, Germany

Variational principles for strongly correlated Fermi systems.

**Lectures:**

1. Correlated electrons: Basic models and methods [1,2,3]

a) Electron correlations

b) Second quantization

c) Exact diagonalization

2. Variational wave functions [4,5]

a) Ritz variational principle

b) Hartree-Fock approximation

c) Gutzwiller wave function and Mott transition

d) Variation of matrix-product states

e) Variational density matrix

f) General approximation strategies

3. Green functions and perturbation theory [1,2,3,6]

a) Diagrammatic perturbation theory

b) Properties of Green functions

c) Luttinger-Ward functional

d) Dynamical functionals

4. Dynamical variational approximations [7,8,9,10]

a) Cluster-perturbation theory

b) Variational cluster approach

c) Dynamical mean-field theory

d) Cluster mean-field theories

5. Applications [4,6,7,8]

a) Collective magnetism

b) Mott transition

c) High-Tc superconductivity

d) Luttinger sum rule

**References:**

[1] J.W. Negele and H. Orland: "Quantum Many-Particle Systems" (Addison-Wesley)

[2] A.A. Abrikosow, L.P. Gorkov and I.E. Dzyaloshinski: "Methods of Quantum
Field Theory in Statistical Physics" (Prentice-Hall)

[3] A.L. Fetter, J.D. Walecka: "Quantum Theory of Many-Particle Systems"
(McGraw-Hill)

[4] F. Gebhard: "The Mott Metal-Insulator Transition" (Springer)

[5] U. Schollwoeck: "The density-matrix renormalization group in the age of
matrix product states", arXiv:1008.3477

[6] M. Potthoff: "Non-perturbative construction of the Luttinger-Ward
functional", Condens. Mat. Phys. 9, 557 (2006), cond-mat/0406671

[7] A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg: "The Local Impurity Self
Consistent Approximation (LISA) to Strongly Correlated Fermion Systems and the
Limit of Infinite Dimensions", Rev. Mod. Phys., 68, 13 (1996), cond-mat/9510091

[8] Th. Maier, M. Jarrell, Th. Pruschke, M.H. Hettler: "Quantum Cluster
Theories", Rev. Mod. Phys. 77, 1027

**Training:** Blackboard discussions. Problem solving.