IX Training Course in the Physics of

Correlated Electron Systems and High-Tc Superconductors


Vietri sul Mare (Salerno) Italy

4 - 15 October 2004


Lecture Topics and Background References


Prof. Kazumi Maki

Department of Physics and Astronomy, University of Southern California, LA 

BCS theory of Nodal Superconductors



1. Introduction

2. QP spectrum in nodal superconductors 

3. Effect of impurity scattering 

4. Universal Heat conduction 

5. Vortex state in nodal superconductors 

6. Quasiclassical approximation 

7. QP spectrum in a planar magnetic field 

8. Gap symmetries in Sr2RuO4, CeCoIn5, k-(ET)2Cu(Ncs)2 and UPd2Al3 

9. hybrid g+s wave superconductivity in YNi2B2C and p+h-wave in PrOs4Sb12

10. Conclusions.




[1] Abrikosov, Gor'kov, Dzyaloshinskii, Method of Quantum Field Theory in Statistical Physics, Chap.1 Sec.2, Chap.3 and Chap.7 

[2] G.E. Volovik, JETP Lett. 58, 496 (1993) 

[3] H. Won and K.Maki, cond-mat/0004105 

[4] H. Won et al., Brazilian J. Phys. 33, 675 (2003)





Prof. Manfred Sigrist

ETH-Institute for Theoretical Physics, Department of Physics, Zurich

Phenomenological aspects of unconventional superconductivity



I. Conventional versus unconventional superconductivity. 

II. Generalized BCS-theory: basic concepts and issue of symmetry for Cooper pairing. 

III. Effects of impurities and spin susceptibility. 

IV. Generalized Ginzburg-Landau formulation. 

V. Exotic phenomena in unconventional superconductors based on Ginzburg-Landau theory.



[1] M. Tinkham: Introduction to Superconductivity, McGraw-Hill 

[2] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991) 

[3] V.P. Mineev and K.V. Samokhin, Introduction to Unconventional superconductivity, Gordon and Breach Science Publisher. 

[4] A.P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003).



Prof. Hideki Matsumoto

Institute of Physics, University of Tsukuba 

New Theoretical Tools for Correlated Insulators and Metals



1. Introduction 

2. Models of Highly Correlated Particle Systems 

2.1. Fermionic System 

2.1.1. Hubbard Model 

2.1.2. Anderson Model 

2.1.3. p-d Model 

2.1.4. Model of Transition Metal Oxide 

2.2. Bosonic System 

2.2.1. Bose-Hubbard Model in Josephson Junction 

2.2.2. Bose Hubbard Model of the Bose Gas in an Optical Lattice. 

3. Theoretical Methods 

3.1. Hubbard Approximation 

3.2. Non-Crossing Approximation 

3.3. Slave Boson Method 

3.4. Composite Operator Method 

4. Analysis of the p-d Model in the Composite Operator Method 

5. Analysis of the Hubbard Model in the Composite Operator Method 

6. Remarks




[1] J. Hubbard, Proc. R. Soc. London Ser A 276,238 (1963); 277, 237 (1964), 281,401 (1964) 

[2] P. W. Anderson, Phys. Rev. 124, 41 (1961). 

[3] V. J. Emery, Phys. Rev. Lett. 58 2794 (1987). 

[4] S. Ishihara, H. Matsumoto, and M. Tachiki, Phys. Rev. B 42, 10041 (1990). 

[5] S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B 55, 8280 (1997).

[6] S. Ishihara, M. Yamanaka, and N. Nagaosa, Phys. Rev. B 56, 686 (1997). 

[7] M. P. A. Fisher, P. B. Weichman, G. Grinstein and D. S. Fisher, Phys. Rev. B 40, 546 (1989). 

[8] E. Roddick and D. Stroud, Phys. Rev. 48, 16600 (1993). 

[9] D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. Rev. A 63, 053601 (2001). 

[10] Y. Kramoto, Z. Phys. B 53, 37 (1983). 

[11] P. Coleman, Phys. Rev. B 29, 3035 (1984). 

[12] H. Matsumoto, M. Sasaki, S. Ishihara, and M. Tachiki, Phys. Rev. B 46, 3009. [13] M. Sasaki, H. Matsumoto, and M. Tachiki, Phys. Rev. B 46, 3022 (1992). 

[14] S. Ishihara, H. Matsumoto, S. Odashima, M. Tachiki and F. Mancini, Phys. Rev. B 49, 1350 (1994). 

[15] F. Mancini, S. Marra and H. Matsumoto, Physics C 252 361 (1995). 

[16] H. Matsumoto, T. Saikawa, and F. Mancini, Phys. Rev. B 54, 14445 (1996). 

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

[18] F. Mancini and A. Avella, ”A Review of the Hubbard Model within the Equations of Motion Approach”, (to be published in Advanced Physics), and references refered there.




Prof. Reinhard M. Noack

Arbeitsgruppe Vielteilchennumerik, Fachbereich Physik Philipps, 

Universitaet Marburg

Diagonalization- and Numerical Renormalization-Group-based Methods for Interacting Quantum Systems



I. Exact Diagonalization 

(i) Introduction to interacting quantum systems 

(ii) Representation of many-body states 

(iii) Complete Diagonalization 

(iv) Iterative Diagonalization (Lanczos and Davidson) 

(v) Dynamics 

(vi) Finite temperature 

II. Numerical Renormalization Group 

(i) Kondo problem 

(ii) Numerical RG for the Kondo problem 

(iii) Numerical RG for quantum lattice problems 

(iv) Numerical RG for a noninteracting particle 

III. From the NRG to the Density Matrix Renormalization Group 

(i) Better methods for the noninteracting particle 

(ii) Density Matrix Projection for interacting systems 

(iii) DMRG Algorithms 

(iv) DMRG-like algorithm for the noninteracting particle 

IV. The DMRG in Detail 

(i) Efficient programming 

(ii) Measurements 

(iii) Wavefunction transformations 

(iv) Extensions to higher dimension 

V. Recent Developments in the DMRG 

(i) Finite temperature 

(ii) Dynamics 

(iii) Quantum chemistry 

(iv) Time evolution 

(v) Matrix product states 

(vi) Quantum information



[1] Density Matrix Renormalization: A New Numerical Method in Physics, Eds. I. Peschel, X. Wang, M. Kaulke, and K. Hallberg, Springer Verlag, Berlin, June 1999. 

[2] E. Dagotto, “Correlated electrons in high-temperature superconductors”, Rev. Mod. Phys. 66, 763-840 (1994). 

[3] A. Booten and H. van der Vorst, “Cracking large-scale eigenvalue problems, part I: Algorithms”, Computers in Phys. 10, No. 3, p. 239 (May/June 1996); “Cracking large-scale eigenvalue problems, part II: Implementations, Computers in Phys. 10, No. 4, p. 331 (July/August 1996). 

[4] K. G. Wilson, “The renormalization group and critical phenomena”, Rev. Mod. Phys. 55, 583-600 (1983). 5. U. Schollw¨ock, cond-mat/0409292.




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