daily from Wednesday, Sep 6, until Friday, Sep 29
10:00 - 11:30 (lectures) and 13:00 - 14:30 (tutorials)
Seminarraum 0.01 TP (new theory building)
(Wednesday, Sep 20, until Friday, Sep 22: Seminarraum Theorie in the old theory building)

Please note: The lecture consists of two parts. Part II (Sep 18-29) will be given by Prof. S. Trebst. The following description of content etc. refers to part I.

This lecture gives an introduction to numerical methods for the investigation of quantum many-particle systems. The focus in on models of strongly correlated electron systems (Hubbard model, single-impurity Anderson model) and quantum spin models (Heisenberg model, Kitaev model). The physical phenomena (Mott transitions, Kondo physics, spin liquid physics, etc.) these models are supposed to describe, are quite often out of the reach of analytical techniques - this triggered the development of very powerful numerical approaches, see Sec. 3 in the table of contents. The lecture also includes a brief introduction to basic theoretical concepts, such as Green functions, continued fraction expansions, reduced density matrices, entanglement measures (see Sec. 2).

Module description of the primary area of specialization `Solid State Theory/Computational Physics'

Contents:

1. Introduction
   1.1 many-particle systems in solid state theory
   1.2 the basic models
   1.3 physical quantities
2. Quantum Many-Particle Systems: Basics
   2.1 single-particle and many-particle spectra
   2.2 Green functions
   2.3 reduced density matrix and entanglement
3. Quantum Many-Particle Systems: Methods
   3.1 Exact Diagonalization
   3.2 Numerical Renormalization Group
   3.3 Density-Matrix Renormalization Group
   3.4 Quantum Monte Carlo
   

script

Part II will cover the following topics:

• percolation
  Monte-Carlo sampling
  Ising model/cluster approaches
  extended ensemble techniques
  quantum Monte-Carlo
  entanglement

Literature:

Here is a selection of review articles, covering the topics in Section 3:

3.2 Numerical Renormalization Group

• R. Bulla, T.A. Costi, and Th. Pruschke
  Numerical renormalization group method for quantum impurity systems
  Rev. Mod. Phys. 80, 395 (2008)

3.3 Density-Matrix Renormalization Group

• U. Schollwöck
  The density-matrix renormalization group
  Rev. Mod. Phys. 77, 259 (2005)

The Autumn School on Correlated Electrons, held every year at the Forschungszentrum Jülich, contains lots of useful overview articles on many-body techniques which are all available online.

Tutorials:

The exercise sheets contain both analytical and programming exercises. We recommend to use the Julia programming language (templates for some of the exercises will be provided).

Exercises:

• sheet 1
• sheet 2
• sheet 3
• sheet 4
• sheet 5
• sheet 6
• sheet 7

Please note: we recommend to go through all the exercise sheets, but in the tutorials the focus will be on the following exercises (this is a preliminary plan for the first eight tutorials):

- tutorial 1 (Sep 6): s1/ex1 single-particle and many-particle spectra, notebook
- tutorial 2 (Sep 7): s2/ex3 tight-binding chain; density of states; s2/ex4 spin-models on a three-site cluster, notebook
- tutorial 3 (Sep 8): s3/ex5 Hamilton matrices for spin models; s4/ex4 the reduced density matrix, notebook
- tutorial 4 (Sep 11): s3/ex1 Hamilton matrix of the tight-binding chain, notebook
- tutorial 5 (Sep 12): s4/ex3 spin correlations of one-dimensional spin models, notebook
- tutorial 6 (Sep 13): s5/ex4 reduced density matrix and entanglement entropy, notebook
- tutorial 7 (Sep 14): s6/ex1 entanglement entropy for one-dimensional spin models
- tutorial 8 (Sep 15): s5/ex3 Lanczos algorithm, notebook

Spinconventions.ipynb
Sheet 5 - Exercise 1
Sheet 5 - Exercise 2
2D_Ising_Notes.pdf, 2D_Ising_Notes.ipynb