# Numerical Methods for Many-Particle Systems

## Short graduate lecture course: Institute for Theoretical Physics, University of Cologne, Germany

WS 2012

In conjunction with the Bonn-Cologne Graduate School of Physics and Astronomy.

### Lecturers

Dr. Andrew Mitchell, Priv.-Doz. Dr. Ralf Bulla, Prof. Dr. Simon Trebst### Course Summary

This intensive course is intended to provide both a working understanding and real hands-on experience with the essential numerical techniques of solid state many-body physics.Rather than a 'black-box' philosophy, the course aims to discuss the theory and physics underpinning numerical approaches. Lectures will introduce models of central importance, such as the Ising model, the Anderson impurity model, the Hubbard model and the Heisenberg model. Using these as concrete examples, the Monte Carlo, Exact Diagonalization, Numerical Renormalization Group and Density Matrix Renormalization Group techniques will be discussed. Students will also gain supervised practical hands-on experience writing, using and modifying simple computer codes to solve real problems.

### Outline / Timetable

**Lecture 1:**Introduction to many particle methods. Ising model and Monte Carlo.

**Computer practical:**Monte Carlo for the classical Ising model.

**Lecture 2:**Basic toolbox for many-particle quantum systems.

Non-interacting quantum systems: diagonalization by orthogonal transformation of operators, calculation of physical quantities.

**Lecture 3:**Quantum Monte Carlo.

**Computer practical:**Basic toolbox for quantum systems.

**Lecture 4:**Green function methods, spectral functions, broadening discrete numerical data, Kramers-Kronig transformations etc.

**Lecture 5:**Exact diagonalization for intereacting quantum systems. Hubbard model.

**Computer practical:**Exact diagonalization for the 1d Hubbard model. Calculations of spectral functions.

**Lecture 6:**Discretization / coarse-graining methods. Quantum impurity problems.

**Lecture 7:**Numerical Renormalization Group (NRG).

**Computer practical:**NRG for the Anderson model.

**Lecture 8:**Renormalization and universality in the Kondo problem.

**Lecture 9:**Introduction to the ALPS project. Density Matrix Renormalization Group (DMRG).

**Computer practical:**DMRG for Heisenberg models.

**Lecture 10:**Summary of numerical methods for many-particle systems.

### Resources:

Instructions and exercise sheet for computer practical 1 can be found hereLecture slides on Monte Carlo for the Ising model can be downloaded here and here

Lecture slides on Quantum Monte Carlo can be downloaded here

Instructions and exercise sheet for computer practical 2 can be found here

Additional information about the C matrix class can be found here

Lecture slides for 'Quantum Systems and the basic toolbox' can be found here

Lecture slides for 'Green functions and equations of motion' can be found here

Instructions and exercise sheet for computer practical 3 can be found here

Lecture slides for 'Exact Diagonalization' can be found here

Lecture slides for 'Discretization and the Anderson Impurity Model' can be found here

Instructions and exercise sheet for computer practical 4 can be found here

Lecture slides for 'Numerical Renormalization Group' can be found here

Lecture slides for 'NRG for quantum impurity problems' can be found here

Computer practical 5: online tutorials for the ALPS package. Start off with the DMRG calculation for the Heisenberg spin chain, located here

Lecture slides for 'The ALPS project: open source software for strongly correlated systems' can be found here

Lecture slides for 'Summary: numerical methods for many-particle systems' can be found here