Computational Methods in Many-Body Physics

This module provides an introduction to numerical methods for the simulation of classical and quantum many-particle systems. A focus lies on the investigation of model systems that describe strongly correlated quantum matter. The emergent physical phenomena in such system are often out of reach for analytical approaches and thus numerical approaches are essential for their understanding. The following methods will be covered in the course: 

• Classical Monte Carlo simulations

• Finite size scaling analysis

• Exact diagonalization 

• Many body entanglement 

• Matrix product states 

• Tensor product states 

• Quantum Monte Carlo methods 

• Non-equilibrium field theory

After successful completion of the module the students are able to:

1. know and reflect the recent developments and open questions in computational many-body physics
2. understand state-of-the-art numerical techniques applied in condensed matter theory
3. judge which numerical method is best suited to sovle a new problem
4. program non-trivial codes in python

Quantum mechanics and statistical physics of the Physics Bachelor.

The modul consists of a lecture and exercise classes.

The lecture is designed for the presentation of the subject, usually by blackboard presentation. The focus resides on theoretical foundations of the field, presentation of methods and simple, illustrative examples. Command of basic methods is deepened and practised through homework problems, which cover important aspects of the field. The homework problems should develop the analytic skills of the students and their ability to perform calculations. The homework problems are discussed in the exercises by the students themselves under the supervision of a tutor in order to develop the skills to explain a physics problem logically.

Oral presentation, blackboard work, lecture notes as PDF for download, beamer presentation, exercise sheets, problems to solve on the PC, accompanying website to the lecture

Introduction to Python, www.scipy-lectures.org
Lecture Notes by Anders W. Sandvik, arxiv.org/abs/1101.3281v1
Lecture Notes by Johannes Hauschild, Frank Pollmann, arxiv.org/abs/1805.00055
Review on DMRG by Ulrich Schollwoeck, arxiv.org/abs/1008.3477
Lecture Notes by Juergen Berges, arxiv.org/abs/1503.02907
Online book by Michael Nielsen, neuralnetworksanddeeplearning.com

The achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using presentations independently prepared by the students. The exam of 25 minutes consists of the presentation and a subsequent discussion.

For example an assignment in the exam might be:

• Implement the N-state Potts model using Monte Carlo.
• Study the critical behavior of the (classical) 3D Ising model using the Swendsen-Wang algorithm.
• Program the Krylov time evolution for a random Heisenberg chain.
• Construct the reduced density matrix from MPS.

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.