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 module: 

• 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 (PH0007) and statistical physics (PH0008) 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 supervision 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 examination for the module is taken in the form of an examination course.

The exam course consists of a written exam (90 minutes) and a programming task. In the run-up to the exam, each student is given a problem to be solved with a program. The program is handed in with the exam. In the programming task, the students show that they are able to find the best numerical method for the problem and implement it in a functioning program.

The subsequent written exam of 90 minutes covers all topics of the module. In it, the achievement of the competencies presented in the section Learning Outcomes is tested at least in the level of knowledge indicated there by means of knowledge and comprehension questions and by means of examples. Examination task could be, for example:

•     Which algorithm is suitable to study classical Ising phase transitions?
•     What are the implications of the so-called "area law"?
•     Describe the backpropagation algorithm for training neural networks?

The grade is based on 2/3 of the exam and 1/3 of the programming task.