## Lecture "Tensor network methods in many-body physics" (summer term 2023)

##### and Lecture: Mathematical Aspects of Tensor Networks in Many-Body Physics.

Lecturers: Norbert Schuch, Bram Vanhecke, Andras Molnar, Jose Garre Rubio

ufind entries:

- 260061 VO Tensor network methods in many-body physics
- 250148 VO Mathematical Aspects of Tensor Networks in Many-Body Physics

Overview

Tensor networks are a powerful framework for the study of many body-systems. Most importantly, they form the right language to study quantum many-body systems, both analytically and numerically. This in particular includes systems which exhibit exotic types of order (so-called "topological order"), which cannot be described by the standard framework of symmetry breaking and local order parameters, as well as other types of systems where quantum correlations play an important role.

The key reason for their success is that tensor networks are precisely built to capture the complex entanglement (i.e., the quantum correlation) which govern the behavior of such quantum many-body systems. On the one hand, this makes tensor networks a powerful analytical tool to analytically understand and characterize the different unconventional phases and to build exactly solvable models. On the other hand, it also makes them a powerful ansatz for the numerical simulation of complex quantum many-body problems which are not susceptible to other methods due to their intricate quantum correlations.

Beyond that, tensor networks also naturally appear in the description of problems in classical statistical mechanics, where they give rise to extremely accurate numerical methods, as well as e.g. in the modeling of high-dimensional data.

This lecture will provide a comprehensive introduction to tensor networks, with a focus on their use in modeling quantum many-body systems.

Structure of the course

The lecture consists of two parts, which are given in the first and second half of the term, respectively.

The first part of the lecture will give a comprehensive introduction to the field of tensor networks. This will include an introduction to the key concepts, as well as the basics of both the analytical and the numerical use of tensor networks. The first part will consists of 4h lecture per week, i.e. both Thursday and Friday, and last for the first half of the semester (until early May).

For the second part of the lecture, there will be two tracks. It will be possible to either choose one track, or to take both tracks (see below). Each track will consists of 2h lecture per week, starting in the middle of the semester.

##### Track A: "Mathematical theory of tensor networks"

This part will specialize on mathematical aspects of tensor networks. This in particular covers the use of tensor networks in the classification of exotic phases with topological order, and their representation theory. The topics in this specialization will be mostly algebraic.

##### Track B: "Numerical simulations with tensor networks"

This part will give an detailed introduction to the different use of tensor networks for the numerical simulations of quantum many-body systems, as well as problems in statistical mechanics, in one, two and three dimensions. This track will in particular also include hands-on programming exercises.

##### Organization of Tracks

Track A will be held in the Friday slot, and Track B will be held in the Thursday slot, starting at the middle of the semester. Students who attend one of the tracks will earn ECTS points for this course. Students who wish to attend both tracks will additionally earn ECTS points for the course 250148 VO Mathematical Aspects of Tensor Networks in Many-Body Physics.

The first part of the lecture will be taught by Norbert Schuch (Faculty of Physics and Faculty of Mathematics). Track A will be taught by Jose Garre Rubio and Andras Molnar (both Faculty of Mathematics), and Track B will be taught by Bram Vanhecke (Faculty of Physics).

### Course material

#### Lecture notes

- Introduction
- Entanglement
- Matrix Product States
- Simulations with MPS
- Solvable models and the classification of phases
- Tensor Networks in two and higher dimensions

### Organisatorial issues

##### Time and Place

The lecture takes place Thursday 14:45-16:15 in the Boltzmann Lecture Hall (Boltzmanngasse 5, ground floor) and Friday 9:00-10:30 in the Schrödinger Lecture Hall (Boltzmanngasse 5, 5th floor). It is planned that specialization Track A will take place in the Friday slot, and Track B in the Thursday slot. See the ufind entry for more details.

##### Exam

The exam will be an oral exam of 30 minutes duration, or 40 minutes if both tracks are taken.

Beyond the exam dates indicated in u:find, there is also the possibility to make individual appointments for exams. It is recommended that students interested in taking the exam contact Prof. Schuch prior to registering the exam to discuss the date and time of the exam.