## Lecture "Entanglement in quantum many-body systems" (summer term 2024)

##### and Lecture: Entanglement in quantum many-body systems: Mathematical results.

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

u:find entries:

- 260050 VO Entanglement in quantum many-body systems
- 250068 VO Entanglement in quantum many-body systems: Mathematical results

Overview

**Complex quantum many-body systems** exhibit a wide range of **intriguing phenomena**, which are rooted in their **intricate quantum correlations: quantum entanglement**. This includes e.g.

**topological order**, where systems order in their global quantum correlations, whereas this order is invisible in local properties; such systems can be used in quantum computers for robust quantum information storage;**measurement based quantum computing**, where the complex entanglement of a quantum many-body state can be used to carry out quantum computations solely by measurement**symmetry protected order**, where a seemingly trivial state acquire non-trivial properties due to the presence of a symmetry.

All these intricate and unconventional physical phenomena have their root in the **complex quantum entanglement** present in those systems. At the same time, however, this entanglement makes these systems **challenging to study, both analytically and numerically**.

Fortunately, **understanding the entanglement** in these systems allows to settle this problem: Their entanglement

structure gives rise to a succinct description, called **tensor network states**. Tensor network states **precisely capture the complex entanglement** which govern the behavior of such quantum many-body systems. This makes them both a **powerful analytical tool** to understand unconventional quantum matter, and 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.

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

Unless discussed otherwise, 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 250068 VO Entanglement in quantum many-body systems: Mathematical results.

The lecture will be taught jointly by Norbert Schuch (Faculty of Physics and Faculty of Mathematics), Jose Garre Rubio and Andras Molnar (Faculty of Mathematics), and Bram Vanhecke (Faculty of Physics).

### Course material

#### Lecture notes

Lecture notes for the course will be published here.- 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 and Friday 11:00-12:30 in the Schrödinger Lecture Hall (Boltzmanngasse 5, 5th floor). It is planned that the 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.