260050 VO Entanglement in quantum many-body systems (2024S)

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 acquires non-trivial properties due to the presence of a symmetry.
  

All these intricate and unconventional physical phenomena have their roots 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 governs 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 provides a comprehensive introduction to tensor networks, with a focus on their use in modeling quantum many-body systems.

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 gives a comprehensive introduction to the field of tensor networks. This includes 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 consists of 4h of lecture per week, i.e. both Thursday and Friday, and lasts for the first half of the semester (until early May).

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

Track A: "Mathematical theory of tensor networks". This part specializes 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 are mostly algebraic.

Track B: "Numerical simulations with tensor networks". This part gives a 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 in particular also includes hands-on programming exercises.

Unless agreed otherwise, Track A is held in the Friday slot, and Track B is held in the Thursday slot, both starting at the middle of the semester. Students who attend one of the tracks can earn ECTS points for this course [260050 VO Entanglement in quantum many-body systems (2024S)]. Students who wish to attend both tracks can additionally earn ECTS points for the course https://ufind.univie.ac.at/en/course.html?lv=260050&semester=2024S

The lecture is taught jointly by Norbert Schuch (Faculty of Physics and Faculty of Mathematics), José Garre Rubio and András Molnár (Faculty of Mathematics), and Bram Vanhecke (Faculty of Physics).

For further information, see the lecture's website at https://schuch.univie.ac.at/nschuch/ent-qmb-ss24/