## Quantum Spin Systems [MA5020]

### Sommersemester 2016

### Quantum Spin Systems - An introduction to the general theory, Frustration-Free models, and Gapped Quantum Phases

The lecture course will be on the following dates:

- April: 14, 26, 28
- May: 3, 12, 19, 24, 31
- June: 2, 7

### Content

The lecture course consists of three parts of approximately equal length, four ninety minute lectures each:

1. The first part is devoted to introducing the basic mathematical framework for the
study of quantum spin systems in a form suitable for applications in condensed
matter physics as well as in quantum information and computation theory. This
includes the construction of infinite systems by taking the thermodynamic limit,
Hilbert space techniques based on the GNS representation, Lieb-Robinson bounds,
a survey of the main questions the theory aims to address, and a discussion of several
important model Hamiltonians.

2. The introduction of the AKLT model in 1988 by Affleck, Kennedy, Lieb, and Tasaki
set in motion a series of new developments in the study of quantum spin systems that
continue to have a profound impact on research on quantum spin models today. We
will discuss the theory of Matrix Product States (aka Finitely Correlated States),
Tensor Networks, the Density Matrix Renormalization Group, and techniques to
estimate the spectral gap above the ground state.

3.The third part of the course will focus on specific properties of gapped ground
states and their phase structure, guided by the analysis of specific models. This will
include models with topological order. In each case we will study the ground states,
the spectral gap above the ground state and the nature of the elementary excitations.
Of particular interest are the anyonic excitations associated with topological order
in two dimensional models.

### Prerequisites

The main prerequisite for the course will be familiarity with the basics of quantum
mechanics and elementary properties of linear operators on Hilbert space.

### Literature

Will be provided during the course

### Lecture notes