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Quantum Spin Systems [MA5020]

Sommersemester 2016

Prof. Dr. Bruno Nachtergaele

Dozent: Prof. Dr. Bruno Nachtergaele
Vorlesung: Tuesday 10:15-11:45, room 03.08.011 and Thursday, 10:15-11:45, room 03.06.011 Anmeldung

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:


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.


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


Will be provided during the course

Lecture notes

Lecture 1
Lecture 2 and 3
Notes for May 12-19
Notes for May 24
Notes for May 31 and June 2