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

Module MA5020

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information

MA5020 is a semester module in English language at Master’s level which is offered once.

This Module is included in the following catalogues within the study programs in physics.

  • Catalogue of non-physics elective courses
Total workloadContact hoursCredits (ECTS)
90 h 30 h 3 CP

Content, Learning Outcome and Preconditions


i. 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. ii. The introduction of the AKLT model in 1988 by Affeck, 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. iii. 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.

Learning Outcome

Students will master the foundations of the mathematical theory of quantum spin systems and will become familiar with a number of important examples of such systems and with the central questions of the field. They will be able to formulate physically relevant properties in mathematically precise terms and be able to demonstrate this knowledge by solving problems related to the material.


The main mathematical prerequisite is familiarity with the basic elements of linear operators on Hilbert spaces. Some background in the elements of quantum mechanics will also be helpful.

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VO 2 Quantum Spin Systems Nachtergaele, B. Dienstag, 10:15–11:45
Donnerstag, 10:15–11:45

Learning and Teaching Methods





Lecture notes will be made available as the course progresses. These will also include references to some additional literature.

Module Exam

Description of exams and course work

Students will be evaluated in a written exam (120 minutes). Because problem solving questions probe the depth of understanding most directly, the majority of the exam questions will be of that type. The use of course notes distributed by the instructor in association with the lectures will be allowed during the exam.

Exam Repetition

There is a possibility to take the exam at the end of the semester.

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