Quantum Spin Systems


This is an online TCC course between the Universities of Bath, Bristol, Imperial, Oxford, Warwick and Swansea. More information here, including how to register.

Schedule: Mondays 14-16, from 10 October to 28 November 2022. Assessment: Options include taking an oral exam or writing an essay.

Description:
This course is related to the recent TCC courses Exactly Solvable Models in Statistical Mechanics of Thomas Bothner and Statistical Mechanics with Continuous Symmetries of Bálint Tóth, although knowlege of these courses will not be assumed and overlap will be minimal.

We will review the basic setting of quantum lattice systems, with emphasis on their statistical mechanical properties at equilibrium. We will introduce the notion of states and of phase transitions. Then we will discuss specific models of quantum spins, namely the Heisenberg and XYZ model.

Students are expected to have basic notions of algebra and analysis and some knowledge of statistical mechanics. The course will be mainly self-contained.

Below is a set of lecture notes, that are I am writing with the help of Jakob Björnberg. They give you an idea of the contents of the module. Comments are welcome!

Lecture notes:
. Table of contents (incomplete, version of 4 October)
1. Introduction (complete, version of 4 October)
2. General setting (complete, version of 4 October)
3. Pressure and tangent functionals (complete, version of 6 October)
4. Gibbs variational principle (complete, version of 5 October)
5. Evolution operator and KMS conditions (incomplete, version of 5 October)
6. Extremal Gibbs states (empty)
7. At high temperatures (complete, version of 6 October)
8. Spin systems (empty)
9. The Ising model (empty)
10. Two-dimensional systems with continuous symmetry (empty)
11. Long-range order via reflection positivity (empty)
A. Mathematical supplement (incomplete, version of 4 October)
B. Solutions to some exercises (incomplete, version of 5 October)
. Bibliography (incomplete, version of 4 October)
. Index (incomplete, version of 4 October)