Quantum Lattice 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: Tuesdays 10-12, from 14 October to 9 December 2025 (excepted 25 November).

Assessment: Options include taking an oral exam or writing an essay.

Description:
Quantum lattice models have been introduced by physicists in order to better understand the electronic properties of condensed matter. Their importance for physics cannot be overstated even though the description is qualitative rather than accurate. They offer attractive challenges for mathematicians, prompting them to develop tools in algebra, analysis, functional analysis, probability theory, and combinatorics.

We will review the basic setting of quantum lattice systems, with emphasis on their statistical mechanical properties at equilibrium. Several models of interests will be discussed in details, notably: (quantum) Ising, Heisenberg, Hubbard, and Bose-Hubbard. We will introduce the notion of states and of phase transitions.

Basic notions of algebra and analysis are enough and the course will be self-contained. Many arguments are creative and challenging.

The main reference is a set of lecture notes co-written with Jakob Björnberg.

Lecture notes:
. Table of contents (complete, version of 10 December)
. Introduction (complete, version of 22 September)
1. Spin systems (complete, version of 13 October)
2. Fermionic and bosonic systems (complete, version of 29 November)
3. Equilibrium states (complete, version of 29 November)
4. Uniqueness and non-uniqueness of Gibbs states (complete, version of 2 December)
5. Mean-field systems (complete, version of 2 December)
6. 2D systems with continuous symmetry (complete, version of 10 December)
A. Mathematical supplement (complete, version of 9 December)
B. Solutions to some exercises (complete, version of 9 December)
. Bibliography (complete, version of 10 December)
. Index (complete, version of 10 December)

Slides:
Week 1, 14 October: Background, spin operators, spin systems, phase diagrams.
Week 2, 21 October: Lattice systems of particles, bosonic spaces, creation and annihilation operators, Bose-Einstein condensation.
Week 3, 28 October: Lattice fermion systems + states and Gibbs states (finite volume).
Week 4, 4 November: Infinite volume setting: observables, interactions, states.
Week 5, 11 November: KMS condition (infinite volume), extremal states, decomposition of states.
Week 6, 18 November: Unique Gibbs state at high temperatures, long-range order at low temperatures in the Ising model.
Week 7, 2 December: Long-range order at low temperatures in the XXZ model; mean-field systems.
Week 8, 2 December: No breaking of continuous symmetry in 2D; quantum Pinsker inequality.