Autumn 2025 • TCC module
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.
A set of lecture notes (co-written with Jakob Björnberg) are being prepared. They are not finished yet, but the current draft gives a good idea of the contents of the module. Comments are welcome!
Lecture notes:
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.
A set of lecture notes (co-written with Jakob Björnberg) are being prepared. They are not finished yet, but the current draft gives a good idea of the contents of the module. Comments are welcome!
. Table of contents (incomplete, version of 25 October)
. Introduction (complete, version of 22 September)
1. Spin systems (complete, version of 13 October)
2. Fermionic and bosonic systems (complete, version of 28 October)
3. Equilibrium states (incomplete, version of 28 October)
4. Uniqueness and non-uniqueness of Gibbs states (incomplete, version of 22 September)
5. Mean-field systems (incomplete, version of 22 September)
6. 2D systems with continuous symmetry (incomplete, version of 22 September)
A. Mathematical supplement (incomplete, version of 22 September)
B. Solutions to some exercises (incomplete, version of 26 October)
. Bibliography (incomplete, version of 22 September)
. Index (incomplete, version of 22 September)
. Introduction (complete, version of 22 September)
1. Spin systems (complete, version of 13 October)
2. Fermionic and bosonic systems (complete, version of 28 October)
3. Equilibrium states (incomplete, version of 28 October)
4. Uniqueness and non-uniqueness of Gibbs states (incomplete, version of 22 September)
5. Mean-field systems (incomplete, version of 22 September)
6. 2D systems with continuous symmetry (incomplete, version of 22 September)
A. Mathematical supplement (incomplete, version of 22 September)
B. Solutions to some exercises (incomplete, version of 26 October)
. Bibliography (incomplete, version of 22 September)
. Index (incomplete, version of 22 September)
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 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).
