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!

. Table of contents (incomplete, version of 22 September)
. Introduction (complete, version of 22 September)
1. Spin systems (incomplete, version of 22 September)
2. Fermionic and bosonic systems (incomplete, version of 22 September)
3. Equilibrium states (incomplete, version of 22 September)
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 22 September)
. Bibliography (incomplete, version of 22 September)
. Index (incomplete, version of 22 September)