PhD course, funded by INdAM “Programma Professori Visitatori 2023”.
When/where: November 21 – December 21, 2022. Università di Bologna.
 Mondays 14h16h, Seminario II
 Wednesdays 16h18h, Aula Arzelà
Exercise sheets
Lecture notes
 Lesson 1, 20221121
 Lesson 2, 20221123
 Lesson 3, 20221128
 Lesson 4, 20221130
 Notes on LiebRobinson bounds
 Lesson 5, 20221205
 Lesson 6, 20221207
 Lesson 7, 20221212
 Notes on projections lemma
 Lesson 8, 20221214
Recordings
Recordings of the lectures are available on YouTube.
Content
Quantum spin systems constitute a very rich and complex landscape in which many interesting mathematical problems arise. Traditionally studied as a model for condensed matter physics, in the last two decades they have gained prominence also in the field of quantum information and computation, as they describe many of the proposed “quantum hardware” implementations.
The aim of the course is to give an introduction to the mathematical formalism of quantum spin systems on a lattice. After recalling some elementary facts about quantum mechanics, I will present the construction of the thermodynamic limit via the so called LiebRobinson bounds. We will then explore other consequences of these bounds, such as the relationship between spectral gap and clustering of correlations. At the end of the course, I will present and discuss some of the techniques used to estimate spectral gap of local Hamiltonians.
Prerequisites
No previous knowledge of physics or quantum mechanics will be required. You should be familiar with linear algebra and properties of operators on finite dimensional vector spaces (such as the spectral theorem and functional calculus, tensor products of Hilbert spaces), as well as a basic knowledge of analysis.
Bibliography
These are some useful references for the topics presented in the course. We will not follow any source from beginning to end, but we will instead jump from one of the other as we need. This is not a list of required reading material.

Spin systems: Mathematical framework
 Lecture Notes “Introduction to Quantum Spin Systems”, B. Nachtergaele and R. Sims, available online https://www.math.ucdavis.edu/~bxn/
 Physics and Mathematics of Quantum ManyBody Systems, H. Tasaki, Springer International Publishing, 2020
 Quantum Spin Systems on Infinite Lattices, P. Naaijkens, Lecture Notes in Physics 933, Springer International Publishing (Cham), 2017. Available at https://arxiv.org/abs/1311.2717
 Operator algebras and quantum statistical mechanics, O. Bratteli and D. W. Robinson, 2 ed., vol. 1 & 2, Springer Verlag, 1987 & 1997

LiebRobinson Bounds:
 LiebRobinson Bounds in Quantum ManyBody Physics, B. Nachtergaele, R. Sims. In “Entropy and the Quantum”, R. Sims and D. Ueltschi (Eds), Contemporary Mathematics, volume 529, American Mathematical Society (2010) pp 141176. Available at https://arxiv.org/abs/1004.2086

Spectral gap
 Spectral gaps of frustrationfree spin systems with boundary, M. Lemm and E. Mozgunov, J. Math. Phys. 60, 051901 (2019).
 Divide and conquer method for proving gaps of frustration free Hamiltonians, M. J. Kastoryano and A. Lucia, Journal of Statistical Mechanics: Theory and Experiment, Volume 2018, March 2018