Last updated: Dec 2022

Mathematics of quantum spin systems (Università di Bologna)

PhD course, funded by INdAM “Programma Professori Visitatori 2023”.

When/where: November 21 – December 21, 2022. Università di Bologna.

  • Mondays 14h-16h, Seminario II
  • Wednesdays 16h-18h, Aula Arzelà

Exercise sheets

Lecture notes


Recordings of the lectures are available on YouTube.


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 Lieb-Robinson 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.


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.


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

  • Lieb-Robinson Bounds:

    • Lieb-Robinson Bounds in Quantum Many-Body 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 141-176. Available at
  • Spectral gap

    • Spectral gaps of frustration-free 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