Abstract

The course will focus on rigorous methods for the analysis of correlated systems in quantum statistical mechanics. The first part will be devoted to quantum spin systems and to the emergence of long-range order. Topics will include an introduction to the Heisenberg model, the absence of phase transitions at high temperature, the Hohenberg-Mermin-Wagner theorem, and the method of reflection positivity, culminating in a proof of long-range order in the Heisenberg antiferromagnet. Along the way, we will also discuss the Lieb-Robinson bounds and the propagation of locality in quantum dynamics, as well as the proof of exponential decay of correlations for gapped ground states. In the second part of the course, we will develop a rigorous renormalization group framework for interacting fermionic systems. After a review of the many-body formalism, we will address the convergence of fermionic perturbation theory and set up a renormalization group analysis of interacting semimetals. As an application, we will discuss the universality of conductivity in interacting graphene.

More information can be found at: https://www.math.sissa.it/course/phd-course/mathematics-many-body-quantum-systems