Mathematics

Mathematics of many-body quantum systems

The course will discuss rigorous methods for the study of many-body systems of importance in quantum statistical mechanics and in condensed matter physics. The focus will be on analytic techniques that allow to describe in a quantitative way the large scale behavior of physically relevant systems. Specifically, we will discuss the following topics.

Part 1: continuous symmetry breaking in quantum spin systems.

• Quantum spin systems. Heisenberg model.
• Continuous symmetry breaking and phase transitions. Heuristics, spin waves.
• Long-range order and spontaneous symmetry breaking in the quantum Heinsenberg antiferromagnet in $d \geq 3$. The method of reflection positivity and infrared bounds.
• Absence of continuous symmetry breaking in d=2, Hohenberg-Mermin-Wagner theorem.
• Ground states of antiferromagnetic Heisenberg chains, Haldane’s conjecture, AKLT model.

Part 2: rigorous renormalization group and quantum transport.

• Fermionic lattice models, grandcanonical description, Fock space.
• Perturbation theory, Wick’s theorem, diagrammatics.
• Exponential decay of correlations for interacting gapped models, Brydges-Battle-Federbush formula.
• Infrared divergences in gapless models. The case of graphene. Rigorous renormalization group analysis: Gallavotti-Nicolò tree expansion.
• Transport at criticality: universality of graphene’s conductivity.

References:

• Hal Tasaki, Physics and Mathematics of Quantum Many-Body Systems. Springer (2020).
• F. J. Dyson, E. H. Lieb and B. Simon. Phase Transitions in Quantum Spin Systems with Isotropic and Non isotropic Interactions. Journal of Statistical Physics, Vol. 18, No. 4, (1978)
• Jürg Fröhlich, Robert Israel, Elliot H. Lieb & Barry Simon. Phase transitions and reflection positivity. I. General theory and long range lattice models. Communications in Mathematical Physics, Volume 62, 1–34, (1978) 
• Jürg Fröhlich, Robert B. Israel, Elliott H. Lieb & Barry Simon. Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions. Journal of Statistical Physics, Volume 22, 297–347, (1980).
• G. Gentile, V. Mastropietro. Renormalization group for one-dimensional fermions. A review on mathematical results. Physics Reports 352 (4-6), 273-437 (2001)
• A. Giuliani and V. Mastropietro. The Two-Dimensional Hubbard Model on the Honeycomb Lattice. Communications in Mathematical Physics, Volume 293, 301–346, (2010). 
• A. Giuliani, V. Mastropietro, M. Porta. Universality of conductivity in interacting graphene. Communications in Mathematical Physics 311, 317-355 (2012)