Mathematics

The course will discuss rigorous methods for the study of quantum statistical mechanics models of importance in condensed matter physics. The focus will be on analytic techniques that allow to describe in a quantitative way the large scale behavior of many-body systems in the thermodynamic limit.

Program:

• Quantum spins on a lattice. Heisenberg model.
• Ground states of the Heinsenberg model. Marshall-Lieb-Mattis theorem.
• Long range order and spontaneous symmetry breaking. Emergence of gapless excitations. Kaplan-Horsch-von der Linden and Koma-Tasaki theorems. 
• Cluster expansion at high temperature, absence of long-range order.
• Lieb-Robinson bounds, propagation of locality in quantum dynamics. Fast decay of correlations for gapped quantum systems.
• Continuous symmetry breaking and phase transitions. Heuristics, spin waves.
• Absence of continuous symmetry breaking in d=2. XY model, Berezinskii harmonic approximation, Hohenberg-Mermin-Wagner theorem. Proof by complex deformation.
• Long-range order and spontaneous symmetry breaking in the quantum Heinsenberg antiferromagnet in d≥3. The method of reflection positivity and infrared bounds.
• Fermions on a lattice. Fock space, second quantization. The Hubbard model. Connection with antiferromagnetism. General properties of the ground state: Lieb’s theorems for the attractive and for the repulsive Hubbard models.
• Absence of magnetic ordering for the attractive Hubbard model via reflection positivity, Kubo-Kishi theorem. Absence of superconducting and of charge ordering for the repulsive Hubbard model at half-filling (Shiba transformation).
• Approach to criticality: introduction to the rigorous renormalization group for interacting fermionic models at zero and at low temperatures. 
• Perturbation theory, Wick’s theorem, diagrammatics.
• Convergence of fermionic perturbation theory. Brydges-Battle-Federbush formula and determinant bounds.
• Infrared divergences in gapless models. The case of graphene. Rigorous renormalization group analysis, expansion in Gallavotti trees.
• 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)