Order, disorder and phase transitions in quantum many body systems

Abstract

In this paper, I give an overview of some selected results in quantum many body theory, lying at the interface between mathematical quantum statistical mechanics and condensed matter theory. In particular, I discuss some recent results on the universality of transport coefficients in lattice models of interacting electrons, with specific focus on the independence of the quantum Hall conductivity from the electron-electron interaction. In this context, the exchange of ideas between mathematical and theoretical physics proved particularly fruitful, and helped in clarifying the role played by quantum conservation laws (Ward Identities), together with the decay properties of the Euclidean current-current correlation functions, on the interaction-independence of the conductivity.

35 pages, 7 figures. These notes are based on a presentation given at the Istituto Lombardo, Accademia di Scienze e Lettere, in Milano (Italy) on May 5, 2016, as well as on the notes of a course given at the EMS-IAMP summer school in mathematical physics `Universality, Scaling Limits and Effective Theories', held in Roma (Italy) on July 11-15, 2016. Final version, accepted for publication

References in corpus (8)

• Coulomb interaction, ripples, and the minimal conductivity of graphene
• Propagation of Correlations in Quantum Lattice Systems
• Minimal conductivity in graphene: interaction corrections and ultraviolet anomaly
• Conductivity of interacting massless Dirac particles in graphene: Collisionless regime
• Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
• Topological phase transitions and universality in the Haldane-Hubbard model
• Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory
• Spin Hall insulators beyond the Helical Luttinger model