Institut für Mathematik


Summer School on Current Topics in Mathematical Physics

Satellite meeting to the "International Congress on Mathematical Physics", Geneva, August 2-7, 2021.

19.7 - 23.7.2021

Organisiert von: A. Deuchert, C. Hainzl, B. Schlein, R. Seiringer

Main Courses

• Speaker: Yoshiko Ogata (Tokyo)

Title: Classification of symmetry protected topological phases in quantum spin systems

Abstract: The notion of symmetry protected topological (SPT) phases was introduced by Gu and Wen [GW]. It is defined as follows: we consider the set of all Hamiltonians with some symmetry, which have a unique gapped ground state in the bulk, and can be smoothly deformed into a common trivial gapped Hamiltonian without closing the gap. We say two such Hamiltonians are equivalent, if they can be smoothly deformed into each other, without breaking the symmetry. We call an equivalence class of this classification, a symmetry protected topological (SPT) phase. I explain about our result, proving the physicist's conjecture that there are group cohomology valued invariants in one- and two-dimensional quantum spin systems.

• Speaker: Marcello Porta (SISSA, Trieste)

Title: The correlation energy of mean-field Fermi gases.

Abstract: In this course I will discuss the ground state properties of three-dimensional, homogeneous Fermi gases, in the mean-field regime. For large values of the number of particles, Hartree-Fock theory provides an effective description of such systems, both for the ground state and for the dynamical properties. Being based on the restriction of the space of fermionic wave functions to the set of Slater determinants, the main limitation of the Hartree-Fock approximation is that it neglects the correlations among the particles, besides those implied by the Pauli principle. In this course I will discuss a description of fermionic correlations based on a rigorous bosonization scheme. This approach allows to study the low-energy excitations of the Fermi gas in terms of emergent, quasi-free bosonic particles. I will use the method to compute the correlation energy, defined as the difference between many-body and Hartree-Fock ground state energies. As the number of particles goes to infinity, the correlation energy converges to the ground state energy of a quasi-free Bose gas. The final expression rigorously justifies the prediction based on the random-phase approximation. The material presented in the course is based on joint works with C. Hainzl and F. Rexze, and with N. Benedikter, P. T. Nam, B. Schlein and R. Seiringer.

• Speaker: Vincent Tassion (ETH)

Title: An introduction to Russo-Seymour-Welsh theory/p>

Abstract: Percolation models were originally introduced to describe the propagation of a fluid in a random medium. Over the years and via geometrical representations, percolation techniques have also become powerful tools to study other physical systems, such as Ising or Potts models.
In dimension two, the percolation properties of a model are encoded by so-called crossing probabilities (probabilities that certain rectangles are crossed from left to right). In the eighties, Russo, Seymour and Welsh obtained general bounds on crossing probabilities for Bernoulli percolation (the most studied percolation model, where edges of a lattice are independently erased with some given probability 1-p). These inequalities rapidly became central tools to analyze the critical behavior of the model. Recently, the theory has been extended to more general models, leading to important results in other fields, such as the rigorous derivation of the order of the phase transition for spin systems.
In this course, I will present the Russo-Seymour-Welsh theory and consequences for Bernoulli percolation. Then I will discuss recent generalizations and their consequences in other fields.

• Speaker: Stefan Teufel (Tuebingen)

Title: Adiabatic theorems and linear response in the thermodynamic limit

Abstract: One of the fundamental problems of quantum mechanics is to determine the response of systems, initially in their ground state, to external perturbations. I first discuss the linear response formalism and its connection to adiabatic theorems in the context of extended gapped quantum systems at zero temperature. The main application is transport in quantum Hall systems. After a short review of such adiabatic theorems for non-interacting many-body systems, I explain mathematical tools, many of which have been developed in recent years, that allow us to prove suitable adiabatic theorems for interacting quantum lattice systems (spins or fermions) even in the thermodynamic limit. I conclude with a discussion of open mathematical and conceptual problems.

Contributed Talks