Current Topics in Mathematic Physics

17.07.2017-21.07.2017

**Organized by:** G.M. Graf, C. Hainzl, B. Schlein

Venue: room Y24 G55 in the Irchel Campus of the University of Zurich

Contact: for question, email benjamin.schlein@math.uzh.ch

Sponsor: SwissMAP

Schedule

Monday, July 17

10:00 - 10:30 coffee and registration

10:30 - 12:00 Seiringer

14:00 - 15:30 Knowles

15:30 - 16:00 coffee

16:00 - 17:30 Giuliani

18:00 - 21:00 reception/dinner

Tuesday, July 18

09:30 - 11:00 Lewin

11:00 - 11:30 coffee

11:30 - 13:00 Giuliani

Afternoon: contributed talks

14:30 - 14:50 Levitt

14:50 - 15:10 Olgiati

15:10 - 15:30 Moser

15:30 - 15:50 Young

15:50 - 16:30 coffee

16:30 - 16:50 Shapiro

16:50 - 17:10 Chen

17:10 - 17:30 Giacomelli

17:30 - 17:50 Lemm

Wednesday, July 19

09:30 - 11:00 Lewin

11:00 - 11:30 coffee

11:30 - 13:00 Seiringer

15:00 - 18:00 free afternoon (sport events)

18:00 - 21:00 barbecue

Thursday, July 20

09:30 - 11:00 Lewin

11:00 - 11:30 coffee

11:30 - 13:00 Giuliani

15:00 - 16:30 Knowles

Friday, July 21

09:30 - 11:00 Seiringer

11:00 - 11:30 coffee

11:30 - 13:00 Knowles

Main Courses

• Speaker: Alessandro Giuliani (Roma 3)

Title: Perturbed 2D Ising models at the critical point

Abstract: The 2D Ising model is probably the most studied statistical mechanics model. As well known, the `standard' 2D Ising model, i.e., the one with translationally invariant nearest neighbor interactions, admits an exact solution, which provides explicit formulas for the free energy and several correlation functions. From this formulas, one can read essentially all the relevant properties of the phase diagram and of the critical point, including the critical exponents characterizing the second order phase transition from the paramagnetic to the ferromagnetic phase. The `universality hypothesis' predicts that these exponents should be invariant under a large class of perturbations of the original Hamiltonian.

In this course I will give an introduction to the Renormalization Group methods used to rigorously prove the robustness of the energy critical exponents and of the scaling limit of the energy correlation functions, under the simplest possible perturbations of the Hamiltonian, i.e., addition of weak next-to-nearest neighbor interactions of strength $\lambda$. I will give a complete proof of the analyticity in $\lambda$ of the interacting critical free energy. I will also explain how to adapt the free energy expansion to the computation of the multi-point energy correlations, of the sub-leading corrections to the free energy and the central charge.

Based on joint works with Rafael Greenblatt and Vieri Mastropietro.

• Speaker: Antti Knowles (Geneve)

Title: Local laws and universality in random matrix theory.

Abstract: I give an introduction to the local semicircle law from random matrix theory, as well as some of its applications. I focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. I state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner’s semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds.

I then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide.

• Speaker: Mathieu Lewin (Paris Dauphine)

Title: An introduction to critical point theory, with applications in quantum
mechanics

Abstract: In this lecture, I will review some basic mathematical tools from
critical point theory, which can be used to construct solutions of equations
that are not necessarily energy minimizers. In the first part, I will explain
the mountain pass lemma and its link with chemical reactions. Then I will
discuss general min-max methods that can be used to construct excited states
for nonlinear effective equations in quantum mechanics, and illustrate with
examples from Hartree-Fock theory and its extensions. In all cases I will
mention open problems.

• Speaker: Robert Seiringer (IST Austria)

Title: The polaron

Abstract: The polaron is a simple model of a quantum particle interacting with the phonon field of a polarizable medium. Models of this kind also appear in various forms as toy models in quantum field theory. We shall discuss in detail the Froehlich model of a polaron, and present some recent results and open problems. The lectures will focus on the mathematical structure of the model, its strong-coupling (or semi-classical) limit, as well as stability and binding properties of multi-polaron systems.

Contributed talks

• Speaker: Antoine Levitt (Inria Paris)

Title: Robust construction of Wannier functions

Abstract: Wannier functions are a localized basis for spectral subspaces
of periodic Schrödinger operators. In this talk, I will present a new
numerical method for their computation, and outline ongoing work into
its extension to the case of Z2 topological insulators and metals. This
is joint work with E. Cancès, G. Panati and G. Stoltz

• Speaker: Alessandro Olgiati (SISSA Trieste)

Title: Composite BEC and effective Gross-Pitaevskii dyamics: mixtures and pseudo-spinors.

Abstract: Composite Bose-Einstein condensation represents one of the current frontiers in cold atoms experiments, as well as a recently developing and already quite active field in mathematical physics. In my talk I will introduce two main types of composite condensates: multi-component mixtures of different atomic species and pseudo-spinorial BEC (different hyperfine levels coupled to an external magnetic field). For both classes of systems, the effective dynamics in the Gross-Pitaevskii limit is well described by suitable systems of coupled non-linear Schroedinger equations; I will present the rigorous derivation of such equations from the many-body linear Schroedinger dynamics, recently obtained in joint works with Alessandro Michelangeli.

• Speaker: Thomas Moser (IST Austria)

Title: Stability of a fermionic N + 1 particle system with point
interactions

Abstract:
Unlike the bosonic case where point interactions lead to instability
because of the Thomas/Efimov effect, stability can be proven for a
system of fermions under suitable conditions. In particular, the 2+1
fermionic system is well understood, and it turns out that there is a
critical mass ratio determining stability. I will talk about our recent
result where we showed stability for the $N+1$ fermion model, allowing for
arbitrary fermions of one kind interacting with one particle of another
kind. This extends preliminary results about stability which were
restricted so far to few body problems.

• Speaker: Amanda Young (University of Arizona, Tucson)

Title: On Stability of Frustration-free Ground States of Quantum Spin Systems.

Abstract: Gapped ground state phases of quantum spin systems have received a renewed interest because of their potential to support topological order. A key feature is the existence of a spectral gap above the low-lying energy states as well as the stability of the spectral gap in the presence of small perturbations. Stability results were developed for models with topologically ordered ground states by Bravyi-Hastings-Michalakis (2010), Bravyi-Hastings (2011), and Michalakis-Pytel (2013). In this talk, we will discuss several new generalizations of the Michalakis-Pytel result. Specifically, we discuss how to extend these results to models with discrete symmetry breaking, more general boundary conditions, and models defined on more general lattices.

• Speaker: Jacob Shapiro (ETH Zurich)

Title: Bulk-Edge Duality and Complete Localization for Chiral Chains

Abstract: We study 1D insulators obeying a chiral symmetry in the single-particle picture where the Fermi energy is assumed to lie within a mobility gap. Topological invariants are defined for infinite (bulk) or half-infinite (edge) systems, and it is shown that for a given bulk system with N.N. hopping, the invariant is equal to the induced-edge-system's invariant. We also give a new formulation of the topological invariant in terms of the Lyapunov exponents of the system, which sheds light on the conditions for topological phase transition extending to the mobility gap regime. Finally we give a proof of complete dynamical localization for our model via Furstenberg's theorem and the fractional moments method, which justifies the deterministic assumptions we make.

• Speaker: Li Chen (University of Toronto)

Title: Linear Bogoliubov de Gennes Equation and Its Spectral Property

Abstract: In the interest in the derivation of Ginzburg-Landau equation
from the Bogoliubov de Gennes (BdG) equation, we consider the linearization of the latter at the normal state. We work in the setting where $\delta << 1$ is the microscopic to macroscopic ratio and $T < T_c$ with $T_c - T = O(\delta^2)$ (here $T_c$ is the critical temperature for superconducting phase transition). Our superconducting sample is taken to be a large box with side length $\delta^{-1}$. We want to understand the spectrum of the linear BdG operator near its ground state energy by perturbation. In the translation invariant case, the linearized operator has a fiber decomposition. This allows us to perform regular perturbation theory. When a weak magnetic field is included, we loose this decomposition to some extend. However, under favorable assumptions, we give a proof that the lowest band of eigenvalues is of the form $\lambda_0 + \lambda_1 \delta^2 k +$ higher order in $\delta$, where $0 > \lambda_0 = O(\delta^2)$ and $\lambda_1 > 0$. This is a joint project with I.M. Sigal.

• Speaker: Emanuela Giacomelli (La Sapienza, Roma)

Title: Surface Superconductivity in Presence of Corners

Abstract: We consider an extreme type-II superconducting wire with non-smooth cross section, i.e., with one or more corners at the boundary, in the framework of the Ginzburg-Landau theory. We prove the existence of an interval of values of the applied field, where superconductivity is spread uniformly along the boundary of the sample. More precisely the energy is not affected to leading order by the presence of corners and the modulus of the Ginzburg-Landau minimizer is approximately constant along the transversal direction. The critical fields delimiting this surface superconductivity regime coincide with the ones in absence of boundary singularities. We will also discuss some recent results about the corner contribution to the first order correction to the energy asymptotics. Joint work with Michele Correggi.

• Speaker: Marius Lemm (Caltech, Pasadena)

Title: On the entropy of reduced density matrices

Abstract: One way to describe the entanglement inherent to an N-particle quantum state is to consider the von Neumann entropy of its k-body reduced density matrices. When the quantum state is bosonic or fermionic, we prove two general facts about these entropies: As functions of k, they are monotone for $k< N/2$ and concave. The proof uses only permutation-invariance and the monotonicity of the relative entropy under the partial trace channel.