# Seminars & Events for Mathematical Physics Seminar

##### Construction and Analysis of a hierarchical massless QFT

I will discuss how rigorous renormalization group methods can be used to construct a massless QFT over a space with hierarchical structure. The model under study is a phi-4 perturbation of a three dimensional fractional Gaussian Free Field and is meant to capture some of the behavior of the Wilson-Fisher 4-epsilon fixed point. Among our results is anomalous scaling for the composite field in this model. I will also show how methods from statistical mechanics can be used to prove the full scale invariance of this QFT. (Joint work with Abdelmalek Abdesselam and Gianluca Guadagni.)

##### THIS SEMINAR HAS BEEN MOVED TO OCTOBER 15. PLEASE SEE NEW POSTING.The quantum Shannon-McMillan theorem and rank of spectral projections of macroscopic observables

The classical Shannon-McMillan theorem states that an ergodic system has typical sets satisfying the asymptotic equipartition property. This theorem demonstrates the significance of entropy which gives the size of the typical sets. There has recently been great progress in the quantum version of the Shannon-McMillan theorem .In particular, Bjelakovic et al. proved Shannon-McMillan theorem for ergodic quantum spin systems, by reducing the quantum setting to a classical one, and applying the classical Shannon-McMillan theorem. In this talk, I would like to introduce a direct proof of the quantum Shannon-McMillan theorem, without relying on the classical theory. Our proof is based on the variational principle, which is a well-known thermodynamic property of quantum spin systems.

##### The quantum Shannon-McMillan theorem and rank of spectral projections of macroscopic observables

The classical Shannon-McMillan theorem states that an ergodic system has typical sets satisfying the asymptotic equipartition property. This theorem demonstrates the significance of entropy which gives the size of the typical sets. There has recently been great progress in the quantum version of the Shannon-McMillan theorem .In particular, Bjelakovic et al. proved Shannon-McMillan theorem for ergodic quantum spin systems, by reducing the quantum setting to a classical one, and applying the classical Shannon-McMillan theorem. In this talk, I would like to introduce a direct proof of the quantum Shannon-McMillan theorem, without relying on the classical theory. Our proof is based on the variational principle, which is a well-known thermodynamic property of quantum spin systems.

##### Structure of large bosonic systems: the mean-field approximation and the quantum de Finetti theorem

I will discuss a general strategy to derive Hartree's theory for the ground state of a generic interacting many-bosons system with mean-field scaling. The validity of the mean-field approximation is interpreted as a consequence of the structure of the set of bosonic density matrices with large number of particles, in particular of the so-called quantum de Finetti theorem. The approach is general and applies for example to famous examples such as the trapped Bose gas, bosonic atoms and boson stars.

##### On the Bogolubov-Hartree-Fock (BHF) Approximation for the Pauli-Fierz Model

The minimal energy of the nonrelativistic one-electron Pauli-Fierz model within the class of quasifree states is studied. It is shown that this minimum is unchanged if one restricts the variation to pure quasifree states, which simplifies the variational problem considerably. Existence and uniqueness of minimizers are established under the assumption of infrared and ultraviolet cutoffs, as well as, a sufficiently small coupling constant and a small momentum of the dressed electron. This is joint work with S. Breteaux and T. Tzaneteas and with S. Breteaux, H.K. Knoerr, and E. Menge.

##### CANCELLED: Topological recursion in random matrices, random maps, and differential systems

**THIS SEMINAR HAS BEEN CANCELLED.**

##### Physical Principles Underlying the Fractional Quantum Hall Effect

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. I review an approach to the theory of the Quantum Hall Effect (QHE) somewhat analogous to Landau's theory of phase transitions

##### Remainder terms for some entropy inequalities and entanglement for fermions

A number of entropy inequalities have proved useful for quantifying the degree of entanglement of bipartite (or multipartite) quantum states. The problem of determining all of the cases of equality in these inequalities is largely complete but less is known about lower bounds on the deficit in these inequalities in terms of the distance to the set of states for which equality obtains. We prove several theorems giving remainder terms that quantify the degree of inequality in various entropy inequalities in terms of the distance to the states for which equality obtains. We apply them to the problem of quantifying the degree of entanglement in fermionic states, which can never be separable.; i.e., unentangled, in the usual sense.

##### Quantum Hall Phases, plasma analogy and incompressibility estimates

When a 2D many-particle system with a repulsive interaction is subject to a suff iciently strong magnetic field, that can also be produced by rapid rotation, str ongly correlated many-body states in the lowest Landau level LLL may emerge. In the talk conditions for the ground state to include the Laughlin state as a fact or will be presented, together with an analysis of the particle density in such states. This is joint work with Sylvia Serfaty and Nicolas Rougerie.

##### Entropic Functionals in Quantum Statistical Mechanics

The mathematical theory of non-equilibrium quantum statistical mechanics has developed rapidly in the last decade. The initial developments concerned the theory of non-equilibrium steady states, the entropy production observable, and linear response theory (Green-Kubo formulas, Onsager reciprocity relations) for open systems driven by thermodynamical forces (say temperature differentials). This line of development was a natural direct quantization of the classical theory. In contrast, extensions of the classical fluctuation relations of Evans-Searles and Gallavotti-Cohen to the quantum domain have led to some surprises and novel classes of entropic functionals with somewhat striking mathematical structure and physical interpretation.

##### From BCS-theory to the Gross-Pitaevskii equation for the evolution of dilute fermion pairs

We consider a system of fermions interacting through a potential admitting a bound state in the low density limit. Assuming the energy to be sufficiently small, we show that particles form pairs, exhibiting Bose-Einstein condensation and evolving according to the time-dependent Gross-Pitaevskii equation.

##### Threshold effects of the two and three-particle Schroedinger operators on lattices

Efimov's effect for a system of three particles moving on three-dimensional lattice and interacting via pairwise short-range potentials is studied. The following new results will be presented: (i). Infinitude of the number of eigenvalues (Efimov's effect) for zero quasi-momentum, and its finiteness for non-zero values of the quasi-momentum. (ii).The corresponding asymptotics for the number of eigenvalues.

##### From conformal invariance of parafermionic observables to conformal invariance of interfaces of planar random-cluster models

In this talk we will explain how the determination of the scaling limit of parafermionic observables can be used to deduce the conformal invariance of interfaces in planar random-cluster models with cluster-weight 1 ≤ Q ≤ 4 (1\leq Q \leq 4). The strategy was introduced in the context of the loop-erased random walk by Lawler-Schramm-Werner, and was implemented for the FK Ising model (a.k.a. the random-cluster model with cluster-weight equal to 2) by Chelkak, Duminil-Copin, Hongler, Kemppainen and Smirnov, based on a main contribution of Smirnov.