Author: quantumseminar

13/02/2023 – 15/03/2023: Matteo Gallone @ UniMi

quantumseminarSeminar UniMi 
13/02  14:00-16:00   Sala di rappresentanza, ground floor, Mathematics Department, Via C. Saldini 50
14/02  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
15/02  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
1/03   14:00-16:00   Aula Mp, Via Mangiagalli 32
2/03   10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
3/03   10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
13/03  14:00-16:00   Aula C10, Via Mangiagalli 25
14/03  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50
15/03  10:00-12:00   Aula dottorato, first floor, Mathematics Department, Via C. Saldini 50

SPEAKER: Matteo Gallone (SISSA Trieste)

Introduction to Renormalisation Group for Fermionic Models

This series of seminars presents techniques used to rigorously approach the analysis of statistical mechanical systems of fermions. These include:
i) Gaussian integration using Feynman graphs
ii) Grassmann variables and Grassmann Gaussian integration
iii) ​Grassmann representation of the 2D Ising Model with quasi-periodic disorder
iv) Decay of the 2-point correlation function.

Students may get credit (in the category of seminar type F) for this course. Please contact Niels Benedikter if you intend to receive credit: niels.benedikter__A_T__unimi.it

20/02/2023: Jan Derezinski @ Polimi

quantumseminarSeminar PoliMi 
February 20, 2023
Aula Consiglio VII Piano, Politecnico di Milano
Edificio 14 (Nave), Campus Leonardo

SPEAKER: Jan Derezinski (University of Warsaw)

Schrödinger operators with singular boundary conditions

In order to define a differential operator on a domain with boundary one often needs to specify boundary conditions. In the presence of singular terms this can be quite tricky. It may lead to interesting and surprising phase diagrams. I will illustrate this with the so-called perturbed Bessel operators, that is one-dimensional Schrödinger operators on the half-line of the form $ – \partial_x^2 + (m^2 – 1/4) 1/x^2 + Q(x) $.


The talk will be based on my joint work with Jeremy Faupin.

5/12/2022: Davide Fermi @ PoliMi

quantumseminarSeminar PoliMi 
December 5, 2022
Aula Seminario VI Piano, Politecnico di Milano
Edificio 14 (Nave), Campus Leonardo

SPEAKER: Davide Fermi (Politecnico di Milano)

Vacuum fluctuations with zero-range potentials

The Casimir effect is a manifestation of van der Waals polarization forces
acting between atoms and molecules, in a regime where relativistic retardation
effects cannot be neglected. An effective description can be given in terms of
quantum fields influenced by classical objects, such as boundaries or external
potentials. This leads to the emergence of ultraviolet divergences which can be
treated by means of the so-called zeta-function regularization. In this seminar
I first present a rigorous implementation of the zeta-function technique, using
the language of canonical quantization for a scalar field. To this purpose,
complex powers of the elliptic operator describing the dynamics are used to
define smeared versions of the Wightman field and renormalization of
expectation values is ultimately attained by analytic continuation. Next, I
address the issue of boundary anomalies, discussing some explicitly solvable
models with zero-range potentials modeling semi-transparent walls.

Based on joint works with Livio Pizzocchero (University of Milano)

24/11/2022: Luca Fresta @ UniMi

quantumseminarSeminar UniMi 
November 24, 2022, 15:45
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Luca Fresta (Universität Bonn)

Stochastic Analysis of Subcritical Euclidean Fermionic Field Theories

In my talk, I will introduce a forward-backward stochastic differential equation
which provides a stochastic quantisation of subcritical Grassmann measures. The method is inspired by the so-called continuous renormalisation group, but
avoids the technical difficulties encountered in the direct study of the Polchinski’s flow equation for the effective potentials. If time permits, I will also show how to prove the exponential decay of correlations by a coupling method.
Work in collaboration with De Vecchi and Gubinelli.

21/11/2022: Nikolai Leopold @ UniMi

quantumseminarSeminar UniMi 
November 21, 2022, 11:15
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Nikolai Leopold (Universität Basel)

Norm approximations for the Fröhlich dynamics

In this talk I will discuss recent results about the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. For large particle number and initial data in which the particles are in a Bose-Einstein condensate and the excitations of the phonon field are in a coherent state I will show that the time evolved many-body state can be approximated in norm by an effective dynamics. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics.
If time permits I will, in addition, present a joint work with D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer about the Fröhlich model in the strong coupling limit and compare the Bogoliubov dynamics in the strong coupling and mean-field regime.

Norm-approximations-for-the-Frohlich-dynamicsDownload

20/12/2022: Thematic Day @ Como

quantumseminarThematic Day Insubria 
December 20, 2022, 11:15
Dipartimento di Scienza e Alta Tecnologia
Università dell’Insubria
Via Valleggio 11
Como

Program

  • 11:15 – Annalisa Panati
  • 12:00 – Markus Lange
  • 12:45 – Lunch Break
  • 14:45 – Andrea Mantile
  • 15:30 – Cristina Caraci
  • 16:15 – Closing

Titles and Abstracts

CARACI: Fluctuations of N-particle quantum dynamics around the Gross-Pitaevskii equation

We consider the quantum dynamics of N interacting bosons in the Gross-Pitaevskii regime. We obtain a norm-approximation for the many-body evolution of initial states exhibiting Bose-Einstein condensation in terms of a unitary Fock space evolution with a quadratic generator for the fluctuations. In addition, using this result, we provide the proof of a central limit theorem for the fluctuations of bounded one-particle observables. This is a joint work with Jakob Oldenburg and Benjamin Schlein.

MANTILE: Scattering theory with both regular and singular perturbations

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of a scattering couple (A,B), where both A and B are self-adjoint operators and B formally corresponds to adding to A two terms, one regular and the other singular. This abstract construction applies to perturbations of the free Laplacian with a Kato-Rellich potential and a singular part modelling boundary or interface conditions at the boundary of a open, bounded Lipschitz domain. We will possibly discuss applications of these models to classical scattering problems.

LANGE: Adiabatic Evolution of Low-Temperature Many-Body Systems

We consider the evolution of thermal equilibrium state of finite-range, many-body fermionic lattice models after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. As an application, the expansion allows to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. The proof is based on a rigorous version of the Wick rotation.

PANATI: Entropic Fluctuations in Quantum Two-time Measurement Framework

Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, thank to the pioneering works of Evans, Cohen, Morris and Searles on the violation of the second law, soon followed by the ground-breaking formulation of the Fluctuation Theorem by Gallavotti and Cohen for entropy fluctuation in the early nineties. Their work was by vast literature, both theoretical and experimental. The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an undergoing effort. Kurchan’s seminal work (2000) showed the measurement role has to be taken in account, leading to the introduction of the so called two-time measurement statistics (also known as full counting statistics). However introducing this frameworks leads to surprising phenomena with no classical counterpart. In this talk, I will present some work in progress, where we attempt to introduce a quantum equivalent of Gallavotti-Cohen (steady) entropic functional and compare it with the Evans-Searls (transient) entropic functional. We show that, due to the invasive measurement role, the situation differs considerably to its classical counterpart.
Joint work with T. Benoist, L. Bruneau, V. Jakšic, C.A.Pillet.

12/12/2022: Vojkan Jakšić @ UniMi

quantumseminarSeminar UniMi 
December 12, 2022, 11:15
- Aula Dottorato, 1st floor -
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Vojkan Jakšić (McGill University Montreal)

Some remarks on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

This talks concerns slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice Z^d . This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states, we will discuss a number of general structural results regarding such realizations.

This talk is based on a joint work with C-A Pillet and C. Tauber.

milan_adiabaticDownload

26/10/2022: Jérémy Faupin @ PoliMi

quantumseminarSeminar PoliMi 
October 26, 2022, 11:15 (new time this semester)
Aula Seminario - III piano
Politecnico di Milano
Edificio 14 (Nave), Campus Leonardo
P.zza da Vinci 32, Milano, Italy

SPEAKER: Jérémy Faupin (U de Lorraine @Metz)

Quasi-classical ground states in non-relativistic QED and related models

We will consider in this talk a non-relativistic particle bound by an external
potential and coupled to a quantized radiation field. This physical system is
mathematically described by a Pauli-Fierz Hamiltonian. We will study the energy
functional of product states of the form u⊗Ψ_f, where u is a normalized state
for the non-relativistic particle and Ψ_f is a coherent state in Fock space for
the field. This gives the energy of a Klein-Gordon-Schrödinger system in the
case of a spinless particle linearly coupled to a scalar field, or the energy
of a Maxwell-Schrödinger system in the case of an electron coupled to the
photon field. In both cases, we will discuss results concerning the existence
and uniqueness of a ground state, under general conditions on the external
potential and the coupling form factor. In particular, neither an ultraviolet
cutoff nor an infrared cutoff needs to be imposed. We will also discuss the
convergence in the ultraviolet limit and the second-order asymptotic expansion
in the coupling constant of the ground state energy.
This is joint work with J. Payet and S. Breteaux.

30/5/2022: Laurent Lafleche @ UniMi

quantumseminarSeminar UniMi 
May 30, 2022, 16:30 (non-standard time)
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Laurent Lafleche (U Texas at Austin)

Semiclassical regularity and mean-field limit
with singular potentials

In this talk I will present several techniques and concepts used in the context of the mean-field and the classical limit allowing to go from the N -body Schrödinger equation with singular potential to the Hartree–Fock and Vlasov equations, linked to works in collaboration with Chiara Saffirio and Jacky Chong. At the level of the Vlasov–Poisson equation, typical mean-field techniques from quantum mechanics for pure states can be translated to a weak-strong stability estimate in L^1 for the Vlasov equation. Another weak-strong stability can be obtained for the difference of the square roots of the solutions in L^2. They allow to better understand the mean-field and semiclassical estimates. These estimates are weak-strong in the sense that they require only the regularity of one of the solutions. This requires the propagation of a semi-classical notion of regularity uniformly in N and h. A typical obstacle is the lack of positivity of the Wigner transform and its few conserved quantities. A solution to this problem is to consider operators as the right generalization of the phase space distribution, and a quantum analogue of Sobolev spaces defined using Schatten norms. The advantage of these techniques is that they allow to obtain regularity estimates without higher order error terms.

16/5/2022: Robin Reuvers @ UniMi

quantumseminarSeminar UniMi 
May 16, 2022, 16:30 (non-standard time)
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Robin Reuvers (U Roma Tre)

Ground state energy of dilute Bose gases in 1D

In 1963, Lieb and Liniger formulated an exactly solvable model for interacting bosons in 1D. Thanks to its exact, Bethe ansatz solution, the model and its generalizations soon became popular objects of study in mathematical physics. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, the Lieb-Liniger model found experimental use and became even better known.

In the meantime, Lieb and collaborators had moved on, and were rigorously studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to acquire, and a popular goal was the rigorous derivation of the ground state energy of gases of bosons in various settings in 2 and 3D. Many of the results focused on the dilute limit, in which the density of the boson gas is very low.

Somehow, Bose gases in 1D were excluded from this development. Of course, the original Lieb-Liniger model provided a solvable example, but we can nevertheless use insights from the 2 and 3D approaches to prove new results about the ground state energy of dilute Bose gases in 1D.

In the talk, I will review the developments above, and explain the new results.