11/04/2025 : Annalisa Panati @UniMi

Friday, April 11, 2025

Aula 6,
Dipartimento di Matematica,
Università degli Studi di Milano,
Via Cesare Saldini 50.

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SPEAKER: Annalisa Panati (Université de Toulon, Centre de Physique Théorique)

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Entropic fluctuations in quantum two-time measurement framework

Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, since the ground-breaking formulation of the transient and steady entropic Fluctuation Relations (FR) in the early nineties.
The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an ongoing effort. Kurchan and Hal Tasaki’s seminal works (2000) showed quantum formulation of the transient version of FR is possible by introducing the two-time measurement framework.
In this talk, we present some results in a recent series of papers, where we attempt to introduce a quantum equivalent of steady entropic functional and compare it to the transient version for open quantum system. We consider both the case of idealised direct measurement on the reservoirs and experimentally accessible indirect measurement through coupling with an ancilla. We analyse in particular stability with respect to the initial state. In order to deal with the thermodynamic limit and to have general results, we use methods of $C^*$- algebras and modular theory.


(Joint work with T. Benoist, L. Bruneau, V. Jakšić, C.A. Pillet)

03/04/2025 : Antoine Borie @UniMi

Thursday, April 3, 2025 - 11:00

Aula dottorato,
Dipartimento di Matematica,
Università degli Studi di Milano,
Via Cesare Saldini 50.

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SPEAKER: Antoine Borie (Université de Rennes)

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Scattering for the positive density Hartree equation

In this talk, we explore the long-time behavior of solutions to of the positive density Hartree equation, which models the evolution of a homogeneous quantum gas. Our focus is the stability of certain stationary states, extending the original result introduced by Mathieu Lewin and Julien Sabin to higher dimensions and more singular interaction potentials. Using tools from dispersive partial differential equations, such as Strichartz estimates and fractional Leibniz rules, we develop a new approach tailored to density matrices.

This talk is based on joint work with Julien Sabin (Rennes University ) and Sonae Hadama (Kyoto University).

06/12/2024 Pierre Roux @UniMi

Wednesday, December 6, 2024 - 14:00

Aula Dottorato,
Dipartimento di Matematica,
Università degli Studi di Milano,
Via Cesare Saldini 50.

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SPEAKER: Pierre Roux (Institut Camille Jordan, École Centrale de Lyon)

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Disentangling pulse-coupled oscillators in the mean-field regime through the pseudo-inverse in a dilated timescale

Systems of pulse-coupled oscillators model synchronization through singular interactions occurring at discrete times, when particles reach a specific firing phase. They have numerous applications in physics, biology and engineering, for example to cardiac cells, neurons and fireflies. In the mean-field limit, the probability density in phase satisfies a singular continuity equation prone to finite-time blow-up, for which very few theoretical results are available. With José A. Carrillo, Xu’an Dou and Zhennan Zhou, we have introduced a reformulation of the mean-field system based on the inverse distribution function seen in a dilated timescale. It allows to show a hidden contraction/expansion mechanism and to propose simple and rigorous proofs of the long-time behaviour, the existence of steady states, the rates of convergence and the occurence of finite time blow-up for a large class of monotone phase response functions.

05/06/2024 : Umberto Morellini @UniMi

Wednesday, June 5, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Umberto Morellini (Université Paris Dauphine, CEREMADE, PSL) ________________________________________________________________________

The free energy of Dirac’s vacuum in purely magnetic fields

The Dirac vacuum is a non-linear polarisable medium rather than an empty space. This non-linear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at non-zero temperature needs to be better understood.
In this talk, we will present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a non-linear functional describing the free energy of quantum vacuum in a classical magnetic field. After introducing our model, we will properly define the free energy functional using the Pauli-Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well known issue of the theory. The study of the properties of this functional will be addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, one can prove the convergence of this functional to the Euler-Heisenberg
formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich in 1979. The talk is based on the work available at arXiv:2404.12733.

07/05/2024: Cornelia Vogel @ UniMi

Tuesday, May 7, 2024 - 11:30
Sala di rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Cornelia Vogel (Universität Tübingen)


Concentration of measure for thermal distributions of quantum states

We generalize Lévy’s Lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a more general class of measures, so-called GAP measures. For any given density matrix ρ on a separable Hilbert space H, GAP(ρ) is the most spread out probability measure on the unit sphere of H that has density matrix ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue ||ρ|| of ρ is small. With the help of this result we generalize the well-known and important phenomenon of ”canonical typicality” to GAP measures. Canonical typicality is the statement that for ”most” pure states ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a ψ-independent matrix. So far, canonical typicality is known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble. Our result shows that canonical typicality holds in general for systems described by a density matrix with small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles. The talk is based on joint work with Stefan Teufel and Roderich Tumulka.

22/04/2024: Antoine Prouff @UniMi

Monday, April 22, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Antoine Prouff (Université Paris-Saclay, Laboratoire de Mathématique d’Orsay) _________________________________________________________________________

Egorov’s theorem in the Weyl-Hörmander calculus and application to the control of PDEs

It is known that geometric optics can be derived as the high-frequency limit of the wave equation, from both experimental and theoretical perspectives. This fact can be regarded as an instance of a “quantum-classical correspondence principle”, made rigorous by Egorov’s theorem, which relates the evolution of a linear PDE (e.g. the wave equation) to the natural underlying classical dynamics (e.g. the geodesic flow).

We will present a version of Egorov’s theorem in the Euclidean space, in the setting of the “Weyl-Hörmander calculus”. This general framework of microlocal analysis involves Riemannian metrics on the phase space adapted to the dynamics under consideration, and allows for a fairly large range of applications (study of Schrödinger, wave and transport equations).

If time allows, we will discuss in more detail an application to the observability of the Schrödinger equation with a confining potential in the Euclidean space.

18/03/2024: Zhituo Wang @ UniMi

Monday, March 18, 2024 - 11:15
Aula Dottorato (first floor)
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Zhituo Wang (Harbin Institute of Technology)


Constructive renormalizations of the 2-D Honeycomb-Hubbard model

In this talk I will present some recent progress on the construction of ground state of the 2-dimensional Hubbard model, which is a prototypical model for studying phase transitions in quantum many-body system. Using fermionic cluster expansions and constructive renormalization theory, we proved that the ground state of the 2-d Hubbard model on the honeycomb lattice with triangular Fermi surfaces is not a Fermi liquid in the mathematical precise sense of Salmhofer. I will also discuss the crossover phenomenon in the 2-d square Hubbard model and universalities. This presentation is based on the work arXiv:2108.10852, CMP 401, 2569–2642(2023) and arXiv:2303.13628.

04/03/2024: Joachim Kerner @ UniMi

Monday, March 4, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Joachim Kerner (FernUniversität in Hagen)


On Bose-Einstein Condensation in the Random Kac-Luttinger Model

This talk is concerned with a random many-particle model originally considered by Kac and Luttinger in 1973 in order to study a well-known quantum phase transition known as Bose–Einstein condensation (BEC). Generally speaking, to understand this phase transition in interacting many-particle systems is a current hot topic in mathematical physics. However, due to the complexity of the underlying random one-particle model, the nature of the BEC in the non-interacting Kac-Luttinger model was understood only recently based on results obtained by Alain-Sol Sznitman (ETH). In this talk, our goal will be to understand the impact of repulsive two-particle interactions on this condensate. We will see that, due to the spatial localization of the condensate, strong enough interactions will immediately destroy it. On the other hand, for two-particle interactions of a mean-field type, we prove BEC in the interacting Kac–Luttinger model into a minimizer of a Hartree-type functional. This talk is based on joint work with C. Boccato (Milan), M. Pechmann (Tennessee), and W. Spitzer (Hagen).

The seminar is part of the activities of the project PRIN 2022AKRC5P “Interacting Quantum Systems: Topological Phenomena and Effective Theories” financed by the European Union – Next Generation EU.

25/01/2024: Amirali Hannani @ UniMi

Thursday, January 25, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Amirali Hannani (KU Leuven)


Localization and Poisson statistics in the “avalanche model”

The “avalanche model” aka “quantum sun model” has been introduced as a toy model to study the stability of the MBL (Many-body localized) phase. Strong numerical and theoretical heuristics suggest a localization-delocalization transition in this family of models varying a natural parameter $\alpha$. We prove localization (in the many-body sense) and Poisson statistics for this model given $\alpha$ sufficiently small. In this talk, first I give some general preliminaries about MBL (Many-body localization) which motivate the above-mentioned model. Then I introduce the model and recall certain numerical “facts” about the localized phase. Finally, I state our theorem concerning localization and Poisson statistics and give some ideas about the proof which rests on showing certain weak information about the absence of level-attractions in this model. This is a joint work with Wojciech De Roeck (KU Leuven).

26/02/2024: Emanuela Giacomelli @ UniMi

Monday, February 26, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Emanuela Giacomelli (LMU München)


The low density Fermi gas in three dimensions

In recent decades, the study of many-body systems has been an active area of research in both physics and mathematics. In this talk we consider a system of N interacting fermions with spin 1/2 confined in a box in the dilute regime. We are interested in studying the correlation energy, defined as the difference between the energy of the fundamental state and that of the free Fermi gas. We will discuss some recent results on a first-order asymptotic for the correlation energy in the thermodynamic limit, where the number of particles and the size of the box are sent to infinity while keeping the density fixed. In particular, we will present some recent results for the correlation energy that go in the direction of a rigorous proof of the well-known Huang-Yang formula of 1957.