Category: Seminar 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.

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

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SPEAKER: Cornelia Vogel (Universität Tübingen)

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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.

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.

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

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SPEAKER: Zhituo Wang (Harbin Institute of Technology)

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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.

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

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SPEAKER: Joachim Kerner (FernUniversität in Hagen)

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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.

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

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SPEAKER: Amirali Hannani (KU Leuven)

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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).

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

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SPEAKER: Emanuela Giacomelli (LMU München)

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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.

Monday, January 15, 2024 - 11:15
Aula 9
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Clara Torres Latorre (Universitat de Barcelona)

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Regularity theory for elliptic and parabolic PDE

In this talk, we’ll explore how regularity theory is crucial for understanding partial differential equations (PDEs), and how it has consequences in physics and numerical analysis. We’ll first focus on why regularity matters, then take elliptic and parabolic PDEs as examples to talk about classical and recent regularity results. The goal is to give a practical overview, explaining when PDE solutions are smooth or singular.

Thursday, July 13, 2023 - 15:00
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Diwakar Naidu (Universität Tübingen)

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Existence of Bell-type pure jump process for the Klein-Gordon Hamiltonian

In this talk I will present my work on Bell-type jump processes. J.S. Bell in
1984 gave a jump rate formula that predict the probability of configurational
jumps and in turn define a stochastic (Markov) jump process that governs the
evolution of particle configurations. The standard method (by Tumulka et al)
for proving existence of such processes does not work for the Klein-Gordon (KG) Hamiltonian as the jump rates for it are unbounded. We show the existence
of a stationary and independent (Markov) pure jump process (i.e. where the
configurational motion occurs only via jumps) for the particle configuration that
is equivariant, i.e. |Ψ_t|² distributed at every time t, where Ψ evolves with the KG
Hamiltonian, using elements from the theory of Lévy processes. Next, we also
want to extend this obtained process to a broader class of Markov process which also depend on the particle configurations and time using the general theory of Markov processes.

October 9, 2023 - 11:15
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

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SPEAKER: Asbjørn Bækgaard Lauritsen (IST Austria)

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Ground state energy and pressure of a dilute spin-polarized Fermi gas

Recently the study of dilute quantum gases have received much interest, in particular regarding their ground state energies and pressures/free energies at positive temperature. I will present recent work on such problems. Namely that of the ground state energy of a spin-polarized Fermi gas and the extension to the pressure at positive temperature. Compared to the free gas, the energy density/pressure of the interacting gas differs by a term of order a^3 \rho^{8/3} with a the p-wave scattering length of the interaction. One of the main ingredients in the proofs is a rigorous version of a formal cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237-260). I will discuss this expansion and the analysis of its absolute convergence.

Joint work with Robert Seiringer.