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.

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).

11/07/2023: Davide Lonigro @ UniMi

July 11, 2023 - 15:00
Sala di Rappresentanza (ground floor on the left)
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Davide Lonigro (Università degli Studi di Bari)

Self-adjointness of a class of spinboson models with ultraviolet divergences 

We study a class of quantum Hamiltonian operators describing a family of two-level systems (spins) coupled with a structured boson field, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences (hence, non-normalizable). Spin–spin interactions which do not modify the total number of excitations are also included. Starting from the single-atom case, and eventually reaching the general scenario, we shall provide explicit expressions for the self-adjointness domain and the resolvent operator of such models. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors by normalizable ones, for example an ultraviolet cutoff.