The program starts on Monday at 14:00 and closes on Wednesday at 13:00.

chair: local

14:00-14:15: opening

14:15-15:00: Giacomelli

coffee break

15:45-16:30: Langella

16:45-17:30: Greenblatt

chair: Zanelli

9:00-9:45 Schlein

coffee break

10:30-11:15 Maspero

11:30-12:15 Marcelli

lunch break

14:15-15:00: Giuliani

coffee break

15:45-16:30: Fermi

16:45-17:30: Ponno

20:00: social dinner

chair: local

9:00-9:45 Gallone

coffee break

10:30-11:15 Porta

11:30-12:15 Olgiati

12:15-12:30 closing

The workshop will take place in the "Sala rappresentanza" at the Department of Mathematics of the University of Milan (Via Cesare Saldini 50).

How to get to the department:
To reach us from the train stations "Milano Centrale" or "Milano Porta Garibaldi", take the green metro line (ticket for 2 Euro, ticket machines in the metro station) to the metro stop "Piola". From there it's a short walk: find the direction to Piazza Leonardo da Vinci/Politecnico, cross the square and continue straight. When you see a big construction site on your left, turn left before that and find the department entrance (historic building) in the next small street (map).

Some trains stop at the smaller train station "Milano Lambrate". From there you can either walk directly to the department (21min), or take the tram 19 (ticket for 2 Euro) until the stop "Via Pascoli" and walk from there (6min), or take the bus 93 until P.le Gorini (Istituto Dei Tumori) from where it is just 240m by foot.

Most international and national rail tickets to Milano can be bought on Trainline or check Seat61.com.

Social dinner: There is a social dinner on Tuesday evening at 20:00 at restaurant "Casa Lodi", in the center of Milan. All participants are welcome to join. Speakers and external organizers will be covered by our funds. The most comfortable way of getting there is by Tram 19 (taking about 30 minutes), both from the department and from Hotel Viva (there are no ticket machines at the tram stop, buy a spare 2-Euro ticket when you are in a metro station).

Hotel: For all speakers and chairs who have indicated the need for a hotel room, a booking (2 nights unless requested differently) was made at Hotel Viva (Via Giacinto Gallina 12, 20129 Milano). This is about 9 minutes by foot from the department.

Please find below the abstracts of the lectures (in alphabetical order of the speakers):

Fermi, Davide (U Rome 3): Homogenization limit for multiple Aharonov-Bohm fluxes

We investigate the effects produced by several singular magnetic fluxes of Aharonov-
Bohm type on an electrically charged quantum particle. Using quadratic form techniques, we
firstly characterize the Friedrichs Hamiltonian and a family of singular perturbations in terms of boundary conditions on the flux lines. Next, we analyze the so-called homogenization limit, assuming the Aharonov-Bohm fluxes to be placed at the vertexes of a 2D square lattice. In a suitable scaling regime where the lattice pitch and the intensity per flux simultaneously go to zero, we show that the Friedrichs Hamiltonian converges in the sense of Gamma convergence to a regular magnetic Hamiltonian, with a uniformly bounded vector potential. Joint work with Michele Correggi (Politecnico di Milano).

Gallone, Matteo (U Milano): 2D Ising model with quasi-periodic disorder

Statistical mechanics models with quasi-periodic modulations describe several system of physical interest like quasi-crystals or the setting of a wide class of experiments where quasi-periodicity is a practical realisation of a disordered background. In the latters, the amplitude of the modulation is often tunable and during its variation one observes a transition between a delocalised and a localised phase.

The two-dimensional Ising model is a paradigmatic example of critical model, and it is therefore interesting to study the effect of a quasi-periodic coupling. The application of Harris criterion to this system suggests that small amplitudes quasi-periodic modulation should irrelevant in the renormalisation group sense, that is critical exponents are the same. Since the quasi-periodic model lies between the random Ising model for which disorder is relevant, no matter how strong, and the family of non-integrable "deterministic" Ising models for which a large class of small interactions is proved to be irrelevant, it is of interest to establish its critical behavior. Using rigorous renormalisation group techniques we prove that small amplitudes quasi-periodic disorder is irrelevant for a class of intrinsically two-dimensional disorders that is larger than the ones considered in the preceeding literature. This is a joint work with Vieri Mastropietro.

Giacomelli, Emanuela (LMU Munich): The dilute Fermi gas via Bogoliubov theory

We consider N spin-1/2 fermions interacting with a positive and regular enough potential in three dimensions. We compute the ground state energy of the system in the dilute regime making use of the almost-bosonic nature of the low-energy excitations of the systems.

Giuliani, Alessandro (U Rome 3 & Centro Linceo Interdisciplinare B. Segre): Renormalization at all orders for lattice infrared QED_{4} with massless electron

One of the goals of constructive Quantum Field Theory (QFT) is to provide a convergent algorithm for computing a consistent set of Euclidean correlation functions starting from a given bare action and, next, to reconstruct the corresponding real-time correlations via analytic continuation in the time variable. This program proved successful for constructing several low dimensional toy QFT models but results in 3 and 4 dimensions are still scarce.
Embarrassingly, until now, not even a consistent construction of infrared QED_{4} with small electron mass at all orders in renormalized perturbation theory was available, unless a loop regularization scheme is employed or a number of non-gauge-invariant counter-terms are included in the bare action.

In this talk I will describe such a consistent construction, at all orders in renormalized perturbation theory, in a lattice gauge theory model of QED_{4} with massless electron and no other counter-term than the one for the electron mass. We also prove that, in the presence of an infrared (IR) cutoff on the photon propagator, the model is non-perturbatively well-defined, provided the electron charge is sufficiently small (a priori, non-uniformly in the IR cutoff).

The proof is based on a Wilsonian Renormalization Group (RG) scheme and uses ideas developed in the last decade in the context of lattice gauge theory models of graphene and Weyl semimetals. In particular, we use Ward Identities at each RG step to control the flow of the effective couplings, including the non-gauge-invariant ones produced at intermediate steps by the multiscale procedure, and prove their infrared asymptotic freedom. I will comment on the perspectives opened by this and related works on the full construction of infrared QED_{4}, on the non-perturbative computation of the chiral anomaly and on the spontaneous emergence of Lorentz symmetry. Joint work with Marco Falconi and Vieri Mastropietro.

Greenblatt, Rafael (SISSA Trieste): The 2D Ising universality class via the constructive renormalization group

The planar Ising model is one of the best-known exactly solved models in statistical mechanics, however many important properties which have been shown using the exact solution (like conformal invariance of the scaling limit) should apply to a broader class of (generically non-solvable) models obtained by perturbing the Hamiltonian with an arbitrary finite-range term which respects the symmetries of the model.

Some rigorous results in this direction have been obtained using constructive renormalization group methods originally developed for the study of interacting Fermionic quantum field theories. I will present a recent result on energy correlations on the discrete cylinder, obtained in a joint work with G. Antinucci and A. Giuliani (arXiv:2006.04458), and discuss the prospects of further results.

Langella, Beatrice (SISSA Trieste): Growth of Sobolev norms for unbounded perturbations of the Laplacian on flat tori

In this talk I will present a recent result on a class of linear time dependent Schrödinger equations on arbitrary flat tori. In particular, I will prove a $|t|^\epsilon$ upper bound for any $\epsilon>0$ on the growth of Sobolev norms of all the solutions. As a main novelty, this result enables to deal with unbounded perturbations of the Laplacian, thus covering for instance the case of a particle moving in a time dependent electromagnetic field. The proof is based on a normal form technique and exploits ideas coming from dynamical systems: in particular, it is obtained as a quantum version of the proof of the classical Nekhoroshev theorem. This work has been done in collaboration with D. Bambusi and R. Montalto.

Marcelli, Giovanna (SISSA Trieste): A new approach to purely linear response of the quantum Hall current and to transport coefficients in the quantum spin Hall effect

Slides Marcelli

Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost-stationary state, we show that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime and derive general formulas for spin conductivity. For the spin transport, we consider both the choice of the conventional and of the proper spin current operator, and we isolate a subclass of discrete periodic models (including the Kane-Mele model) where the conventional and the proper spin conductivity agree. These results are proved in the following framework, including both discrete and continuum models: the perturbation to a periodic spectrally gapped equilibrium one-particle Hamiltonian is modeled by a linear potential. This seminar is based on joint works with D. Monaco, G. Panati and S. Teufel.

Maspero, Alberto (SISSA Trieste): Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

We consider linear, time dependent Schrödinger equations of the form $\text{i} \partial_t \psi = (H+V(t))\psi$, where $H$ is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and $V(t)$ a time periodic potential. We describe a large class of potentials which provoke unbounded growth of Sobolev norms. The main condition is that the resonant average of $V(t)$ has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay.

Olgiati, Alessandro (U Zurich): Upper bound for the ground state energy of hard-core bosons in the Gross-Pitaevskii regime

I will present a result on the ground state energy of N bosons interacting through
a hard-core potential (i.e., infinitely repulsive and with finite radius). The radius of
the interaction scales as 1/N, and the bosons are confined on the three dimensional unit torus (Gross-Pitaevskii regime). Under these conditions we prove an upper bound for the ground state energy up to an error which vanishes as N grows. Our result matches the known expression for the energy in the case of integrable potentials, which depends universally on the scattering length of the interaction, thus confirming the prediction of Bogoliubov’s
theory. Joint work with G. Basti (GSSI), S. Cenatiempo (GSSI), G. Pasqualetti (University of Zürich), and B. Schlein (University of Zürich).

Ponno, Antonio (U Padova): Coherent Dynamics of Cold Atoms in Optical Lattices

We provide rigorous estimates on the quantum dynamics of the Bose-Hubbard model (in any dimension). In particular, we consider the quantum evolution of the Wick symbol of polynomials in the Dirac variables defining the model, and compare it with the ``classical'' evolution of the symbol at time zero along the dynamics of the discrete Gross-Pitaevskii equation. The comparison is made in the $L_2 (\mu)$ norm, where $\mu$ is the Gauss-Planck measure on the coherent states at temperature T. We then prove that the $L_2 (\mu)$ distance of the two evolutions is bounded a priori, uniformly in time, and grows towards the bound at most linearly. The bound turns out to be the smaller, the lower is the temperature. For a finite number of neutral atoms in a linear array the estimate for simple monomials is given as an explicit expression of the four physical quantities characterizing the optical lattice: temperature, laser frequency, well depth, mass of the atom. This is a joint work with Lorenzo Zanelli (Padova) and Alberto Maiocchi (Milano-Bicocca).

Porta, Marcello (SISSA Trieste): Multi-Channel Luttinger Liquids at the Edge of Quantum Hall Systems

We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory. Joint work with Vieri Mastropietro.

Schlein, Benjamin (U Zurich): Bosonization of Fermionic Many-Body Systems

We consider fermionic systems in a joint mean field and semiclassical limit. We show how bosonic Bogoliubov theory can be used to describe corrections to the predictions of Hartree-Fock theory, both for the computation of the ground state energy and for the approximation of the time-evolution. The talk is based on joint works with N. Benedikter, P.T. Nam, M. Porta and R. Seiringer.

Zanelli, Lorenzo: session chair Wednesday

Speakers have been invited directly. If you would like to participate in the workshop, please write an email to Niels Benedikter (see contact information below).

Covid tracing: The organizers will keep a list of participants to ensure contact tracing in case of Covid infections. Note also that for travel to Italy, for attendance to the workshop, and in almost all public places a valid EU Covid certificate (vaccination, tested, or recovered) with QR code, in Italy called the "Green Pass", has to be presented. Before entering Italy, you will **additionally** have to fill out the EU Passenger Locator Form. This is not done by the organizers, but has to be done individually by each participant! Furthermore, to travel to Italy, now a negative Covid test has to be shown (a PCR test taken max. 48h before or an antigenic test taken max. 24h before).

For questions, please write an email to: niels.benedikter "[a t]" unimi.it

Organizing committee:

Dario Bambusi, Niels Benedikter, Chiara Boccato, and Vieri Mastropietro

Dipartimento di Matematica "Federigo Enriques"

Università degli Studi di Milano

Michele Correggi and Marco Falconi

Politecnico di Milano

We gratefully acknowledge financial support by Università degli Studi di Milano, Politecnico di Milano, and the MIUR PRIN 2017 project MaQuMA (PRIN201719VMAST01).