Monday 9/1
10.30-11.15 Giuliani
11.20-12.05 Greenblatt
12.10-12.55 Gallone
14.00-14.45 Caravenna
14.50-15.35 De Vecchi
15.40-16.25 Falconi
17.00-17.45 Scola
17.50-18.35 Benedikter
Tuesday 10/1
9.40-10.25 Gentile
10.30-11.15 Porta
11.20-12.05 Sotiriadis
12.10-12.55 Lerose
14.00-14.45 Bonetto
14.50-15.35 Moosavi
15.40-16.25 Renzi
17.00-17.45 Fresta
17.50-18.35 Pizzo
Wednesday 11/1
10.30-11.15 Boccato
11.20-12.05 Langella
12.10-12.55 Reuvers
We consider a gas of bosonic particles confined in a box with Neumann boundary conditions. We prove Bose-Einstein condensation in the Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our lower bound for the ground state energy in the box implies (via Neumann bracketing) a lower bound for the ground state energy of the Bose gas in the thermodynamic limit. (Joint work with Robert Seiringer.)
In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, we worked to extend the analysis to more "realistic" versions of the original model.
I will introduce the Kac model and present some standard and more recent results. These results refer to a system with a fixed number of particles and at fixed kinetic energy (micro canonical ensemble) or temperature (canonical ensemble). I will introduce a "Grand Canonical" version of the Kac system and discuss new results on it.
We study the 2d Stochastic Heat Equation, that is the heat equation in two space dimensions with a multiplicative random potential (space-time white noise). Due to the singularity of the potential, this equation is ill-posed and its solution is not clearly defined, but its moments are closely connected to the multi-body delta Bose gas. We regularise the Stochastic Heat Equation by discretising space-time and we prove that, in the limit when discretisation is removed, and the noise strength is rescaled in a critical way, the solution has a well-defined and unique limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow.
(joint work with R. Sun and N. Zygouras)
Stochastic quantization is a method to build and study Euclidean quantum field theories introduced by Parisi and Wu, and based on the analysis of the measure related to the quantum field through suitable infinite dimensional stochastic differential equations. This method has received attention in recent years, mainly for Bosonic systems, thanks to the new developments in the investigation of singular stochastic PDEs. After a short review of the results in the Bosonic case, we show how to extend some of stochastic quantization techniques in the anti-commutative (Fermionic) setting. The talk is based on [1] in collaboration with S. Albeverio, L. Borasi and M. Gubinelli and on [2] in collaboration with L. Fresta and M. Gubinelli.
[1] Albeverio, S., Borasi, L., De Vecchi, F. C., & Gubinelli, M. (2022). Grassmannian stochastic analysis and the stochastic quantization of Euclidean Fermions. Probability Theory and Related Fields, 1-87.
[2] De Vecchi, F. C., Fresta, L., & Gubinelli, M. (2022). A stochastic analysis of subcritical Euclidean fermionic field theories. arXiv preprint arXiv:2210.15047.
In this talk, I will describe a synergy between the renormalization group (RG) in the form of the Polchinski's equation and the stochastic quantisation in the form of a forward-backward stochastic differential equation (FBSDE). This approach can be used for constructing subcritical Grassmann Gibbsian measures and is based on controlling the solution of the FBSDE by means of a flow equation with respect to a scale parameter. However, unlike the standard RG approach, we only need to solve the Polchinski's equation in an approximate way, resulting in great simplification of the analysis. Based on joint work with F. De Vechhi and M. Gubinelli.
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.
We consider the 2D Ising Model with a weak bidimensional quasi-periodic disorder in the hopping. Harris-Luck criterion predicts that the diorder is irrelevant, and previous first order analysis for 1d disorder analysis confirm such expectation. In this talk I will show that that the singularity of the specific heat and the exponent of the energy correlations are the same as in the integrable two dimensional model, both in the case of 1d and 2d disorder. The proof is based on Renormalizaion Group analysis and small divisors are controlled assuming a suitable Diophantine condition. This is a joint work with V. Mastropietro.
We consider classical O(N) vector models in dimension three and higher and investigate the nature of the low-temperature expansions for their multipoint spin correlations. We prove that such expansions define asymptotic series, and derive explicit estimates on the error terms associated with their finite
order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from
reflection positivity and the chessboard estimate, with an integration-by-parts method applied systematically
to a suitable integral representation of the correlation functions. Our method generalizes an approach, proposed originally by Bricmont-Fontaine-Lebowitz-Lieb-Spencer in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables. Joint work with S. Ott.
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 and boundary spins correlations on the discrete cylinder, obtained in a joint work with G. Antinucci and A. Giuliani and work in progress with G. Cava and A. Giuliani, and discuss the prospects of further results.
In this talk I will present some results on prethermalization for systems on a lattice perturbed with quasi-periodic external driving in time. The unperturbed model is the sum of mutually commuting terms with well separated energy scales; the time quasi-periodic perturbation can be either small in size or with fast frequency.
I will show that, up to exponentially long times in the proper perturbative parameter, these systems evolve accordingly to a time independent effective Hamiltonian, and that within these times one can find a certain number of quasi-conserved quantities.
I will also present the physical consequences of these dynamical constraints to quasi-periodic perturbations of the quantum Ising model and the generalized Fermi Hubbard model.
The theory of non-equilibrium dynamics of isolated quantum many-body systems plays a central role in contemporary research in theoretical physics, heavily stimulated by both foundational theory questions and future technological applications.
I will give a guided overview of aspects of this subject — some of which contributed by my work — of interest for future rigorous studies, including (many) conjectures and (few) proved statements in the literature. The focus of my talk will be on thermalization, mechanisms hindering thermalization, and non-equilibrium phases of matter.
If time permits, I will finally discuss a new influence-functional approach, introduced by myself and collaborators, which allowed some progress and holds some promises.
I will discuss a pair of coupled partial differential equations identified as Bogoliubov-de Gennes equations known from superconductivity but with Dirac operators replacing the usual Laplace operators and show that the Dirac version appears naturally for inhomogeneous quantum liquids. The equations feature an effective local gap due to inhomogeneities, coupling right- and left-moving degrees of freedom, leading to scattering, and so far had not been solved in general. I will show that one can obtain analytical solutions using Magnus expansions and study their properties to obtain information about the dynamics. The main physical motivation comes from so-called inhomogeneous Tomonaga-Luttinger liquids, but the equations also arise in descriptions of, for instance, superconductor-normal-metal interfaces, polymer chains, and coupled fractional quantum Hall edges
I will consider families of quantum lattice systems that have attracted much interest amongst people studying topological phases of matter. Their Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian consisting of strictly local terms and with a (strictly positive) energy gap above its ground-state energy.
I will review the main ideas of a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the lattice. By this method fermions and bosons are treated on the same footing, and our technique does not face a large field problem, even though bosons are involved.
In this talk I will discuss recent rigorous results about the charge transport properties for gapless interacting fermionic lattice models, with a particular emphasys on the emergence of universality at the macroscopic scales. I will outline a general strategy, based on the combination of: analytic continuation of real-time response functions to imaginary times; RG analysis of imaginary-time correlations; resolution of the scaling limit; lattice conservation laws and Ward identities. I will focus on the application to the edge response function of two-dimensional interacting topological insulators, defined starting from the linear response ansatz. In the last part of the talk, I will discuss how the framework can be used go beyond the linear response approximations, and to fully control the real-time dynamics for gapped systems at zero and at small positive temperature. If time permits, open problems will be discussed.
Based on collaborations with G. Antinucci, A. Giuliani, R. Greenblatt, M. Lange, G. Marcelli, V. Mastropietro.
The 2D dimer model is a simple model of discrete random surfaces and its exact solvability have been known since the 60's. In the so called "liquid phase", the height field scales to a log correlated Gaussian Free Field (GFF), with a universal stiffness coefficient, independent of the height's average slope and of the choice of the underlying lattice. In recent years, Giuliani, Mastropietro and Toninelli (GMT) addressed the question of universality for a class of perturbations breaking integrability, showing that it survives in a different, deep, form: while the critical exponent of the oscillating part of the dimer-dimer correlation changes continuously with the strength of the perturbation, the height field still scales to a log correlated GFF with an amplitude equal to the dimer-dimer critical exponent, irrespective of the choice of the height's slope. The GMT results are restricted to planar lattices with minimal elementary cell. In this talk we discuss an extension of their results to a class of periodic "weakly non-planar" lattices, with elementary cell of arbitrary size. Notwithstanding the loss of planarity, the scaling limit of the height field is still well defined and given by a GFF, in agreement with a conjecture by S. Sheffield. Based on joint work with A. Giuliani and F. Toninelli
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. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, Lieb and Liniger's model found experimental use and became even better known. In the meantime, mathematical physicists had been studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to obtain, and a popular goal was the derivation of the ground state energy of Bose gases in various settings in 2 and 3D. Many of these efforts focused on the dilute limit, in which the density of the gas is very low. Somehow, Bose gases in 1D were not studied in this way, most likely because the Lieb-Liniger model provides an exactly solvable set-up. Nevertheless, we can use insights from the 2 and 3D approaches to prove results about the ground state energy of dilute Bose and Fermi gases in 1D. In the talk, I will review the developments above and explain our results.
We construct the critical exponents for the interacting Green’s function and the densitydensity correlation function associated with the non-trivial fixed point theory of a fermionic φ 4 d model with fractional kinetic term in dimension d = 1, 2, 3. In this model, the exponent of the fractional Laplacian in the kinetic term is fixed so that the quartic interaction is weakly relevant, with scaling dimension ε > 0. The non-trivial fixed point potential has been constructed in previous work by Giuliani-Mastropietro-Rychkov and proved to be analytic in ε. Here we construct the fixed point generating function in the presence of two external auxiliary fields coupled to the fermionic field and to the density field, respectively, and the corresponding two point correlations. In particular, we prove that the density-density critical exponent is anomalous, that is, different from the naive one suggested by dimensional analysis, and analytic in ε. Time permitting, I will comment about the non-perturbative definition of scaling operators, which readily follows from our construction. Joint work with A. Giuliani, V. Mastropietro and S. Rychkov.
The analytical study of quantum many-body dynamics is a challenging mathematical problem. Considering a process in which a far from equilibrium initial state is let to evolve unitarily under an interacting Hamiltonian, the physically relevant quantities are the asymptotics of observables in the thermodynamic limit and at large times. However, even when the expansion of the initial state in the energy eigenstate basis is known, the evaluation of sums over all energy eigenstates and the derivation of the asymptotics is a formidable task. I will present a strategy to tackle this problem in a case study, the expansion of a one-dimensional Bose gas in the limit of point-like interactions, the Lieb-Liniger model, which offers the advantage of integrability.
References: arxiv:2007.12683