The ground state energy of the interacting Bose gas

Chiara Boccato (Università di Milano)

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.)

On the 2d Stochastic Heat Equation and delta Bose gas

Francesco Caravenna (Università Milano Bicocca)

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)

Renormalization at any order of lattice infrared QED_4

Marco Falconi (Politecnico di Milano)

Two dimensional Ising model with weak quasi-periodic disorder

Matteo Gallone (SISSA)

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.

Low temperature asymptotic expansion for classical O(N) vector models

Alessandro Giuliani (Università Roma Tre)

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.

Prethermalization in quasi-periodically driven quantum systems

Beatrice Langella (SISSA)

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.

Dirac-Bogoliubov-de Gennes equations and inhomogeneous quantum liquids

Per Moosavi (ETH Zurich)

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

Aspects of Transport in Many-Body Quantum Systems

Marcello Porta (SISSA)

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.

Ground state energy of dilute Bose and Fermi gases in 1D

Robin Reuvers (Università Roma Tre)

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.

Expansion of a one-dimensional interacting Bose gas

Spyros Sotiriadis (Crete)

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