Japan-Italy Quantum Meeting

Program

Titles & Abstracts

C. Caraci. The Φ⁴₂ theory as limit of interacting inhomogeneous Bose gases

Abstract. Euclidean field theories have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. Formally, they can be described by Gibbs measures associated with Euclidean action functionals over spaces of distributions. In the latest years it has been shown how such theories emerge as high-density limit of interacting Bose gases at positive temperature, giving a rigorous derivation from a realistic microscopic model of statistical mechanics.  In this talk, I will present a result providing the derivation of such a field theory, in other words of the invariant Gibbs measure, with a quartic local interaction in two dimensions as a limit of an inhomogeneous interacting Bose gas, extending the previous work on the torus by Fröhlich-Knowles-Schlein-Sohinger. Based on a joint work with Antti Knowles, Alessio Ranallo and Pedro Torres Giesteira.

M. Correggi. Quasi-classical Limit of the Renormalized Nelson Model

Abstract. We investigate the quasi-classical limit of the Nelson model, describing nucleons interacting with a scalar bosonic field, i.e., when the field degrees of freedom becomes classical while the nucleons retain their quantum nature. It is well known that such a model admits an energy renormalization of the ultraviolet divergence via the so-called dressing transformation. We then study the interplay between such a renormalization and the quasi-classical limit in both the stationary and dynamical pictures.

Joint work with M. Falconi (Politecnico di Milano) and M. Olivieri (UPC, Barcelona).

D. Fermi. Homogenization of Schrödinger operators with singular potentials

Abstract. We discuss the behaviour of a non-relativistic quantum particle interacting with a regular electromagnetic field and a large collection of singular zero-range potentials supported on a discrete set of points. We consider a scaling regime in which the number of points goes to infinity, while their mutual distances and the associated intensities simultaneously vanish, keeping the total interaction strength finite. Assuming the singular potentials have negative scattering lengths and are uniformly distributed, we prove that the corresponding Hamiltonians convergence in the strong resolvent sense to a limit effective operator comprising an additional attractive electric potential. The convergence can be strengthened to norm resolvent convergence when an external trapping mechanism is present. Our approach relies on quadratic forms and Γ-convergence methods. In the final part of the talk, I will outline some related recent developments and open problems.

Based on joint work with Domenico Cafiero and Michele Correggi (Politecnico di Milano).

E.L. Giacomelli. On the Huang-Yang conjecture for a repulsive low density Fermi gas

Abstract. In 1957, Huang and Yang predicted an asymptotic formula for the ground state energy of a dilute Fermi gas in the thermodynamic limit up to the third order in the expansion. This formula highlights a remarkable universality, showing that the correlation energy depends solely on the interactions scattering length. In a joint work with C. Hainzl, P. T. Nam and Robert Seiringer we give a proof of this conjecture.

Y. Goto. Violation of Parity and Flavor Symmetries in a Nambu-Jona-Lasinio Model

Abstract. In this talk, I discuss a lattice Nambu–Jona-Lasinio model with certain continuous chiral and two-flavor symmetries. For the Hamiltonian of this model, we construct a ground state that simultaneously breaks parity and flavor symmetries. This phase structure may be analogous to the Aoki phase, a well-known phenomenon in lattice field theory.

R. Greenblatt. Constructing a weakly-interacting fixed point of the Fermionic Polchinski equation

Abstract. The renormalization group is one of the most important tools for the description of critical points in theoretical physics.  Most mathematically rigorous treatments are based on implementing the renormalization group as a discrete dynamical system, which introduces a number of theoretical complications, and in particular obscures the emergence of full scale invariance, which is one of the most important features of a critical point.  In the last few years, however, there have been several new results about solutions of the Polchinski equation (a nonlinear differential equation which implements the renormalization group as a continuous dynamical system) for Fermionic systems.  I will present one in particular, the construction of a nontrivial fixed point of a family of continuous renormalization group flows corresponding to certain weakly interacting Fermionic quantum field theories with a parameter in the propagator allowing the scaling dimension to be tuned in a manner analogous to dimensional regularization.

K. Higuchi. Semiclassical Analysis of Resonant Tunneling via Quantum Walks

Abstract. Resonant tunneling is a well-known quantum-mechanical phenomenon. In a typical setting, for a one-dimensional Schrödinger operator with two symmetric potential barriers, the transmission probability for a particle with a real energy close to the real part of a quantum resonance is almost one. This is well known from WKB analysis of the Schrödinger equation and from physical experiments. Since the transmission probability through a single potential barrier is extremely small, resonant tunneling is contrary to the intuition of classical probability theory. 

Nevertheless, the fact that resonant tunneling has been known for a long time does not mean that its relationship with quantum resonances is fully understood. Indeed, the existence of a quantum resonance does not by itself imply a high transmission probability: there are many examples in which the transmission probability remains small even at real energies close to a quantum resonance. 

In this talk, we study the relationship between quantum resonances and resonant tunneling in the setting of quantum walks. In particular, we explain how a symmetry in the behavior of resonant states at infinity causes the resonant tunneling effect. 

A. Ishida. On inverse scattering for time-decaying harmonic and repulsive potentials

Abstract. We study quantum inverse scattering for Schrödinger equations with time-dependent quadratic Hamiltonians. The unperturbed system contains either harmonic or repulsive quadratic potentials whose strength decays at a critical rate in time. This decay produces scattering states that differ from both the free and time-independent cases. For real-valued interaction potentials satisfying suitable spatial decay conditions, the existence of wave operators has been established in previous work, and the associated scattering operator is well defined. The main result presented here proves uniqueness in inverse scattering: the interaction potential is uniquely determined by the scattering operator. In the harmonic case, scattering arises despite the confining nature of the quadratic potential. In the repulsive case, the time decay modifies the long-time behavior and allows decay comparable to the long-range class in the Stark effect, while remaining short-range within this framework.

A. Lucia. Spectral gap bounds for quantum Markov semigroups via correlation decay

Abstract. The return-to-equilibrium dynamics for quantum spin systems at positive temperatures, in certain regimes, is described by an ergodic quantum Markov semigroup, whose mixing time can be controlled by estimating the spectral gap of its generator. In the case of semigroups which satisfy a strong quantum detailed balance condition, I will show how to estimate the spectral gap of the generator via a measure of decay of correlations, evaluated on the invariant state. Examples of models for which the correlation decay can be explicitly computed are every 1D model and the family of 2D topological quantum error correcting codes introduced Kitaev. Based on a joint work with D. Pérez-García and A. Pérez-Hernández (arXiv:2505.08991)

T. Miyao. Bloch-Fibre Hamiltonians for a Periodic ℤ₂ Gauge–Fermion Chain: An XXZ-Type Reference Spectrum and Spectral Comparison

Abstract. We study a periodic one-dimensional ℤ₂ gauge–fermion chain using a Bloch-fibre formulation. In the strong-coupling regime, a quasi-local Schrieffer–Wolff reduction yields an effective low-energy operator on the rigid-dimer manifold. A discrete Schrödinger operator with a linear (string) potential emerges as a central comparison object. After a compression step, we relate the low-energy spectrum to an explicit XXZ-type spin-chain reference model and obtain uniform finite-volume spectral bounds. We also prove existence of the thermodynamic limit of the ground-state energy per site at fixed density, and briefly discuss the corresponding asymptotic behaviour of ground-state particle-density profiles in this regime.

T. Mine. IDS for random point interactions

Abstract. We study the asymptotics of the integrated density of states (IDS) for the Schrödinger operators on the Euclidean space ℝ³ with point interactions supported on some random point configurations.

Especially, we show that the decay rate of IDS N(λ) as λ→∞ is slower than that for the random scalar potential supported near the same random point configurations. 

We also show the numerical result about the IDS for several random point interactions.

This talk is based on the joint work with Masahiro Kaminaga (Tohoku Gakuin Univ.), Fumihiko Nakano (Tohoku Univ.) and Tomoyuki Shirai (Kyushu Univ.).

D. Monaco. Exactness of Ohm's law in extended quantum Hall systems

Abstract. In the quantum Hall effect, when a 2D electron gas is brought out of equilibrium by an electric field of small intensity, the transverse current responds exhibiting a quantized conductivity. This linear response coefficient can be expressed through the so-called double commutator formula, involving only the equilibrium ground state of the system, and furthermore all higher-order terms in this response vanish, establishing exactness of Ohm's law. I will show how these results can be proved by means of a recently developed technique which characterizes a non-equilibrium almost-stationary state (NEASS) of the perturbed quantum system. This technique applies to continuum models of quantum Hall systems in the 1-body approximation, as well as to weakly interacting lattice fermions. (Based on joint works with G. Marcelli, T. Miyao, S. Teufel, and M. Wesle.)

M. Moscolari. Bulk-edge correspondence in the absence of ergodicity

Abstract. We prove the zero-temperature bulk-edge correspondence for projections on spectral islands of 2d magnetic Schrödinger operators, without any ergodicity assumptions. A key ingredient in the proof is a Streda-like formula for projections which may not have a proper integrated density of states. Our approach relies on a careful analysis of the magnetic derivative of the finite volume integrated density of states obtained by gauge covariant magnetic perturbation theory, and on the index of pair of projections. 

The talk is based on an ongoing joint work with H. Cornean and S. Teufel.

F. Nakano. A 1D random system with 2-body interaction

Abstract. The effect of 2-body interaction in random systems is not well understood. In fact, there is still no agreement on the existence of many body localization. In this talk, we discuss a 1D system of spin 1/2 fermions with random potential. We consider the 2-body density matrix and show that this does not decay at large distance in a special case: near the half-filling with infinite on-site interaction. 

M. Olivieri. Effective Scattering theory for the Nelson-Yukawa model in the Semi-Classical regime

Abstract. Scattering and transition amplitudes are fundamental observables that allow us to extract crucial information about quantum systems in experimental physics. In this talk, we compute their semiclassical approximation for the Nelson–Yukawa model, which describes the strong nuclear interaction between mesons and nucleons and represents a linearized model for Quantum Electrodynamics. We show that, in the semiclassical limit of a large number of particles and a high number of field excitations, the quantum scattering and transition amplitudes of the Yukawa model converge to the corresponding classical scattering and transition amplitudes for the Schrödinger–Klein–Gordon equations. This establishes a direct connection between the quantum and classical scattering theories, proving Bohr's correspondence principle in this framework. This talk is based on a series of joint works with B. Alvarez, Z. Ammari, and M. Falconi.

M. Ozawa. TBA

Abstract. TBA

G. Panati. The Localization Dichotomy for periodic and non-periodic insulators

Abstract. We aim at investigating the localization properties of (independent) electrons in solids, possibly including a periodic magnetic field, as e. g. in Chern insulators and in Quantum Hall systems. The dynamics of the electrons is modeled by a magnetic Schrödinger operator,  or by its discrete analog, possibly with ergodic disorder.

In 2016, we proposed to describe the localization properties of the system by the decay of the Wannier Bases associated with the spectral projection below the Fermi energy, which is supposed to be in a spectral gap. In the periodic case, we proved the validity of a localization dichotomy, for dimension d ≤ 3, stating the following: either there exist an exponentially localized Wannier Basis, and correspondingly the system is in a Chern-trivial topological phase with vanishing Hall conductivity, or the decay of any Wannier basis is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is infinite. In the latter case, the Chern number of the Fermi projector is non-zero.

While in the non-periodic (disordered) case the mathematical picture is still incomplete, several new results have been proved in the last few years, based on a variety of techniques. 

In my talk, I will first review the results concerning periodic systems, and I will later explain some recent achievements in the non-periodic setting.  

The talk is based on joint works with D. Monaco, A. Pisante, and S. Teufel for the periodic setting, and with G.Marcelli, G. Moscolari, and V. Rossi for the non-periodic case.

M. Porta. Z2 lattice gauge theory coupled to fermionic matter

Abstract. I will discuss a model for fermions on a two-dimensional square lattice, minimally coupled to a Z2-valued dynamical gauge field, living on the bonds of the lattice. As observed numerically, this system displays a rich phase diagram, depending on the model parameters. In particular, at half-filling, the model at low temperature exhibits a semimetallic phase, in which the low-energy charge excitations are effectively described by 2+1 dimensional massless Dirac fermions. I will discuss the rigorous proof of this fact, building on Lieb’s seminal work about the solution of the pi-flux phase conjecture. In presence of a staggered mass term, the ground state of the gauge theory turns out to be four-fold degenerate, and separated by the rest of the spectrum by a gap. In particular, it supports anyonic excitations, equivalent to the ones of the toric code. The proofs are based on reflection positivity, chessboard estimates, fermionic cluster expansion and Hastings’ quasi-adiabatic flow. Based on collaborations with Sven Bachmann (UBC Vancouver) and Leonardo Goller (SISSA)

A. Sakai. Lace expansion for the quantum Ising model

Abstract. The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. Recently we proved that, for the nearest-neighbor model in d>4 with sufficiently small quantum parameter q>0, the susceptibility (defined as the sum of the Duhamel 2-point function) diverges as 1/(β_c(q) - β) near the critical point β_c(q), yielding the critical exponent γ=1. A key ingredient in our proof is the infrared bound on the 2-point function, which arises from reflection positivity. We expect that the infrared bound holds for a wider class of models, including next-nearest-neighbor and spread-out models.

In this talk, I will introduce the lace expansion for the Duhamel 2-point function, present the exact expression for the critical point β_c(q), and explain how to bound the expansion coefficients in high dimensions. Additionally, I will demonstrate how the infrared bound can be obtained without assuming reflection positivity.

This presentation is based on ongoing joint work with Yoshinori Kamijima (Toyo University).

K. Taira. A geometric obstruction in phase space for essential self-adjointness of pseudodifferential operators

Abstract. In this talk, I will give sufficient conditions for (pseudo)differential operators to fail to be essentially self-adjoint from a geometric perspective in phase space. This result covers many of the results from the recent work by Colin de Verdière and Bihan, as well as from my own previous work. Parts of the assumptions are reminiscent of a justification of the Bohr-Sommerfeld quantization, established by Duistermaat and Weinstein. As an application, we can show that the wave operator on the Schwarzschild spacetime is not essentially self-adjoint due to the behavior near the event horizon.

T. Watanabe. Transition probabilities in tangential avoided crossings: adiabatic vs. non-adiabatic regimes

Abstract. In this talk, we investigate the asymptotic behavior of the transition probability in two-level avoided crossings in the limit where two parameters -the adiabatic parameter and the energy gap parameter- simultaneously approach zero.

This work extends our previous collaboration with M. Zerzeri, which focused on avoided crossings generated by transversal intersections in a non-adiabatic regime, to the case of tangential intersections.

We examine not only the asymptotic expansion of the transition probability, but also quantum interference effects arising from multiple avoided crossings, as well as the coexistence of distinct two-parameter regimes associated with different vanishing orders.

This is a joint work with K.Higuchi (Gifu university).