September 15 - 19: Titles and Abstracts




Tristan Benoist
Title: Ancilla tomography estimation of quantum entropic fluctuations
Abstract: Defining and accessing fluctuations of thermodynamic quantities in quantum systems is still a challenge more than a century after the birth of quantum mechanics. The two-time measurement definition introduced in 2000, while satisfactory from a theoretical point of view, is experimentally unrealistic, as it involves precise projective measurements of macroscopic quantities. This has drastic thermodynamic consequences (see Annalisa Panati’s talk).
In my presentation I will discuss a workaround called ancilla state tomography. Instead of recording entropic fluctuations, an ancilla qubit is used to perform a physical Fourier transform that is then estimated through the tomography of the ancilla. While this technique provides access to the law of the fluctuations, it does not resolve each fluctuation. However, it is experimentally possible to implement it, at least theoretically, as it involves only local perturbations of the thermodynamic system.
I will explain how the ancilla tomography relates to two-time measurement definition and present how we show a form of stability of this estimation technique with respect to the initial state of the system (Principle of Regular Entropic Fluctuations). The proof is based on a resonance analysis of quantum dynamical systems. I will illustrate the results using a spin-fermion model.

Volker Betz
Title: Localization and subdiffusivity for path integrals
Abstract: Quantum particles coupled to quantized fields can be described by path integrals, where Brownian motion is perturbed by a pair potential. For short range potentials, the long time behaviour of the relevant path measure is diffusive, which leads to a description of the effective mass of the quantum particle. This regime is fairly well understood by now.
In this talk, I will mainly address the complementary regime, where the pair potential is so long range that the path integrals behave dub-diffusively. In terms of the quantum model, this can be interpreted as the effective mass being infinite, or, in the most extreme case of path localization, as a self-trapping of the quantum particle through the field. I will talk about some recent progress for such models, and also highlight the still significant part of the picture that is not quite clear. This is based on joint work with Mark Sellke (Harvard) and Tobias Schmidt (Darmstadt).

Jakob Björnberg
Title: Dimerization and exponential decay of correlations in O(n)-invariant spin systems
Abstract: The general O(n)-invariant quentum spin systems has a rich phase diagram, part of which we have recently studied using a probabilistic representation. Here we have established, for large n: (1) dimerization accompanied by exponential decay of all truncated correlations in the ground-state of the one-dimensional model, and (2) exponential decay of two-point correlations, in arbitrary dimension for parameters where reflection-positivity holds. Based on joint works with Kieran Ryan, and Pete Mühlbacher, Bruno Nachtergaele and Daniel Ueltschi.

Yuxuan Chen
Title: Local large deviation principle for dispersive PDEs
Abstract: We present a new general criterion for large deviation principle on non-compact phase spaces, which characterizes the exponentially rare events where empirical distributions deviate from the unique invariant measure at order 1. The primary applications are to weakly damped randomly forced dispersive PDEs. Our approach is based on a novel concept called asymptotic exponential tightness, which captures the asymptotic compactness behavior from dispersive dynamics. This talk is based on joint works with Ziyu Liu, Shengquan Xiang, Zhifei Zhang and Jiacheng Zhao.

Noé Cuneo
Title: Large deviations for quantum trajectories: a look at the Keep–Switch instrument
Abstract: This talk addresses large deviations for quantum trajectories arising from repeated measurements. These trajectories form highly singular Markov processes that escape the scope of classical theory, making their large deviation properties largely an open problem. I will present an example, namely the Keep–Switch instrument, for which T. Benoist, C. Pellegrini, P. Petit and I were able to prove the Large Deviation Principle. This example shows that there is hope, yet still much work to do in order to develop a full theory.

Jan Dereziński
Title: Propagators on curved spacetimes from operator theory
Abstract: I will discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold M. Typically, I will assume that M is globally hyperbolic. Here, the term {\em propagator} refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Classical or, especially, Quantum Field Theory.
The off-shell setting is based on the Hilbert space L^2(M). It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called out-in Feynman and in-out anti-Feynman propagator.
The on-shell setting is  based on the Krein space W_KG of solutions of the Klein-Gordon equation. It allows us to define 2-point functions associated to two, possibly distinct Fock states as the Klein-Gordon kernels of projectors onto maximal uniformly positive subspaces of  W_KG.
After a general discussion, I will review a number of examples. I start with static and asymptotically static spacetimes, which are especially well-suited for Quantum Field Theory. Then I discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schrödinger operators, de Sitter space and the (universal covering of) anti-de Sitter space.
Based on a joint work with Christian Gass.

Louis Garrigue
Title: Higher order macroscopic operators for graphene
Abstract: The macroscopic descritpion of graphene is in general given by a massless Dirac operator. We show how the coupling of three methods (variational approximation, perturbation theory and a multiscale approach) enables to obtain corrected operators, leading to more accurate effective models.

Alain Joye
Title: Fermionic quantum walkers coupled to a bosonic reservoir
Abstract: We analyse the discrete-time dynamics of a model of non-interacting fermions on a finite dimensional sample coupled to an infinite reservoir formed by a bosonic quantum walk on Z in a quasi free state. We derive the Heisenberg dynamics of fermionic observables and obtain a systematic expansion in a large-coupling regime, which we control by using spectral methods. We prove that the reduced state of the fermions converges in the large-time limit to a mixture of infinite-temperature Gibbs state in each particle sector.
This is joint work with O. Bourget and D. Spehner

Joachim Kerner
Title: Atypical spectral and transport properties of non-locally finite crystals (and maybe more)
Abstract: In the first part of the talk we discuss recent results on Schrödinger operators on periodic graphs which are non-standard in the sense that we allow  vertices to have an infinite number of neighbours. It turns out that such non-locally finite graphs exhibit various phenomena which are absent in the locally finite setting: and this is true from a spectral as well as a transport point of view. Using some explicit examples, we shall illustrate such new effects in more detail. Quite surprisingly, it turns out that one of the examples provides us with a negative answer to a question raised by Damanik et al. in a recent paper on ballistic transport (this part of talk is based on joint work with O. Post, M. Sabri and  M. Täufer).
If time allows, we shall also quickly discuss spectral comparison results on discrete graphs. In recent years, various authors have derived such comparison results on Euclidean domains and quantum graphs. Our aim is to present a generalization to the discrete setting. Along the way, we also establish a so-called local Weyl law which is of independent interest (the second part of the talk is based on joint work with P. Bifulco and C. Rose). 

Flora Koukiou
Title: Mean-Field Models: Entropy, Large deviations and Multifractality
Abstract: We prove a universality result relating the low temperature free energy with the relative entropy. This allows the study of the entropy behaviour and the dimension spectrum of the Gibbs measure.

Asbjorn Baekgaard Lauritsen
Title: Cluster expansion for strongly correlated fermionic trial states
Abstract: I will present formulas for the reduced densities of fermionic Jastrow type trial states arising from a cluster expansion known as the Gaudin--Gillespie--Ripka (GGR) expansion. Using such formulas, we were recently able to give precise upper bounds for the energies of hard-sphere interacting Fermi gases. I will discuss the GGR expansion and our analysis of its convergence. Joint with Robert Seiringer.

Mathieu Lewin
Title: The gnocchi phase in nuclear pasta
Abstract: We provide the first rigorous justification for the so-called "gnocchi phase" in the liquid drop model at low density. Joint work with Rupert Frank and Robert Seiringer.

Sascha Lill
Title: Renormalizing Generalized Spin-Boson Models
Abstract: In quantum optics, emission and absorption of light is commonly described using so-called spin-boson type models. From the mathematical point of view, these models are only well-defined if the Hamiltonian of the system is a self-adjoint operator on some Hilbert space. While this is quite easy to prove for sufficiently regular form factors (i.e., functions in L^2), the singular form factors used in the physics literature typically render the construction of a self-adjoint Hamiltonian a challenging task, which requires sophisticated renormalization procedures. We present recent findings on this matter, based on joint work with Benjamin Alvarez, Davide Lonigro, and Javier Martín.

Annalisa Panati
Title: Entropic fluctuations in quantum two-time measurement framework
Abstract: Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, since the ground-breaking formulation of the transient and steady entropic Fluctuation Relations (FR) in the early nineties. The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an ongoing sort. Kurchan and Hal Tasaki's seminal works (2000) showed quantum formulation of the transient version of FR is possible by introducing the two-time measurement framework.
In this talk, we present some results in a recent series of papers, where we attempt to introduce a quantum equivalent of steady entropic functional and compare it to the transient version for open quantum system. In order to deal with the thermodynamic limit and to have general results, we use methods of C*- algebras and modular theory.
In this talk, we will consider idealised direct projective measurement on the reservoirs. We show in particular that the direct measurement has an invasive role, leading to dramatic consequences on stability with respect to the initial state. Therefore it is natural to consider experimentally accessible indirect measurement through coupling with an ancilla. This case will be mentioned briefly as it will be addressed in Tristan Benoist talk.
Joint work with T. Benoist, L. Bruneau, V. Jaksic, C.A. Pillet.

Lubashan Pathirana
Title: Stochastically generated matrix product states: Correlation decay in the thermodynamic limit
Abstract: We introduce and study a class of random matrix product states (MPS) constructed from stochastically generated local tensors that are identically distributed but not necessarily independent. This framework interpolates between two gextremal regimes: fully stochastically correlated random translation-invariant (TI) MPS, and MPS generated from independent and identically distributed (IID) local tensors, i.e. no stochastic correlation between local tensors. Under mild assumptions, we establish almost sure exponential decay of correlations in the thermodynamic limit, with stochastic prefactors and rates. Furthemore, when the maximal stochastic correlations between local tensors decay with distance, one obtains uniform, exponential or polynomial decay of two-point correlations with high-probability. Other high-probabilistic correlation rates can be obtained for specific distributions on local  tensors.
This is based on joint work with Albert H. Werner.

Nathan Réguer
Title: Semiclassical concentration of eigenfunctions of Toeplitz operators in phase space
Abstract: We are interested in semiclassical operators, which are operators depending on a small parameter h, particularly focusing on the concentration behaviour of their eigenfunctions. In the case of pseudodifferential operators, using elementary semiclassical tools, it is possible to find a set outside which the eigenfunctions are small with respect to the parameter h. We say that the eigenfunctions concentrate on this set. Although, it can be more challenging to understand how much they concentrate, which is why we are interested in bounds of the L^p norms of the eigenfunctions. Since the 1980's, a lot of results were found, with various behaviours depending on the geometry. In this talk, I will present a different kind of operators, called Berezin-Toeplitz, which act on functions defined on the phase space. We will see that, for these operators, the bounds of the L^p norms of the eigenfunctions are simpler, as they are less affected by the geometry.

Kieran Ryan
Title: Ground states in the XXZ chain and the Lorentz mirror model with loop weight 2
Abstract: We present some work in progress on infinite volume ground states of the XXZ chain. The work uses a representation of the chain as a probabilistic continuous time model of loops due to Ueltschi. The Lorentz mirror model with loop weight 2 can be thought of as a discrete time version of this loop model, and has a simple coupling with the 6-vertex model. We use a recent result of Glazman and Lammers and a further, new coupling to show that for the Lorentz loop model, certain finite volume measures converge to a common infinite volume measure, which exhibits no infinitely long loops, and has slow decay of connection probabilities. Work is in progress on the same statement in the quantum loop model, and its consequences for the XXZ model: certain finite volume, finite temperature states converge to a common infinite volume ground state, with slowly decaying spin-spin correlations. In particular, these arguments do not use any integrability or exact solutions.

Grega Saksida
Title: The large-mass limit of interacting Bose gases in the continuum
Abstract: We consider Bose gases in thermal equilibrium and show convergence of the grand-canonical Gibbs state to the corresponding large-mass (classical particle) limit. This limit corresponds to a classical theory of point particles with two-body interactions. Our analysis is carried out in the continuum. The analogous result on the lattice was previously shown by Fröhlich, Knowles, Schlein, and Sohinger. A challenge in the continuum is the unboundedness of the heat kernel, which requires us to suitably tune the chemical potential. Unlike in the related study of the mean-field limit from the work of Lewin, Nam, and Rougerie and Fröhlich, Knowles, Schlein, and Sohinger, one does not need to apply renormalisation in higher dimensions.
The main tool of our analysis is the random loop representation of the interactions due to Ginibre. In this framework, we can obtain quantitative estimates on convergence for the partition function and reduced p-particle density matrices. In the finite volume, we are able to work with stable interaction potentials. This is an ongoing joint work with S. Garouniatis and V. Sohinger.

Robert Seiringer
Title: Ubiquity of bound states for the strongly coupled polaron
Abstract: We study the spectrum of the Fröhlich Hamiltonian for the polaron at fixed total momentum. We prove the existence of excited eigenvalues between the ground state energy and the essential spectrum at strong coupling. In fact, our main result shows that the number of excited energy bands diverges in the strong coupling limit. In contrast, at weak coupling no such excited states exist. (Joint work with David Mitrouskas).

Oliver Siebert
Title: On the thermodynamic limit of interacting fermions in the continuum
Abstract: In this talk I will consider the dynamics of non-relativistic fermions in infinite volume in the continuum, interacting through a non-regularized pair potential.  Employing methods developed by Buchholz in the framework of resolvent algebras for bosons, I will show how the CAR algebra can be extended such that the dynamics acts as a group of *-automorphisms, which are continuous in time in all sectors of fixed particle numbers. Using the subalgebra generated by time-averages, one obtains a C*-dynamical system which is dense in the extended CAR algebra with respect to the seminorms of fixed particle numbers. The discussion is significantly shorter than in the bosonic case and provides a potential framework for discussing KMS states.

Vedran Sohinger
Title: Gibbs measures as local equilibrium Kubo-Martin-Schwinger states for focusing nonlinear Schrödinger equations
Abstract: Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure global well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the classical Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation.  In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of Hamiltonian PDEs, including nonlinear Schrödinger equations with defocusing interactions. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari (University of Rennes).
In the second part of the talk, we study (local) Gibbs measures for focusing nonlinear Schrödinger equations. These measures have to be localized by a truncation in the mass in one dimension and in the Wick-ordered (renormalized) mass in dimensions two and three. We show that local Gibbs measures correspond to suitably localized KMS states. This is joint work with Andrew Rout and Zied Ammari (University of Rennes).
 
Anna Szczepanek
Title: Dynamical Conservation Laws in Quantum Spin Systems
Abstract: We discuss new dynamical conservation laws for specific entropy, specific energy, specific relative entropy, and weak-Gibbisanity/state-interaction pair regularity in the lattice quantum spin system. These conservation laws, derived by using the Lieb-Robinson and Ruelle/Araki bounds, will be a starting point in the study of Approach to Equilibrium in these systems (see the talk by C. Tauber). Joint work with V.  Jakšić, C.-A. Pillet, C. Tauber.

Clément Tauber
Title: Approach to equilibrium in translation-invariant quantum systems
Abstract: I will discuss recent structural results about the problem of approach to equilibrium in translation-invariant quantum spin systems on a lattice. The main result is a new characterization in terms of existence and decay properties of some asymptotic interaction, together with the interplay with constant of motions. The proof relies strongly on conservation laws and the regularity property, recently developed in this framework (see the talk by A. Szczepanek).
Joint work with V.  Jakšić, C.-A. Pillet, and A.Szczepanek

Stefan Teufel
Title: Finding spectral gaps in quasicrystals
Abstract: The spectrum of periodic operators can be computed analytically and also approximated numerically using Bloch-Floquet theory, i.e. discrete Fourier transformation. For quasi-periodic operators no such general approach is available. In my talk I will first present a method for approximating the spectrum of general short-range infinite-volume operators on discrete sets with two-sided error control, using only data from finite-sized local patches. For operators with the additional property of finite local complexity this yields an explicit algorithm for approximating the spectrum numerically and allows, in particular, for computer assisted proofs of spectral gaps in such systems. As examples I discuss the p_x p_y-model and the discrete magnetic Laplacian on the two-dimensional quasi-periodic Ammann-Beenker tiling. This is based on joint work with Paul Hege and Massimo Moscolari (Physical Review B 2022 and Mathematics of Computation 2025).

Luc Vinet
Title: Dynamical algebra of the generic superintegrable model on the 2-sphere and bivariate Jacobi polynomials
Abstract: The dynamical algebra of the superintegrable model on S_2 will be introduced. It will be shown to provide an algebraic interpretation of the bivariate Jacobi polynomials. This will offer a framework to review the properties of certain algebras of the Askey-Wilson types that are associated to bispectral hypergeometric polynomials. The covariance of the pentagonal scheme that will emerge will be described.

Shengquan Xiang
Title: Exponential mixing for the randomly forced NLS equation
Abstract: We explore exponential mixing of the invariant measure for randomly forced nonlinear Schrödinger equation, with damping and random noise localized in space. Our study emphasizes the crucial role of exponential asymptotic compactness and control properties in establishing the ergodic properties of random dynamical systems. This talk is based on the recent joint work with Yuxuan Chen, Zhifei Zhang and Jia-Cheng Zhao.