# Programme

## Abstracts

Abstracts of the talks are available in PDF format: Abstracts.

Speaker: Simon Becker

Title: Mathematics of moiré materials

Abstrac: I will review recent mathematical developments in the theory of moiré materials. Moiré materials are new types of 2D lattice systems that due to twists, strain or other mechanical features have proven to show remarkable physical properties. The flagship example is twisted bilayer graphene exhibiting superconductivity at so-called magic twist angles (TBG). I will explain the current status of the field and discuss the main mathematical models.

Speaker: Stephan De Bievre

Title: (Un)certainty, (In)compatibility, (Non)classicality, Decoherence, Entanglement and all that

Abstract: In this talk, we will review recent work on the following question: « How to identify/characterize the nonclassical states of a bose field? » We will show the « quadrature coherence scale » of the state provides an efficient nonclassicality witness and provides bounds linking it to the entanglement of the state.

Speaker: Hakim Boumaza

Title: Localization for quasi-one dimensional models

Abstract: This talk  is about several ergodic families of matrix-valued random operators in dimension one, for which we prove (or expect to prove) Anderson localization. First I discuss how to reduce the question of Anderson localization in dimension 1 to the study of an algebraic object, the Fürstenberg group. For this purpose, I present typical objects of the dimension 1: the transfer matrices, the Lyapunov exponents and a bit of Kotani theory. Then I present the tools from Lie group theory which allow to prove the required properties of the Fürstenberg group in different settings : discrete or continuous matrix-valued Schrödinger operators with Anderson potential, and a unitary model, the scattering zipper.

Speaker: Anton Bovier

Title: Fluctuations of the free energy in p-spin SK models on two scales

Abstract: 20 years ago, Bovier, Kurkova, and L\ »owe \cite{BKL} proved a central limit theorem  (CLT) for the fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model of spin glasses at high temperatures. In this paper we improve their results in two ways. First, we extend the range of temperatures to cover the entire regime where the quenched and annealed free energies are known to coincide. Second, we identify the main source of the fluctuations as a purely coupling dependent term, and we show a further CLT for the deviation of the free energy around this random object. Joint work with Adrien Scherzer.

Speaker: Massimo Campanino

Abstrac: Estimates of the entanglement entropy for the quantum Ising model in one dimension based on stochastic representations are discussed.

Speaker: Margherita Disertori

Title: Supersymmetric method in the study of random matrices

Abstract:The supersymmetric method is basically a combinatorial tool that enables to reformulate averages of (products) of  resolvents (z-H)^-1 for a random matrix H, in terms of a statistical mechanics-type model. This technique involves the use of both real and anticommuting variables. I will give a brief introduction to this technique concentrating on some applications to the density of states for random band matrices.

Speaker: Christoph Fischbacher

Title: Entanglement entropy and localization results in the spin-J XXZ model

Abstract:We study the Heisenberg XXZ chain with local spin J ($J \in\{1/2,1,3/2,…\}$) with background magnetic field. As in the spin-1/2 case, it’s possible to rewrite the Hamiltonian as a direct sum of $N$-particle Schr\ »odinger operators with attractive interaction. We will present a bound on the entanglement entropy for states of arbitrarily high (but fixed) energy. After this, we discuss localization results under the presence of a random magnetic background field, which follow from a modification of work by Elgart and Klein in the spin-1/2 case. This is joint work with Fisher, Klein (Localization) and Ogunkoya (Entanglement Entropy).

Speaker: Peter Hislop

Title: Random band matrices in the localization regime

Abstract: The problem of determining the local eigenvalue statistics (LES) for one-dimensional random band matrices (RBM) will be discussed with an emphasis on the localization regime. RBM are real symmetric $(2N+1) \times (2N+1)$ matrices with nonzero entries in a band of width $2 N^\alpha +1$ about the diagonal, for $0 \leq \alpha \leq 1$.  The nonzero entries are independent, identically distributed random variables. It is conjectured that as $N \rightarrow \infty$ and for $0 \leq \alpha < \frac{1}{2}$, the LES is a Poisson point process, whereas for $\frac{1}{2} < \alpha \leq 1$, the LES is the same as that for the Gaussian Orthogonal Ensemble. This corresponds to a phase transition from a localized to a delocalized state as $\alpha$ passes through $\frac{1}{2}$. In recent works with B.\ Brodie and with M.\ Krishna, we have made progress in proving this conjecture for $0 \leq \alpha < \frac{1}{2}$.

Speaker: Ilya Kachkovskiy

Title: Perturbative diagonalization for quasiperiodic operators with monotone potentials

Abstract:We consider quasiperiodic operators on $\mathbb Z^d$ with unbounded monotone sampling functions (« Maryland-type”), and construct the Rayleigh—Schrodinger formal perturbation series for the eigenvalues and eigenvectors. We discuss the combinatorial structure and cancellations in such series and sufficient conditions for their convergence. This allows to establish Anderson localization for several classes of mononone quasiperiodic operators by constructing explicit converging expansions for eigenvalues and eigenvectors. If time permits, we will discuss cases where the requirement of strict monotonicity or unboundedness can be relaxed. The talk is based on the joint work with S. Krymskii, L. Parnovskii, and R. Shterenberg, both published and in progress.

Speaker: Werner Kirsch

Title: Trimmed Anderson models – revisited.

Abstract: We look at random Schr\ »{o}dinger operators $H$ on $X=M\times Z^{d}$ where $M$ is either $Z^{k}$ or a cube in the $Z^{k}$.The potential $V_{\omega}$ consists of iid random variables on a subset $\Gamma$ of $X$ and vanishes outside $\Gamma$. We investigate the spectrum of $H$ and prove pure point spectrum (e. g.) for small energies. Under certain assumptions we also prove the existence of some absolutely continuous spectrum. Joint work with M. Krishna.

Speaker: Frédéric Klopp

Title: Resonances for large random samples in the localized regime

Abstract:We consider random Schrödinger operators on the euclidean space. It is well known that, under the right assumptions, the spectrum of many such models exhibit exponential localization near spectral edges. The talk is devoted to the study of resonances near these edges when the random potential is restricted to a (large) convex domain which we dub a (large) random sample. While the literature contains rather precise results in one space dimension (that we shall partially present for comparison), in dimensions larger than 1, much less is known. Our main result is that, as the size of the sample goes to infinity, the density of resonances in strips near the real axis is given by the density of states of the full random Schr\”odinger operator. The talk is based on joint work with Martin Vogel.

Speaker: Davide Macera

Title: Anderson localisation for quasi-one-dimensional random operators

Abstrac: I’ll present recent spectral and dynamical localisation results for a general class random operators acting on $\ell^2(\mathbb Z)\times \mathbb C^W $(H\psi)(x)=L_{x-1}\psi(x-1)+L^*_x\psi(x+1)+V_x\psi(x),$ where$\{L_i\}_{i\in\mathbb Z}\inGL(\mathbb R,W) (hopping matrices), $V_i \in Sym(\mathbb R,W)$ (potential matrices) are mutually independent sequences of i.i.d. random matrices satisfying some very relaxed conditions. This model includes, as particular cases, the Anderson Hamiltonian on the product of a finite connected graph times $\mathbb Z$ and the one dimensional Wegner orbital model, describing the motion of a particle with internal degrees of freedom in a one dimensional disordered medium. This is a joint work with Sasha Sodin (QMUL).

Speaker: Chokri Manai

Title: Recent Rigorous Results on Quantum Spin Glasses

Abstract: In the past few decades, the theory of spin glasses has become a major field of interest in condensed matter physics, mathematical physics and proabbility theory. While in the classical spin glass theory many problems remain unsolved, at least a rough understanding of the underlying physics has been established. The milestone so far has been the derivation of a formula for the free energy in the classical Sherrington-Kirkpatrick model by this year’s Nobel price winner Giorgio Parisi based on his replica method and its rigorous proof by Guerra and Talagrand.

The situation is vastly different for quantum spin glasses, where quantum effects are for example incorporated via a transversal field. In this case no closed formula has been found for the Quantum SK-model; and most physical results are based on numerical simulations. In this talk, I will give an introduction to the topic of quantum spin glasses. I will discuss recent results on hierarchical Quantum spin glasses, where one can rigorously prove a formula for the free energy. Moreover, we will have a quick look on the Quantum SK model, where one can at least prove the existing of replica symmetry breaking.

Speaker: Rodrigo Matos

Title: Localization and eigenvalue statistics within Hartree-Fock theory

Abstract:This talk will consist of two parts. In the first part, I will provide background and present localization results for the disordered Hubbard model within Hartree-Fock theory which were obtained in joint work with Jeffrey Schenker. In the second part, I will present recent progress on the eigenvalue statistics for the above model. In particular, under weak interactions and for energies in the localization regime which are also Lebesgue points of the density of states, it is shown that a suitable local eigenvalue process converges in distribution to a Poisson process with intensity given by the density of states times Lebesgue measure. Time allowing, I will also discuss possible extensions to other interacting Hamiltonians and proof ideas

Speaker: Stanislav Molchanov

Title: Non-stationary Anderson parabolic problem with the non-local Laplacian and correlated white noise

Abstract: Twenty-five years ago, R. Carmona and S. Molchanov published in Memoirs of AMS the paper “Anderson parabolic problem and intermittency”. It was a simplified (scalar) version of the physical theory describing the magnetic field in the turbulent flow of the conducting fluid. One of the applications of this theory is the explanation of the intermittent distribution of the “black spots” on the Sun. The talk will present several results about generalized Anderson model (non-local Laplacian, correlated white noise type potential etc.) The phase transitions in this model mainly depend on the spectral theory of the Schr\”odinger type operators with a positively definite potential. The central part of the talk will describe the spectral bifurcations in this class of operators.

Speaker: Peter Müller

Title: Stability of Szegö-type asymptotics

Abstract: We consider a multi-dimensional continuum Schr\ »odinger operator H which is given by a perturbation of the negative Laplacian by a compactly supported bounded potential. We show that, for a fairly large class of test functions, the second-order Szegö-type asymptotics for the spatially truncated Fermi projection of H is independent of the potential and, thus, identical to the known asymptotics of the Laplacian.

Speaker: Bruno Nachtergaele

Title: Dimerization and the ground state gap for a class of $O(n)$ spin chains

Abstract: We consider two families of quantum spin chains with $O(n)$-invariant nearest-neighbor interactions and discuss the ground state phase diagram of this class of models. Using a graphical representation for the partition function, we provide a proof of a spectral gap above the ground states and spontaneous breaking of the translation symmetry for an open region in the phase diagram, for all sufficiently large values of n. (Joint work with Jakob Bjoernberg, Peter Muehlbacher, and Daniel Ueltschi).

Speaker: Fumihiko Nakano

Title: Shape of eigenvectors for the decaying potential model

Abstract: We consider the 1d Schr$\ »$odinger operator with decaying random potential, and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions which is based on the formulation by Rifkind-Virag. As a result,  we have completely different behavior depending on the decaying rate $\alpha > 0$ of the potential: the limiting measure is equal to (1) Lebesgue measure for the super-critical case ($\alpha > 1/2$), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case($\alpha=1/2$), and, (3) the delta measure with its atom being uniformly distributed for the sub-critical case($\alpha<1/2$). We also discuss the local version of this problem.

Speaker: Leonid Pastur

Title: Qubit Dynamics in Random Matrix Environment

Abstract: We consider models of dissipative and decoherent evolution of two qubits embedded in a disordered environment. Unlike the well known spin-boson model, which describes the translation invariant and macroscopic environment, we model both the environment and its interaction with qubits by random matrices of large size, which are widely used to describe multi-connected disordered media of mesoscopic and even nanoscopic size. An important property of our model is that it incorporate so called non-Markovian dynamics, which allows for the backflow of energy and information from the environment to qubits and has been actively studied recently in various settings. We find several interesting cases of the entanglement evolution of qubits, including vanishing their entanglement at a finite moment and, especially, the subsequent entanglement revival. These properties of entanglement dynamics are known in quantum information theory as the entanglement sudden death and entanglement sudden birth and are pertinent to the non-Markovian entanglement evolution. They have been found before in special versions of the macroscopic and translation invariant spin-boson model. Our results, obtained for non-macroscopic and disordered environment, demonstrate the robustness and universality of the above and other essential properties of entanglement evolution. Being combined with other processes of quantum information technology (e.g. entanglement distillation), they can lead to a considerably slower decay of entanglement up to its asymptotic persistence.

Speaker: Matthew Powell

Title: Positivity of the Lyapunov exponent for quasiperiodic operators with a finite-valued background

Abstract: The purpose of this talk is to describe recent and ongoing work regarding Lyapunov exponents for analytic quasiperiodic operators with finite-valued backgrounds. We prove that, for sufficiently large coupling constant, the Lyapunov exponent is positive with a uniform lower bound, based on a method that goes back to Bourgain.

Speaker: Renaud Raquépas

Title: Entropy production in nondegenerate diffusions: the large-time and small-noise limits

Abstract: Entropy production (EP) is a key quantity originating from thermodynamics and statistical physics which quantifies the irreversibility of the time evolution of physical systems. I will start with a general introduction to the different approaches to defining EP. Then, I will focus on the context of nondegenerate diffusions and I will describe the large-deviation properties of EP as time goes to infinity. Finally, I will discuss the behaviour of the corresponding rate function as the intensity of the noise goes to zero.

Speaker: Mostafa Sabri

Title: Transport distributions and dispersion

Abstract: In this talk, we consider Schr\ »odinger operators in a regime of delocalization and investigate two questions on the dynamics of evolution. First, given an initial state $\psi$, we would like to understand its limiting distribution in space. More precisely, we aim to identify the limit of the probability measure $|(e^{-itH} \psi)(x)|^2 dx$, properly normalized.

Second, we seek uniform bounds of the form $\| e^{-itH} \psi\|_\infty \le C t^{-\alpha}$, showing that the wave « flattens » as time goes on, so a real spreading occurs (this excludes the scenario of « sliding bumps »).

The results are mainly obtained for operators having a form of periodicity, though some partial results will be mentioned in the random case. We will conclude with some open questions. Based on joint works with Anne Boutet de Monvel and Ka\ »is Ammari.

Title: Radial transfer matrices for higher dimensional graphs

Abstract : We introduce a dynamics of sets of rectangular radial transfer matrices for discrete Hermitian locally finite hopping operators, such as Schrödinger operators on graphs. Then we generalize a special averaging formula to this setup and obtain criteria for absolutely continuous spectrum and show some applications.

Speaker: Jeffrey Schenker

Title: Theory of Ergodic Quantum Processes

Abstract: Any discrete quantum process is represented by a sequence of quantum channels. In this talk, I will consider general ergodic sequences of stochastic channels with arbitrary correlations and non-negligible decoherence and present a theorem showing that the composition of such a sequence of channels converges exponentially fast to a rank-one (replacement) channel, as well as a law of large numbers and central limit theorem for the expectations of observables.  Time permitting, applications of this formalism to the thermodynamic limit of ergodic Matrix Product States will be described.  (Joint work with Ramis Movassagh and Lubashan Pathirana).

Speaker: Hermann Schulz-Baldes

Title: Partially hyperbolic random dynamics on Grassmannians

Abstract: A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure showing that with high probability the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, even if the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained (joint work with Joris De Moor, Florian Dorsch).

Speaker: Mira Shamis

Title: Upper bounds on quantum dynamics

Abstract: We shall discuss the quantum dynamics associated with ergodic Schroedinger operators with singular continuous spectrum. Upper bounds on the transport moments have been obtained for several classes of one-dimensional operators, particularly by Damanik–Tcheremchantsev, Jitomirskaya–Liu, Jitomirskaya–Powell. We shall present a new method which allows us to recover most of the previous results and also to obtain new results in one and higher dimensions. The input required to apply the method is a large-deviation estimate on the Green function at a single energy. Based on joint work with S. Sodin.

Speaker: Jacob Shapiro

Title: Dynamical localization for random band matrices up to N^{1/4} band width

Abstract: We consider a large class of $N\times N$ Gaussian random band matrices with band-width $W$, and prove that for $W \ll N^{1/4}$ they exhibit Anderson localization at all energies. To prove this result, we rely on the fractional moment method, and on the so-called Mermin-Wagner shift (a common tool in statistical mechanics). Joint with Cipolloni, Peled, and Schenker.

Speaker: Senya Shlosman

Title: Glassy states

Abstract:I will talk about the Ising model on the Cayley trees and the Lobachevsky plane, which undergoes the spin-glass phase transition. I will explain that the free state in the spin-glass regime is a mixture of a continuum of pure phases.

Speaker: Israel Michael Sigal

Title: Many-body dynamics in thermodynamic regime

Abstract:In this talk, I present an extension of some recent results on evolution of many-body quantum systems in the thermodynamic regime. In particular, I discuss transport of states, correlations and particles. The proofs are based on new types of the Lieb-Robinson bound. The results (still work in progress)  are based on a joint work with Jeremy Faupin and Marius Lemm.

Speaker: Wei-Min Wang

Title: Anderson localization for the nonlinear random Schroedinger equations

Abstract:We review results on nonlinear Anderson localization. This talk is based on the joint works with J. Bourgain, and more recently with W. Liu.

Speaker: Sylvain Zalczer

Title: Anderson localization for random Dirac operators in dimension 1

Abstract: The Dirac operator, originally introduced to model relativistic matter, has been recently given a lot of interest  in the context of graphene models. Here, I will look at 1D Dirac operators, which could be a first model for graphene nanoribbons. I study what happens when various types of random potentials are added: some of them produce localization, while some others do not. The proof uses the usual features for random 1D systems: Lyapunov exponent, Thouless formula, multiscale analysis.