# One world IAMP mathematical physics seminar

This online seminar takes place on Tuesdays, starting at UTC 14.

Current organisers are Jan Dereziński (Warsaw) and Daniel Ueltschi (Warwick).

Scientific committee: Nalini Anantharaman (Strassbourg), Mihalis Dafermos (Cambridge), Stephan De Bièvre (Lille), Krzysztof Gawedzki (ENS Lyon), Bernard Helffer (Nantes), Vojkan Jaksic (McGill), Flora Koukiou (Cergy), Antti Kupiainen (Helsinki), Mathieu Lewin (Paris Dauphine), Bruno Nachtergaele (UC Davis), Claude-Alain Pillet (Toulon), Robert Seiringer (IST Austria), Jan Philip Solovej (Copenhagen), Hal Tasaki (Gakushuin).

 January 12, 2021 Fabio Toninelli (Technical University Vienna) The anisotropic KPZ equation and logarithmic super-diffusivity The AKPZ equation is an anisotropic variant of the celebrated (two-dimensional) KPZ stochastic PDE, which is expected to describe the large-scale behavior of (2+1)-dimensional growth models whose average speed of growth is a non-convex function of the average slope (AKPZ universality class). Several interacting particle systems belonging to the AKPZ class are known, notably a class of two-dimensional interlaced particle systems introduced by A. Borodin and P. Ferrari. In this talk, I will show that the non-linearity of the AKPZ equation is marginally relevant in the Renormalization Group sense: in fact, while the 2d-SHE is invariant under diffusive rescaling, for AKPZ the diffusion coefficient diverges (logarithmically) for large times, implying marginal super-diffusivity. [Based on joint work with G. Cannizzaro and D. Erhard] Video link: youtu.be/d6-mN_x8ZO8 January 5, 2021 Søren Fournais (Aarhus University) The ground state energy of dilute Bose gases The rigorous calculation of the ground state energy of dilute Bose gases has been a challenging problem since the 1950s. In particular, it is of interest to understand the extent to which the Bogoliubov pairing theory correctly describes the ground state of this physical system. In this talk I will report on a recent proof of the second term in the energy expansion for dilute gases, the so-called Lee-Huang-Yang term, and its relation to Bogoliubov theory. Video link: youtu.be/JCa_n9nUK2U December 15, 2020 Yoshiko Ogata (University of Tokyo) Classification of symmetry protected topological phases in quantum spin systems A Hamiltonian is in a SPT phase with a given symmetry if it cannot be smoothly deformed into a trivial Hamiltonian without a phase transition, if the deformation preserves the symmetry, while it can be smoothly deformed into a trivial Hamiltonian without a phase transition, if the symmetry is broken during the deformation. We consider this problem for one- and two-dimensional quantum spin systems with on-site finite group symmetries. Video link: youtu.be/cXk6Fk5wD_4 December 8, 2020 Katrin Wendland (Albert-Ludwigs-Universität Freiburg) On invariants shared by geometry and conformal field theory I will recall how some conformal field theories can be given geometric interpretations. This can be useful from a practical point of view, when geometric methods are transferred from geometry to conformal field theory. I will in particular focus on certain invariants that are shared by geometry and conformal field theory, including the complex elliptic genus. As we shall see, this invariant is also useful from a theoretical viewpoint, since it captures information about generic properties of certain theories. Video link: youtu.be/zJ-zF1KBYM0 December 1, 2020 Alessandro Pizzo (University of Rome Tor Vergata) Stability of gapped quantum chains under small perturbations We consider a family of quantum chains that has attracted much interest amongst people studying topological phases of matter. Their Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian consisting of on-site terms and with a strictly positive energy gap above its ground-state energy. We prove stability of the spectral gap, uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved. Furthermore the method can be extended to higher spatial dimensions and to complex Hamiltonians obtained by considering complex values of the coupling constant. (Joint work with S. Del Vecchio, J. Fröhlich, and S. Rossi.) Video link: youtu.be/S5lRLi3W6-E November 24, 2020 Roland Bauerschmidt (University of Cambridge) The Coleman correspondence at the free fermion point Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. I will then present a result that extends this correspondence to the non-Gaussian massless' sine-Gordon field on $\R^2$ at $\beta=4\pi$ and massive Dirac fermions. This is an instance of Coleman's prediction that the massless' sine-Gordon model and the massive Thirring model are equivalent. We use this correspondence to show that correlations of the massless' sine-Gordon model decay exponentially for $\beta=4\pi$. This is joint work with C. Webb (arXiv:2010.07096). Video link: youtu.be/uckzDglTbcM November 17, 2020 Stefan Hollands (University of Leipzig) (In)determinism Inside Black Holes In classical General Relativity, the values of fields on spacetime are uniquely determined by their initial values within its "domain of dependence". However, it may occur that the spacetime under consideration extends beyond this domain, and fields, therefore, are not entirely determined by their initial data. In fact, such a naive failure of determinism occurs inside all physically relevant black holes. The boundary of the region determined by the initial data is called the "Cauchy horizon". Penrose has proposed ("strong cosmic censorship conjecture") that the Cauchy horizon is actually unstable in the sense that the slightest perturbation caused by remnant fields convert it to a final singularity. Whether or not this is the case -- and thus whether there is a real problem with determinism -- has been put into question recently. In this colloquium I ask wheter quantum effects will come to the rescue of strong cosmic censorship. Video link: youtu.be/BmZxDseHnT0 November 10, 2020 Peter Hintz (MIT) Linear stability of slowly rotating Kerr spacetimes I will describe joint work with Dietrich Häfner and András Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. The proof uses a detailed description of the resolvent of an associated wave equation on symmetric 2-tensors near zero energy. Video link: youtu.be/iTBpM_DlGaI November 3, 2020 Nalini Anantharaman (University of Strasbourg) The bottom of the spectrum of a random hyperbolic surface I will report on work in progress with Laura Monk, where we study the bottom of the spectrum of the laplacian, on a compact hyperbolic surface chosen at random, in the limit of growing genus. We pick combinatorial ideas from the study of random regular graphs, to propose a strategy to prove that, with high probability, there are no eigenvalues below $1/4-\epsilon$. Video link: youtu.be/MThzUIIZx2Y October 27, 2020 Bruno Nachtergaele (UC Davis) Gapped ground state phases of lattice systems - Stability of the bulk gap I will give an overview of stability results for the spectral gap above the ground states of quantum lattice systems obtained in the past decade and explain the overall strategy introduced by Bravyi, Hastings, and Michalakis. A new result I will present proves the stability of the bulk gap for infinite systems with an approach that bypasses possible gapless boundary modes (joint work with Bob Sims and Amanda Young). Video link: youtu.be/rKzw0mr9xg4 October 20, 2020 Jeremy Quastel (University of Toronto) Towards KPZ Universality The 1-d KPZ universality class contains random interface growth models as well as random polymer free energies and driven diffusive systems. The KPZ fixed point has now been determined, through the exact solution of a special model in the class, TASEP, and is expected to describe the asymptotic fluctuations for all models in the class. It is an integrable Markov process, with transition probabilities described by a system of integrable PDE's. Very recently, new techniques have become available to prove the convergence of the KPZ equation itself, as well as some non-integrable extensions of TASEP, to the KPZ fixed point. This talk will be a gentle introduction to these developments with no prior knowledge assumed. The results are, variously, joint works with Daniel Remenik, Konstantin Matetski, and Sourav Sarkar. Video link: youtu.be/ecnoyc5Wf-E October 13, 2020 Svitlana Mayboroda (University of Minnesota) The landscape law for the integrated density of states Complexity of the geometry, randomness of the potential, and many other irregularities of the system can cause powerful, albeit quite different, manifestations of localization, a phenomenon of sudden confinement of waves, or eigenfunctions, to a small portion of the original domain. In the present talk we show that behind a possibly disordered system there exists a structure, referred to as a landscape function, which can predict the location and shape of the localized eigenfunctions, a pattern of their exponential decay, and deliver accurate bounds for the corresponding eigenvalues. In particular, we establish the "landscape law", the first non-asymptotic estimates from above and below on the integrated density of states of the Schroedinger operator using a counting function for the minima of the localization landscape. The results are deterministic, and rely on a new uncertainty principle. Narrowing down to the context of disordered potentials, we derive the best currently available bounds on the integrated density of states for the Anderson model. Video link: youtu.be/0f4vaNuLbK8 October 6, 2020 Clotilde Fermanian Kammerer (Université Paris Est - Créteil Val de Marne) Nonlinear quantum adiabatic approximation We will discuss generalization of the quantum adiabatic theorem to a nonlinear setting. We will consider evolution equations where the Hamiltonian operator depends on the time variable and on a finite number of parameters that are fixed on some coordinates of the unknown, making the equation non-linear. Under natural spectral hypotheses, there exist special functions that we call « Instantaneous nonlinear eigenvectors » such that, in the adiabatic limit, the solutions of the nonlinear equations with those initial data remain close to them, up to a rapidly oscillating phase. We will explain why this question arises, discuss the proof of this result and show how it can fail (works in collaboration with Alain Joye and Rémi Carles). Video link: youtu.be/kY-IrHnlw54 September 29, 2020 Alessandro Giuliani (University Roma Tre) Non-renormalization of the chiral anomaly' in interacting lattice Weyl semimetals Weyl semimetals are 3D condensed matter systems characterized by a degenerate Fermi surface, consisting of a pair of `Weyl nodes'. Correspondingly, in the infrared limit, these systems behave effectively as Weyl fermions in 3+1 dimensions. We consider a class of interacting 3D lattice models for Weyl semimetals and prove that the quadratic response of the quasi-particle flow between the Weyl nodes, which is the condensed matter analogue of the chiral anomaly in QED4, is universal, that is, independent of the interaction strength and form. Universality, which is the counterpart of the Adler-Bardeen non-renormalization property of the chiral anomaly for the infrared emergent description, is proved to hold at a non-perturbative level, notwithstanding the presence of a lattice (in contrast with the original Adler-Bardeen theorem, which is perturbative and requires relativistic invariance to hold). The proof relies on constructive bounds for the Euclidean ground state correlations combined with lattice Ward Identities, and it is valid arbitrarily close to the critical point where the Weyl points merge and the relativistic description breaks down. Joint work with V. Mastropietro and M. Porta. Video link: youtu.be/zUhM1k5nPcc September 22, 2020 Ian Jauslin (Princeton University) An effective equation to study Bose gasses at both low and high densities I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a one-particle problem, which incorporates the many-body correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations. Video link: youtu.be/HyRG-PzvpyY September 15, 2020 Victor Ivrii (University of Toronto) Scott and Thomas-Fermi approximations to electronic density In heavy atoms and molecules, on the distances $a \ll Z^{-1/2}$ from one of the nuclei (with a charge $Z_m$), we prove that the ground state electronic density $\rho_\Psi (x)$ is approximated in $\sL^p$-norm by the ground state electronic density for a single atom in the model with no interactions between electrons. Further, on the distances $a \gg Z^{-1}$ from all of the nuclei (with a charge $Z_1,\ldots, Z_m$) we prove that $\rho_\Psi (x)$ is approximated in $\sL^p$-norm, by the Thomas-Fermi density. We cover both non-relativistic and relativistic cases. Video link: youtu.be/O25BT_-XNNE September 8, 2020 Antti Kupiainen (University of Helsinki) Integrability of Liouville Conformal Field Theory A. Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to put a natural measure on the set of Riemannian metrics over a two dimensional manifold. Ever since, the work of Polyakov has echoed in various branches of physics and mathematics, ranging from string theory to probability theory and geometry. In the context of 2D quantum gravity models, Polyakov’s approach is conjecturally equivalent to the scaling limit of Random Planar Maps and through the Alday-Gaiotto- Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories. Through the work of Dorn,Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is believed to be to a certain extent integrable. I will review a probabilistic construction of LCFT developed together with David, Rhodes and Vargas and recent proofs concerning the integrability of LCFT: -The proof in a joint work with Rhodes and Vargas of the DOZZ formula (Annals of Mathematics, 81-166,191 (2020) -The proof in a joint work with Guillarmou, Rhodes and Vargas of the bootstrap conjecture for LCFT (arXiv:2005.11530). Video link: youtu.be/0ms4gEUT2Nw July 28, 2020 Nicolas Rougerie (University of Grenoble Alpes) Two modes approximation for bosons in a double well potential We study the mean-field limit for the ground state of bosonic particles in a double-well potential, jointly with the limit of large inter-well separation/large potential energy barrier. Two one-body wave-functions are then macroscopially occupied, one for each well. The physics in this two-modes subspace is usually described by a Bose-Hubbard Hamiltonian, yielding in particular the transition from an uncorrelated "superfluid" state (each particle lives in both potential wells) to a correlated "insulating" state (half of the particles live in each potential well). Through precise energy expansions we prove that the variance of the number of particles within each well is suppressed (violation of the central limit theorem), a signature of a correlated ground state. Quantum fluctuations around the two-modes description are particularly relevant, for they give energy contributions of the same order as the energy difference due to suppressed variances in the two-modes subspace. We describe them in terms of two independent Bogoliubov Hamiltonians, one for each potential well. Joint work with Alessandro Olgiati and Dominique Spehner Video link: youtu.be/ylb6BWewlpI July 21, 2020 Hugo Duminil-Copin (IHES / University of Geneva) Marginal triviality of the scaling limits of critical 4D Ising and φ_4^4 models In this talk, we will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the λφ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis. Video link: youtu.be/DtLKEQran_Y July 14, 2020 Hal Tasaki (Gakushuin University) 'Topological' index and general Lieb-Schultz-Mattis theorems for quantum spin chains A Lieb-Schultz-Mattis (LSM) type theorem states that a quantum many-body system with certain symmetry cannot have a unique ground state accompanied by a nonzero energy gap. While the original theorem treats models with continuous U(1) symmetry, new LSM-type statements that only assume discrete symmetry have been proposed recently in close connection with topological condensed matter physics. Here we shall prove such general LSM-type theorems by using the "topological" index intensively studied in the context of symmetry protected topological phase. Operator algebraic formulation of quantum spin chains plays an essential role in our approach. Here I do not assume any advanced knowledge in quantum spin systems or operator algebra, and illustrate the ideas of the proof (which I believe to be interesting). The talk is based on a joint work with Yoshiko Ogata and Yuji Tachikawa in arXiv:2004.06458. Video link: youtu.be/q0k1sch56Dk July 7, 2020 Bruno Després (Sorbonne University) Spectral-scattering theory and fusion plasmas Motivated by fusion plasmas and Tokamaks (ITER project), I will describe recent efforts on adapting the mathematical theory of linear unbounded self-adjoint operators (Kato, Lax, Reed-Simon, ....) to problems governed by kinetic equations coupled with Maxwell equations. Firstly it will be shown that Vlasov-Poisson-Ampere equations, linearized around non homogeneous Maxwellians, can be written in the framework of abstract scattering theory (linear Landau damping is a consequence). Secondly the absorption principle applied to the hybrid resonance will be discussed. All results come from long term discussions and collaborations with many colleagues (Campos-Pinto, Charles, Colas, Heuraux, Imbert-Gérard, Lafitte, Nicolopoulos, Rege, Weder, and many others). Video link: youtu.be/lmnm1D3NFp8 June 30, 2020 Laure Saint-Raymond (ENS Lyon) Fluctuation theory in the Boltzmann-Grad limit In this talk, I will discuss a long term project with T. Bodineau, I. Gallagher and S. Simonella on hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behavior are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics. Video link: youtu.be/fLDFA7ZCagA June 23, 2020 Nilanjana Datta (University of Cambridge) Discriminating between unitary quantum processes Discriminating between unknown objects in a given set is a fundamental task in experimental science. Suppose you are given a quantum system which is in one of two given states with equal probability. Determining the actual state of the system amounts to doing a measurement on it which would allow you to discriminate between the two possible states. It is known that unless the two states are mutually orthogonal, perfect discrimination is possible only if you are given arbitrarily many identical copies of the state. In this talk we consider the task of discriminating between quantum processes, instead of quantum states. In particular, we discriminate between a pair of unitary operators acting on a quantum system whose underlying Hilbert space is possibly infinite-dimensional. We prove that in contrast to state discrimination, one needs only a finite number of copies to discriminate perfectly between the two unitaries. Furthermore, no entanglement is needed in the discrimination task. The measure of discrimination is given in terms of the energy-constrained diamond norm and one of the key ingredients of the proof is a generalization of the Toeplitz-Hausdorff Theorem in convex analysis. Moreover, we employ our results to study a novel type of quantum speed limits which apply to pairs of quantum evolutions.This work was done jointly with Simon Becker (Cambridge), Ludovico Lami (Ulm) and Cambyse Rouze (Munich) Video link: youtu.be/gHEjszXSjMQ June 16, 2020 Nicola Pinamonti (University of Genova) Equilibrium states for interacting quantum field theories and their relative entropy During this talk we will review the construction of equilibrium states for interacting scalar quantum field theories, treated with perturbation theory, recently proposed by Fredenhagen and Lindner. We shall in particular see that this construction is a generalization of known results valid in the case of C*-dynamical systems. We shall furthermore discuss some properties of these states and we compare them with known results in the physical literature. In the last part of the talk, we shall show that notions like relative entropy or entropy production can be given for states which are of the form discussed in the first part of talk. We shall thus provide an extension to quantum field theory of similar concepts available in the case of C*-dynamical systems. Video link: youtu.be/excgcO7loj0 June 9, 2020 Andreas Winter (Universitat Autònoma de Barcelona) Energy-constrained diamond norms and the continuity of channel capacities and of open-system dynamics The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the continuity of one-parameter unitary groups and certain one-parameter semigroups of quantum channels. Video link: youtu.be/05ZQPFB0aAc June 2, 2020 Mihalis Dafermos (Cambridge University) The nonlinear stability of the Schwarzschild metric without symmetry I will discuss an upcoming result proving the full finite-codimension non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region. No symmetry is assumed. The work is based on our previous understanding of linear stability of Schwarzschild in double null gauge. Joint work with G. Holzegel, I. Rodnianski and M. Taylor. Video link: youtu.be/6Vh62H0rPiA May 26, 2020 Sven Bachmann (University of British Columbia) Adiabatic quantum transport In the presence of a spectral gap above the ground state energy, slowly driven condensed matter systems may exhibit quantized transport of charge. One of the earliest instances of this fact is the Laughlin argument explaining the integrality of the Hall conductance. In this talk, I will discuss transport by adiabatic processes in the presence of interactions between the charge carriers. I will explain the central role played by the locality of the quantum dynamics in two instances: the adiabatic theorem and an index theorem for quantized charge transport. I will also relate fractional transport to the anyonic nature of elementary excitations. Video link: youtu.be/ErgMuxMR_1A May 19, 2020 Pierre Clavier (University of Potsdam) Borel-Ecalle resummation for a Quantum Field Theory Borel-Ecalle resummation of resurgent functions is a vast generalisation of the well-known Borel-Laplace resummation method. It can be decomposed into three steps: Borel transform, averaging and Laplace transform. I will start by a pedagogical introduction of each of these steps. To illustrate the feasability of the Borel-Ecalle resummation method I then use it to resum the solution of a (truncated) Schwinger-Dyson equation of a Wess-Zumino model. This will be done using known results about this Wess-Zumino model as well as Sauzin's analytical bounds on convolution of resurgent functions. Video link: youtu.be/EzRoLEZhono May 12, 2020 Jan Philip Solovej (University of Copenhagen) Universality in the structure of Atoms and Molecules Abstract: The simplest approximate model of atoms and molecules is the celebrated Thomas-Fermi model. It is known to give a good approximation to the ground state energy of heavy atoms. The understanding of this approximation relies on a beautiful and very accurate application of semi-classical analysis. Although the energy approximation is good, it is, unfortunately, far from being accurate enough to predict quantities relevant to chemistry. Thomas-Fermi theory may nevertheless tell us something surprisingly accurate about the structure of atoms and molecules. I will discuss how a certain universality in the Thomas-Fermi model, indeed, holds approximately in much more complicated models, such as the Hartree-Fock model. I will also show numerical and experimental evidence that the approximate universality may hold even for real atoms and molecules. Video link: youtu.be/FCxkP7CqtQQ May 5, 2020 Martin Hairer (Imperial College London) The Brownian Castle Video link: youtu.be/Ve_EFZDbXTU