Program:

Wednesday, August 25

• 12:00am-1:30pm. Welcome Lunch at Cermics.
  
• 1:30pm-2:10pm. Mathieu Lewin,
  New results on the Lieb-Thirring inequality.
  
  The Lieb-Thirring inequality is a generalization of the Gagliardo-Nirenberg inequality, which plays a central role in the analysis of large quantum systems. In this talk I will first introduce the inequality and then focus the discussion on its best constant. After reviewing what is believed, what is known and what is open, I will present new results on the value of the best constant as well as numerical simulations for periodic systems in 1D and 2D.
  Collaboration with Rupert L. Frank (Munich, Germany & Caltech, USA) and David Gontier (Paris-Dauphine, France).
• 2:10pm-2:50pm. Gian-Michele Graf,
  Topology in shallow-water waves: A violation of bulk-edge correspondence.
  
  A two-dimensional rotating shallow-water model describes a layer of water, in guise of oceans covering the Earth. Its mathematical description parallels that of a band insulator, except for the energy range of a band being unbounded. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a varied phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson’s theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, which is ultimately the reason for the violation.
  (Joint work with H. Jud and C. Tauber.)
• 2:50pm-3:20pm. Break.
  
• 3:20pm-4:00pm. Giovanna Marcelli,
  A new approach to transport coefficients in the quantum spin Hall effect.
  
  We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator does not commute with the spin operator in view of Rashba interactions. A gapped periodic one-particle Hamiltonian is perturbed by adding a constant electric field of small intensity and the linear response, with respect to the strength of the electric field, in terms of a spin current is computed. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We study the independence of the spin conductivity from the choice of the fundamental cell, and we isolate a subclass of discrete periodic models (including the Kane-Mele model) where the conventional and the proper spin conductivity agree. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almost-stationary state, which well approximates the physical state of the system.
  This seminar is based on joint work with G. Panati and S. Teufel.
• 4:00pm-4:40pm. Radu Purice,
  Spectral analysis near a conic point of a 2 dimensional periodic Hamiltonian in a weak magnetic field.
  
  We consider a special type of intersection (conic point) of two Bloch levels of a periodic Schroedinger operator in 2 dimensions and study the behaviour of the associated spectral region in a magnetic field that may have some weak smooth variation around a non-zero constant value.
  This work has been done in collaboration with Horia Cornean and Bernard Helffer.
• 4:40pm-5:10pm. Coffee break.
  
• 5:10pm-5:50pm. Kevin Stubbs,
  The localization-topology correspondence via projected position operators.
  
  Topological materials have generated great interest over the past few years due their remarkable physical properties. Mathematically speaking, we can understand the properties of periodic topological insulators through the Bloch-Floquet transform which maps electronic states to a fiber bundle over the Brillouin torus. This type of analysis has led to the ``localization-topology correspondence'' which states that (1) the Fermi projector for an insulator admits an orthonormal basis with finite second moment if and only if it is topologically trivial and (2) an insulator is topologically trivial if and only if its Fermi projector admits an exponentially localized orthonormal basis. In contrast, in non-periodic systems it is unknown whether a localization-topology correspondence still holds since the Bloch-Floquet transform cannot be applied.
  In this talk, I will discuss some recent work which establishes a weaker version of the localization-topology correspondence for both periodic and non-periodic insulators in two dimensions. In particular, we show by construction that if the Fermi projector for an insulator admits an orthonormal basis with sufficiently fast algebraic decay, then it also admits an orthonormal basis with exponentially fast decay. The proof of this result is based on an algorithm for constructing exponentially localized bases using projected position operators (i.e. operators of the form PXP and PYP).
  Joint work with Jianfeng Lu and Alexander Watson.
• 5:50pm-6:30pm. Jacob Shapiro,
  Does the tight-binding limit commute with calculating the topological index?
  
  We present two case studies, of the integer quantum Hall effect and the Su-Schrieffer–Heeger model, where the answers to the question in the title are yes and no respectively, and discuss some consequences and related open problems.
  Based on joint work with M. I. Weinstein.

Thursday, August 26

• 12:00am-1:30pm. Lunch at Cermics.
  
• 1:30pm-2:10pm. Massimo Moscolari,
  A new paradigm for bulk-edge correspondence.
  
  We consider 2d random ergodic Schroedinger operators with long range magnetic fields on domains with and without boundary. By extending the gauge covariant magnetic perturbation theory to infinite domains with boundary, we prove that the celebrated bulk-edge correspondence is just a particular case of a much more general paradigm, which also includes the theory of diamagnetic currents and of Landau diamagnetism. More precisely, we obtain a formula which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field is equal to the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge.
  Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-boundary correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a ``zero temperature" limit.
  The talk is based on a joint work with H. Cornean and S. Teufel.
• 2:10pm-2:50pm. Domenico Monaco,
  Topology vs localization in synthetic dimensions.
  
  The recent development of quantum simulators using optical lattices and ultracold atoms have made it possible to model higher-dimensional quantum systems in the lab, by exploring so-called synthetic dimensions. This is of particular relevance to the investigation of new exotic phases of topological matter: for example, a four-dimensional (4D) analogue of the quantum Hall effect (QHE) has been effectively probed. The topology of 4D quantum systems features an integer, called the second Chern number, which is responsible for this effect in a similar way to how the "usual" first Chern number controls the two-dimensional QHE. In this talk, I will show how the second Chern number arises as an obstruction to the existence of orthonormal bases of localized Wannier functions in 4D crystals, and how relaxing the orthonormality constraint requiring only a Parseval frame removes this topological obstruction.
  The talk is based on ongoing joint work with Thaddeus Roussigné (ENS Paris Saclay).
• 2:50pm-3:20pm. Break.
  
• 3:20pm-4:00pm. Johannes Kellendonk,
  Diffraction and topological invariants for aperiodic systems.
  
  In this talk we draw a connection between the Bragg peaks in the diffraction spectrum of a solid and its electronic spectrum. Such a connection is known for crystals for which perturbation theory predicts that gaps open up at wave vectors which correspond to Bragg peaks. We look at this connection from a different point of view which highlights its topological nature and applies to aperiodic systems as well: based on the well known formulation of Bragg peak positions as eigenvalues of the dynamical system associated to the solid, we establish a morphism between the exterior module of the topological eigenvalues and the K-theory of the solid. The top degree part of this morphism links the Bragg peak positions to the gap labelling by means of the integrated density of states.
• 4:00pm-4:40pm. Alexis Drouot,
  Dirac operators and topological insulators.
  
  This talk will focus on Dirac operators that emerge when studying macroscopic transport between topological insulators. I will analytically construct canonical edge states: coherent states that propagate along interfaces, but do not admit natural counter-propagating companions. I will illustrate the results with various numerical simulations. 
  Joint work with Bal, Becker, Fermanian Kammerer, Lu and Watson.
• 4:40pm-5:10pm. Coffee break.
  
• 5:10pm-5:50pm. Emil Prodan,
  A Groupoid Approach to Interacting Fermions.
  
  Consider a fermion gas populating a Delone point pattern. For a single fermion, the algebra of local observables consists of the compact operators over the Hilbert space of square summable sequences over the pattern. In a string of outstanding papers, Bellissard and Kellendonk shown that the generators of the dynamics of these local observables fall into a groupoid C*-algebra canonically associated to the pattern. In this talk, I will present an extension of these results, where the algebra of compact operators is replaced by GICAR algebra. In the process, I will introduce natural extensions of the groupoids associated to the pattern on arbitrary N-fermion sectors. The formalism will be exemplified on new topological phases of matter.
  This is based on work in collaboration with Bram Mesland.
• 5:50pm-6:30pm. Hermann Schulz-Baldes,
  Invariants of disordered semimetals via the spectral localizer.
  
  The spectral localizer consists of placing the Hamiltonian in a Dirac trap. For topological insulators its spectral asymmetry is equal to the topological invariants, providing a highly efficient tool for numerical computation. Here this technique is extended to disordered semimetals and allows to access the number of Dirac or Weyl points as well as weak invariants. These latter invariants imply the existence of surface states.
  Joint work with Tom Stoiber

Friday, August 27

• 12:00am-1:30pm. Lunch at Cermics.
  
• 1:30pm-2:10pm. Clotilde Fermanian,
  Some results about the dynamics of an electron in a crystal.
  
  We will discuss the asymptotic analysis of a Schrödinger equation modeling the dynamics of an electron in a crystal for small wave-length comparable to the characteristic scale of the crystal. We will be interested in the description of the limit of time averaged energy densities for rather general  initial data. In particular, we shall authorize in their decomposition  on Bloch bands  leading order contributions on modes that cross.
  Joint work with Victor Chabu and Fabricio Macia
• 2:10pm-2:50pm. Marcello Porta
  Edge States Scattering and Universality in Quantum Hall Systems.
  
  We consider the edge transport properties of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. With respect to previous work, our result allows to consider a generic class of quantum Hall systems, displaying arbitrarily many edge modes. Our proof quantifies the validity and the limitations of the Luttinger liquid effective description for the edge currents. In particular, due to edge states backscattering, the effective description alone is not able to predict the universality of the edge conductance. The exact quantization follows after fully taking into account the bulk degrees of freedom, whose precise contribution to the edge transport is determined thanks to lattice conservation laws.
  Joint work with Vieri Mastropietro.
• 2:50pm-3:20pm. Break.
  
• 3:20pm-4:00pm. Antoine Levitt,
  Numerical computation of response functions and resonances in molecules and solids.
  
  The computation of dynamical response properties of molecules and solids usually proceeds via a space truncation, which turns continuous spectrum into discrete and hampers the effective approximation of response functions. In the first part, about work done with M-S. Dupuy, I will present error estimates aimed at a better understanding of this phenomenon. In the second part, about work in progress with I. Duchemin, L. Genovese, E. Letournel and S. Ruget, I will present a new method based on a complex deformation of the Brillouin zone that allows for an accurate computation of response functions as well as resonances in solids (to our knowledge, the only method able to do so).
• 4:40pm-5:10pm. Coffee break.