The workshop will take place on Thursday 20 October and Friday 21 October, roughly from 9am till 6pm. The detailed schedule is displayed below, and can be downloaded in pdf form here.

Thursday 20 October

9h - 9h20	Morning coffee
9h20 - 9h30	Welcome address
9h30 - 10h20	Nascimbene
10h20 - 11h10	Charles
11h10 - 11h50	Coffee break
11h50 - 12h40	Stéphan
12h40 - 14h20	Lunch break
14h20 - 15h10	Karabali
15h10 - 16h	Klevtsov
16h - 16h40	Coffee break
16h40 - 17h30	Bzdušek
18h - 22h	Conference dinner

Friday 21 October

9h - 9h30	Morning coffee
9h30 - 10h20	Goldman
10h20 - 11h10	Bernevig
11h10 - 11h50	Coffee break
11h50 - 12h40	Trebst
12h40 - 14h20	Lunch break
14h20 - 15h10	Cappelli
15h10 - 16h	Trifunović
16h - 17h	Afternoon tea

• List of talks

  Here is the list of titles and (clickable) abstracts of the talks presented at the workshop, along with their downloadable slides in pdf form.

  
  

  Andrei Bernevig
  Classification of topological flat bands

  Abstract

  TBA

  
  

  Tomáš Bzdušek
  From hyperbolic drum towards hyperbolic topological matter
  Slides

  Abstract

  Hyperbolic lattices correspond to regular tessellations of the sheet of constant negative curvature. In recent years, the study of classical and quantum dynamics on hyperbolic lattices has witnessed a fascinating progress. In particular, hyperbolic heptagon-kagome lattice has been experimentally realized using coupled waveguide resonators, and a theoretical characterization of eigenstates using a hyperbolic extension of Bloch theorem has been developed. Strikingly, owing to the non-commutativity of translations in a curved space, the momentum space associated with two-dimensional hyperbolic lattices is inevitably four- (or higher-)dimensional. In this talk, I will first discuss our simulation of a negatively curved "hyperbolic drum" with an electric-circuit network, and show how fingerprints of the emergent curvature enter both the static and the dynamical properties of the setup. Subsequently, I will clarify selected aspect of the representation theory of hyperbolic (Fuchsian) symmetry groups, and illustrate how the hyperbolic band theory can be applied to study concrete models of topological insulators and flat bands in a curved space.

  
  

  Andrea Cappelli
  Surface excitations of 3D Topological Insulators:
  conformal invariance, self-duality and bosonization
  Slides

  Abstract

  Massless fermions and anyons on the surface of (3+1)-dimensional topological insulators can be described at the semiclassical level by a non-local Abelian gauge theory involving two gauge fields. The theory is non-trivial owing to its solitonic excitations with electric and magnetic charges. Upon rewriting it into a local theory in one extra dimension, we compute the partition function and the solitonic spectrum, thus showing conformal invariance and electric-magnetic self-duality. This theory also provides a framework for semiclassical bosonization of (2+1)d fermions.

  
  

  Laurent Charles
  On Landau levels of magnetic Schrödinger operators
  on compact manifolds
  Slides

  Abstract

  In the limit of a large non-degenerate magnetic field, magnetic Schrödinger operator on compact manifolds have spectral clusters, which are reminiscent of the Landau levels. I will discuss some properties of these clusters: dimension in terms of characteristic classes, dynamics, state density.

  
  

  Nathan Goldman
  Chiral orbital order of ultracold bosons without higher bands
  Slides

  Abstract

  Interactions in higher Bloch bands provide a remarkable setting for realizing many-body states that spontaneously break time-reversal symmetry. Here, we show that the low-energy physics of interacting bosons in a plaquette pierced by pi-flux shares feature of p-band bosons, thus displaying condensates with chiral orbital order. We analyze the excitations of the condensate in terms of two orbital-like degrees of freedom and identify a gapped collective mode corresponding to the out-of-phase oscillations of the relative density and phase of the two orbitals. We further highlight the chiral nature of the ground state by revealing the cyclotron-like dynamics of the density upon quenching an impurity potential on a single site. This single-plaquette construction can be used as a building block for extended dimerized lattices with local orbital order, as we exemplify by using the Benalcazar-Bernevig-Hughes model. Our results provide a distinct direction to realize interacting orbital-like models with broken time-reversal symmetry, without resorting to higher bands nor to external drives, with direct implications for cold gases and nonlinear photonics.

  
  

  Dimitra Karabali
  Aspects of higher dimensional quantum Hall effect:
  Effective actions, entanglement entropy
  Slides

  Abstract

  Complex projective spaces allow for a simple and elegant solution to the Landau problem. I will present the formulation of QHE on such spaces, with both abelian (electromagnetic) and nonabelian background fields and derive the effective actions describing the edge dynamics as well as the responses to changes in the external gauge and gravitational fields (related to the Hall conductivity, Hall viscosity, etc). I will also discuss the calculation of the entanglement entropy. In the case of the LLL, a semiclassical analysis shows that the entanglement entropy is proportional to the phase-space area of the entangling surface with a universal overall constant, same for any dimension as well as abelian or nonabelian background magnetic fields.

  
  

  Semyon Klevtsov
  Geometric test for topological states of matter
  Slides

  Abstract

  We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterize the robustness, or topologicity, of the quantum state of matter and apply our test to the Laughlin states. Laughlin states form a vector bundle, the Laughlin bundle, over the Jacobian - the space of Aharonov-Bohm fluxes through the holes of the surface. The rank of the Laughlin bundle is the degeneracy of Laughlin states or, in presence of quasiholes, the dimension of the corresponding full many-body Hilbert space; its slope, which is the first Chern class divided by the rank, is the Hall conductance. We compute the rank and all the Chern classes of Laughlin bundles for any genus and any number of quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that Laughlin bundles with non-localized quasiholes are not projectively flat and that the Hall current is precisely quantized only for the states with localized quasiholes. Hence our test distinguishes these states from the full many-body Hilbert space. Based on joint work with Dimitri Zvonkine (CNRS, University of Paris-Versaille)

  
  

  Sylvain Nascimbene
  Realization of an atomic quantum Hall system in four dimensions
  Slides

  Abstract

  From 2D quantum Hall systems to 3D topological insulators and Weyl semi-metals, various types of topological states of matter have been realized in condensed matter systems. Thanks to the concept of synthetic dimensions, engineered systems could give access to a larger range of topological systems, predicted to occur in dimensions D > 3.
  In this talk, I will present the experimental realization of a 4D quantum Hall system using ultracold dysprosium atoms. We use the giant electronic spin of this atom to encode two synthetic dimensions, which we couple to two spatial ones using coherent light. This spin-orbit coupling gives rise to an effective magnetic field acting on the 4D atom dynamics. We reveal the topological nature of this system by measuring a quantized non-linear electromagnetic response, and study the nature of 3D hyperedge modes. We also investigate elementary excitations, corresponding to non-planar cyclotron orbits of the center of mass, beyond the circular orbits encountered in lower dimensions.

  
  

  Jean-Marie Stéphan
  Edge modes and counting statistics
  in the 2D and 4D Quantum Hall effects
  Slides

  Abstract

  I discuss simple Slater determinant wave functions generalizing the two-dimensional Integer Quantum Hall effect to four dimensions (4D). The corresponding gapless modes at the edge of the droplet are known to be anisotropic: they only propagate in one direction, foliating the 3D boundary into independent 1d conduction channels. In this talk, I focus on 4D droplets confined by harmonic traps, and show that the nature of the 1d conduction channels depends strongly on the commensurability of the trapping frequencies. In particular incommensurable frequencies produce quasi-periodic, ergodic trajectories; the corresponding correlation functions of edge modes also exhibit fractal-like features. If time permits, I will also explore the consequences on counting statistics for particles in a region of space in 4D but also back in 2D.

  
  

  Simon Trebst
  Topological fracton quantum phase transitions
  Slides

  Abstract

  Gapped fracton phases of matter generalize the concept of topological order and broaden our fundamental understanding of entanglement in quantum many-body systems. However, their analytical or numerical description beyond exactly solvable models remains a formidable challenge. In this talk we discuss recent work where we employ an exact 3D quantum tensor-network approach that allows us to study a Z_N generalization of the prototypical X-cube fracton model and its quantum phase transitions between distinct topological states via fully tractable wavefunction deformations. We map the (deformed) quantum states exactly to a combination of a classical lattice gauge theory and a plaquette clock model, and employ numerical techniques to calculate various entanglement order parameters. For the Z_N model we find a family of (weakly) first-order fracton confinement transitions that in the limit of N→∞ converge to a continuous phase transition beyond the Landau-Ginzburg-Wilson paradigm. We also discover a line of 3D conformal quantum critical points (with critical magnetic flux loop fluctuations) which, in the N→∞ limit, appears to coexist with a gapless deconfined fracton state.
  [Joint work with Guo-Yi Zhu, Ji-Yao Chen, and Peng Ye.]

  
  

  Luka Trifunović
  Topologically localized insulators
  Slides

  Abstract

  Topological insulators are known for obstructing Anderson localization. In this talk, I will introduce a three-dimensional insulating topological phase, dubbed "Topologically Localized Insulator" (TLI), whose existence is enabled by the phenomenon of Anderson localization. Interestingly, although the bulk of TLI is fully localized, it has isotropic and quantized magnetoelectric polarizability tensor while its boundary has quantized Hall conductance.