Program

Title:  Bulk-boundary correspondence in quantum Hall systems and beyond
Abstract:  This talk reviews the bulk-boundary correspondence for quantum Hall systems and shows how the techniques can be extended to other dimensions and chiral systems. One of the outcomes is a proof of the existence of delocalized boundary states in non-trivial topological insulators. This is based on joint work with Emil Prodan.

Title:   Emergent dynamical metric of the FQHE
Abstract:  

Title:  Geometric responses of Quantum Hall systems
Abstract:   The fractional quantum Hall effect (FQHE) is a fascinating physical phenomenon of quantization of the Hall conductance of two-dimensional electron gas in terms of fundamental constants. The phenomenon is observed at low temperatures and in strong magnetic fields. It is characterised by a formation of a collective state of electrons - an incompressible electron liquid. The responses of this rather exotic fluid to external electromagnetic and elastic perturbations are rather geometric in nature. These responses can be compactly described by the effective action of FQHE. In this talk I will consider the geometric part of the effective action for FQHE. It consists of three terms: Chern-Simons, Wen-Zee and gravitational Chern-Simons terms. I will show how to derive the geometric part of the effective action and will discuss the physical meaning of the corresponding linear responses of FQHE. In particular, I will consider the role of framing anomaly in the theory of linear responses for FQHE.

Title:  Geometric response at the edge
Abstract:   It is well established that non-vanishing Hall conductance in the bulk implies existence of chiral edge modes. This relation can be understood without any reference to microscopic physics in terms of Callan-Harvey anomaly inflow. In this talk I will explain the relation between the geometric response coefficients and edge modes. This relation will imply some nontrivial facts about "bulk CFT" which I will comment on. If time permits I will briefly explain recent experimental observation of Haldane-Wen-Zee shift in photonic Landau levels on a cone.

Title:  Thermofield dynamics and gravity
Abstract:   I will start with a presentation of thermofield dynamics in terms of a path-integral using coherent states, equivalently, using a co-adjoint orbit action. A field theoretic formulation in terms of the quantum Hall effect on a manifold $\M \times {\tilde\M}$ where the two components have opposite orientation will also be discussed. Then I shall consider formulating gravitational dynamics for noncommutative geometry using thermofield dynamics, doubling the Hilbert space modelling the noncommutative space. For 2+1 dimensions, which will be the case study, the commutative limit leads to the Einstein-Hilbert action as the difference of two Chern-Simons actions, due to the opposite orientation of $\M$ and ${\tilde \M}$.

Title:  Single particle Green's functions, topological invariants and quantum Hall effect
Abstract:   It has been appreciated for quite some time that topological invariants can be constructed out of single particle Green's functions of gapful electronic systems, regardless of whether they are interacting or not. These invariants can be used to probe the structure of boundary single-particle excitations of such states, if any. I will describe the procedure which allows us to do that, as well as attempts to use these invariants to study the edge and the bulk of two dimensional fractional quantum Hall states.

Title:   Goldstone mode stochastization/relaxation in a quantum Hall ferromagnet in terms of the excitonic representation technique
Abstract:   Theoretical and experimental studies of the coherent spin dynamics of two-dimensional GaAs/AlGaAs electron gas were performed. The system in the quantum Hall ferromagnet state exhibits a spin relaxation mechanism that is determined by many-particle Coulomb interactions. In addition to the spin exciton with changes in the spin quantum numbers of $\delta S=\delta S_z=-1$ the quantum Hall ferromagnet supports a Goldstone spin exciton that changes the spin quantum numbers to $\delta S=0$ and $\delta S_z=-1$, which corresponds to a coherent spin rotation of the entire electron system to a certain angle. The Goldstone spin exciton decays through a specific relaxation mechanism that is unlike any other collective spin state.

Title:  A classification of 2+1D bosonic/fermionic topological orders
Abstract:   2+1d topological orders can be fully characterized reps of modular group. Using such an point of view, we can obtain a classification of topological order.

Title:  Probing quantum Hall states with flux tubes and cones
Abstract:   In this talk, I will review our approach (with M. Laskin and P. Wiegmann) to studying fractional quantum Hall wave functions on surfaces with non-constant Gaussian curvature and magnetic field. As a demonstration of this formalism in action, I will demonstrate how FQH states can be characterized by adiabatic transport of flux tubes and conical singularities (which are delta functional configurations of magnetic field and curvature, respectively). I will also show that such defects can be described by operators in a large N quantum field theoretical description of FQH states.

Title:  Geometric deformation and Berry curvature in quantum Hall states
Abstract:   In this talk, I will present a method for computing the topological central charge and orbital spin variance of a topological phase directly from bulk wave functions, as a Berry curvature produced by adiabatic variation of the spatial metric. I will show explicit results of this computation-as well as a related derivation of the Hall conductivity and Hall viscosity - for trial wave functions that can be represented as conformal blocks in a chiral conformal field theory (CFT). This calculation makes use of the gravitational anomaly in the CFT, as well as the hypothesis of generalized screening. Along the way, I will also show how the Hall conductivity can be obtained in an analogous way from the U(1) gauge anomaly.

Title:  Multipole expansion in the quantum Hall effect
Abstract:   The effective action for low-energy excitations of Laughlin's state is obtained by systematic expansion in inverse powers of the magnetic field. It is based on the W-infinity symmetry of quantum incompressible fluids and the associated fields of increasing spin. Besides reproducing the Wen and Wen-Zee actions and the Hall viscosity, this approach indicates that the low-energy excitations are extended objects with dipolar and multipolar moments.

Title:   Geometry and Large N limits in Laughlin states
Abstract:   I will talk about the recent work on the geometry of Laughlin states on Riemann surfaces. We study the geometric properties and large N limits (for large number of particles) of the generating functional, density of states, adiabatic transport for the Laughlin state on Riemann surfaces. The aim of the talk is to connect some physics and math topics of the workshop: Quantum Hall states, Bergman kernels, Kähler geometry, Quillen metric, free fields. Based on: 1309.7333, 1410.6802, 1504.07198.

Title:  Quantum Hall effect and Quillen metric
Abstract:   We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. Based on: 1510.06720.

Title:  Hierarchy Quantum Hall wave function - one idea and three applications
Abstract:  I first explain how states with quasielectrons can be described within the conformal field theory approach to quantum Hall wave functions. Based on this I show: 1. That the Jain composite fermion states fit in the Haldane-Halperin hierarchy. 2. How to construct abelian hierarchy states on the torus. 3. How to obtain a matrix product state representation of quasielectron wave functions.

Title:  Coulomb gas ensembles, fluctuations, and Laplacian growth
Abstract:   We outline the main features of Coulomb gas ensembles, which model 2D fermions confined by a potential. We study condensation phenomena when the number of fermions increases and the confining potential is jacked up proportionally. The fermions tend to condensate to a droplet, which can be analyzed in terms of potential theory. In the bulk of the droplet, the interaction of the fermions is at the scale of the typical interaction distance of the particles governed by the Ginibre ($\infty$) law. More crudely, we can analyze the stochastic process in terms of linear statistics, without scaling to typical distances. The result is, for inverse temperature $\beta=2$ only, that we can obtain subleading expansion of the ensemble corresponding to the fluctuation field (Ameur, Hedenmalm, Makarov). The process of adding further fermions to the ensemble corresponds to Laplacian growth for the droplet in the semiclassical limit.

Title:  Laplacian growth on a branched Riemann surface
Abstract:   Laplacian growth refers to domain evolution driven by harmonic gradients, for example the gradient of the Green's function with a fixed pole. It makes sense on Riemannian manifolds of arbitrary dimension. In the talk I will discuss a particular case when the manifold is a branched covering surface of the complex plane. There turn out to be unexpected couplings to topics in complex analysis, such as contractive zero divisors in Bergman space. The talk is based on joint work with Yu-Lin Lin.

Title:  Large N expansion for logarithmic gas on a closed contour
Abstract:   We introduce a model of the logarithmic gas (the beta-ensemble) on an arbitrary closed contour in the plane. The 1/N expansion is based on the loop equation which has the form of the boundary condition for an analog of the stress-energy tensor in conformal field theory.

Title:  Geometric disorder and critical behavior at the integer quantum Hall plateau transition
Abstract:   Recent high-precision results for the critical exponent of the localization length at the integer quantum Hall (IQH) transition differ considerably between experimental (\nu = 2.38) and numerical (\nu = 2.6) values. The numerical value is obtained in the Chalker-Coddington (CC) network model. We revisit the arguments leading tot he CC model and consider a more general network where geometric or structural disorder is incorporated. Numerical simulations of this new model lead to the value \nu=2.37 in very close agreement with experiments. We further argue that in a continuous limit the geometrically disordered model maps to the free Dirac fermion coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the geometrically disordered model and may shed more light on the analytical theory of the IQH transition.

Title:  Partial Density Function for Divisors
Abstract:   I will discuss joint work with Michael Singer concerning the behaviour of the partial density function for sections of high powers of hermitian line bundles that are required to vanish to a particular order along a divisor. Under the assumption that all data is invariant under a holomorphic S^1 action, we prove the existence of a full smooth asymptotic expansion that is to be understood as holding on a suitable real blowup. As such one recovers the classical "forbidden region" and can see that the partial density function has a standard "error function" behaviour across its boundary.

Title:  Ward's equation and the distribution of random eigenvalues
Abstract:   Ward's equation is a non-linear identity which can be obtained by rescaling in Ward's identity on the mesoscopic scale. Together with Makarov and Kang, I have studied this equation in the case when beta=1, to study universality of scaling limits.

Title:  Geometric theory of quantum response in open systems
Abstract:   I shall describe joint work with Martin Fraas and G.M. Graaf: 1202.5750.

Title:   Matrix product state representation of quantum Hall quasi-particles
Abstract:  The description of quantum Hall quasi-particles is, due to the Pauli-principle, more complicated than the description of quasi-holes. In this talk, I will discuss the Matrix Product state description of the CFT formulation of quasi-particles, as developed by the Stockholm group.

Title:  Effective field theory of the quantum Hall transition
Abstract:   Understanding the critical behavior of the Anderson transition occurring between integer quantum Hall plateaus is a long-standing problem. In this talk I will discuss a novel approach to the microscopic models and derive an effective conformal field theory description of the multi-fractal geometry of critical wave functions. The methods used are rather general and apply to other two-dimensional Anderson transitions as well. The talk is based on collaborations with M. Zirnbauer and D. Wieczorek.

Title:  Conformal Loop Ensemble for the Integer Quantum Hall Transition
Abstract: A probabilistic geometrical description of the integer quantum Hall transition is proposed. The proposed description focuses on the path of an electron as it moves through the sample. The current work complements previous results that dealt only the exterior perimeter of the path using conformal restriction, and will allow for a description of the whole path including the internal structure. The talk will detail work in progress.

Title:   Plateau transition in Hall effect: Matrix model for this class of problems
Abstract:   We consider random networks of scatterings with arbitrary multiparticle S-matrices and formulate action for them on appropriate lattice. The randomness of network is transforming into the randomness of the geometry of lattices, which can be described by inclusion of quenched (or not) 2D gravity into the problem. Some interesting problems, such as sign factor of 3D Ising model, plateau transitions in Hall effect belongs to this class of problems. We formulate Random Matrix model which describes this random geometries.

Title:  A no-go theorem for tensor network states and Chern bands
Abstract:   Tensor Network States (TNS) are variational wave functions for many-body systems that are constructed by tracing out additional, auxiliary degrees of freedom, coupled locally to the physical degrees of freedom. In one dimension, TNS are nothing but Matrix Product States (MPS), and they are known to provide enough variational freedom to accurately represent all gapped phases of matter. In higher dimensions, however, some gapped phases have appeared to be difficult to represent as TNS. I will discuss a no-go theorem that says that it is not possible to represent a topologically non-trivial band by a free (or gaussian) TNS in any dimension. The talk is based on joint work with N. Read (1307.7726, Phys. Rev. B 92, 205307, 2015).

Title:   Pairing between zeros and critical points of random polynomials
Abstract:   Consider a polynomial p_N(z) in one complex variable. The Gauss-Lucas Theorem says that the critical points of p_N lie inside the convex hull of its zeros. But are critical points actually distributed inside the convex hull if p_N is chosen at random? The purpose of this talk is to explain that in fact each critical point of p_N typically comes paired with a single zero. The distance between a critical point and its paired zero is on the order of N^{-1}, which is much smaller than the typical N^{-1/2} spacing between order of N iid points on the sphere. In the first part of my talk, I will give a heuristic interpretation for this pairing by relating zeros and critical points to electrostatics on the Riemann sphere. In the second part, I will explain how to use Bergman kernel asymptotics to prove rigorous theorems about this pairing.

Title:  AdS/CFT and geometric aspects of Chern-Simons theories
Abstract:   Holographic nature of Quantum Hall Effect and Chern-Simons theories was understood much before the discovery of the AdS/CFT correspondence in string theory. From this perspective they continue to be interesting probes in the study of holographic correspondence. In my talk I will attempt to review both conventional views of holography on the topological matter as well as more recent developments pointing to new possible interpretations of QHE.

Title:   Pieri rules, vertex operators and Baxter Q-matrix
Abstract:   I will explain what Pieri rules are and why they are related to integrability and Vertex operator algebras. I will attempt to relate this work to the subject of this conference. Based on: 1510.08709.