Seminar on mathematical aspects of the quantum Hall effect
and related topics

Mathematical Institute, Institute for Theoretical Physics, Cologne, 2016-2017

Organizers: M. Hermanns (ITP), S. Klevtsov (ITP and MI) and G. Marinescu (MI)


Local Information, Directions



Summer Semester 2017

11.07.2017      Samuel Monnier (Geneva, Switzerland)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: TBA

Abstract: TBA



27.06.2017      Julien Keller (Marseille, France)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: TBA

Abstract: TBA



20.06.2017      Jerome Dubail (Nancy, France)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: TBA

Abstract: TBA



13.06.2017      Benoit Estienne (LPTHE, Paris, France)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: TBA

Abstract: TBA



30.05.2017      Vatsal Dwivedi (University of Cologne, Germany)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: TBA

Abstract: TBA



23.05.2017      Martin Zirnbauer (University of Cologne, Germany)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: The integer quantum Hall plateau transition: a new type of conformal field theory

Abstract: The scaling behavior near the transition between plateaus of the integer quantum Hall effect has traditionally been interpreted on the basis of a two-parameter renormalization group flow conjectured from the non-linear sigma model of Pruisken. Yet, this scaling picture never led to any analytical understanding of the critical point. Here we propose a novel and quantitative description of the critical point by a conformal field theory of unusual type: Pruisken's non-linear sigma model coupled to a maximally gauged Wess-Zumino-Witten model of level 4. This proposal explains the existing numerical data for the multifractal scaling exponents of critical wave functions.



25.04.2017      David Gross (University of Cologne, Germany)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: [pdf] Entanglement Polytopes

Abstract: From a physical point of view, I will describe a method for classifying the global information about a pure state that is accessible from single-particle observations alone. Indeed, it turns out that the eigenvalues of single-particle states compatible with certain global entanglement classes form configurations of convex polytopes. Mathematically, these sets can be identified as non-abelian moment polytopes associated with the action of local symmetry groups on the global state vector. For low-dimensional situations, we can compute the polytopes explicitly and apply them to physical problems. Recently, there have been several experiments that demonstrate that our criteria can be tested in practical situations.



18.04.2017      Alexander Gorsky (IITP, Moscow, Russia)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: [pdf] New critical phenomena in the constrained random networks

Abstract: We review the new critical phenomena found in large N constrained Erdos-Renyi networks(CERN) and regular random graphs (RRG). The unicolor network undergoes the complete defragmentation into the fixed number of almost complete graphs at critical value of the chemical potential for the closed triangles. The multicolor networks develop the finite plateau in the spectral gap (lowest non-vanishing eigenvalue of graph Laplacian) in the interval of the chemical potentials for the open trimers. The critical phenomena are described in terms of the spectral properties of the networks and comparison with the conventional matrix models is performed.




Winter Semester 2016-2017

31.01.2017      Eddy Ardonne (Stockholm University, Sweden)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: Properties of 'metaplectic' anyons

Abstract: Metaplectic anyons have been considered in the context of topological quantum computation. The quantum dimensions of these anyons are either integers, or square roots of integers, as is the case for Ising anyons. In this talk, I will discuss the structure of 'anyon models' in general, and the properties of metaplectic anyons in particular. Namely, I will comment on their classification, and on an interesting connection of their braid properties with number theory.



24.01.2017      Bogdan Georgiev (Max Planck Institute for Mathematics, Bonn, Germany)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: [pdf] Some recent estimates on nodal sets and nodal domains in the high-energy limit

Abstract: Given a closed manifold $M$, the Laplace operator $\Delta$ is known to possess a discrete spectrum of eigenvalues $\lambda$ converging to infinity. We are interested in properties of the corresponding eigenfunctions $\phi_\lambda$ as $\lambda$ becomes large (i.e. the high-energy limit). From a physical point of view, $\{ \phi_\lambda \}$ represent stationary states of a free quantum particle on $M$ - when $\phi_\lambda$ is $L^2$-normalized, it may be interpreted as the probability density of a particle in $M$, having energy $\lambda$. Various questions about the geometry of $\phi_\lambda$ have been studied thoroughly: for example, what is the size of the vanishing (nodal) set of $\phi_\lambda$; how is the nodal set distributed; how big can the $L^p$ norms of $\phi_\lambda$ be and how are they localized; how many nodal domains are there; how large are the nodal domains, etc.
We focus on some classical, as well as recent results along these lines - this includes joint work with M. Mukherjee and S. Steinerberger.




17.01.2017      Maria Hermanns (University of Cologne, Germany)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: [pdf] Nonabelian hierarchies

Abstract: The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past thirty years, has generated many groundbreaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. The Haldane-Halperin hierarchy provides a simple, intuitive picture of fractional quantum Hall liquids in the lowest Landau level and predicts both their properties and relative stability. For the zoo of states in the second Landau level, the situation is far less clear. Many states are in fact believed to be nonabelian, and therefore cannot be described within the usual hierarchical framework. In this talk, I am going to review several attempts to generalize the Haldane-Halperin hierarchy to nonabelian states, and discuss their respective predictions and properties.



20.12.2016      Joaquim Ortega-Cerdà (University of Barcelona, Spain)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: [pdf] Energy of determinantal point processes in the torus and the sphere

Abstract: We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere and the torus. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere and also in the periodic setting.
This is joint work with Carlos Beltran and Jordi Marzo.




08.12.2016      Mikhail Katsnelson (University of Nijmegen, Netherlands)
16:00

Room: (Joint with QM2 Colloquium) Conference room 0.03, Theoretical Physics (new building)

Title: [pdf] Gauge fields and strain engineering in graphene

Abstract: It is demonstrated now both experimentally and theoretically that graphene is usually not flat but covered by ripples resulting from both intrinsic flexural instability of two-dimensional membranes and roughness of substrate. Thus charge carriers are not just Dirac fermions but Dirac fermions moving in a curved space. The effect of the corrugations on the electron spectrum can be described in terms of gauge (pseudo-magnetic) fields [1] which result, in particular, in formation of pseudo-Landau levels predicted theoretically [2] and found experimentally [3]. These gauge fields can be used for "strain engineering", including tunable gap opening [4], quantum pumping [5] and creation of valley-polarized current [6]. Ripples can induce puddles, that is, charge inhomogeneities [7,8] The scattering by the ripples is also one of the limiting factors restricting the charge carrier mobility in graphene [9].

[1] M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea, Phys. Rep. 496, 109 (2010).
[2] F. Guinea, M. I. Katsnelson, and A. K. Geim, Nature Phys. 6, 109 (2010).
[3] N. Levy et al, Science 329, 544 (2010).
[4] T. Low, F. Guinea, and M. I. Katsnelson, Phys. Rev. B 83, 195435 (2011).
[5] T. Low, Y. Jiang, M. I. Katsnelson, and F. Guinea, Nano Lett. 12, 850 (2012).
[6] Y. Jiang, T. Low, K. Chang, M. I. Katsnelson, and F. Guinea, Phys. Rev. Lett. 110, 046601 (2013).
[7] M. Gibertini, A. Tomadin, M. Polini, A. Fasolino, and M. I. Katsnelson, Phys. Rev. B 81, 125437 (2010).
[8] M. Gibertini, A. Tomadin, F. Guinea, M. I. Katsnelson, and M. Polini, Phys. Rev. B 85, 201405 (R) (2012).
[9] M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A 366, 195 (2008).




29.11.2016      Stéphane Nonnenmacher (Université Paris-Sud, France)
10:15

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: [pdf] Random perturbations of non-selfadjoint differential operators, and the Gaussian Analytic Function


Abstract: It is well-known that the spectrum of nonselfadjoint operators may be very unstable w.r.t. perturbations (pseudospectral effect). We study the (discrete) spectra of semiclassical (pseudo-)differential operators, subject to small random perturbations. Previous works by Hager, Sjöstrand and others have shown that, in the semiclassical limit, with high probability the eigenvalues of the perturbed operator spread over the "classical spectrum", and the macroscopic spectral density satisfies Weyl's law. Beyond the spectral density, we investigate the short distance correlations between the eigenvalues. In the case of 1-dimensional operators, these correlations depend on the structure of the energy shell of the unperturbed operator (a finite set of points), and of the type of perburbation (random matrix vs. random multiplicative potential), but otherwise locally satisfy some form of universality. The central object describing these "universal" random point processes is the Gaussian Analytic Function, a well-known statistical object. Introduced by Hannay in the context of quantum chaos, the GAF was originally designed to describe eigenfunctions of quantized chaotic systems; it also showed up in the study of random holomorphic sections on Kaehler manifolds by Bleher-Schiffman-Zelditch. In the present case the GAF appears through the spectral determinant of our randomly perturbed operator, when rescaled to the microscopic scale. Let us notice that such spectral statistics differ from the well-known Ginibre ensemble of complex matrices.



23.11.2016      Steve Zelditch (Northwestern University, USA)
16:00

Room: (Joint with QM2 Colloquium) Hörsaal VIII, Hauptgebaude (click to see the map)

Title: [pdf] How to fill a domain with quantum states in quantum Hall problems

Abstract: Let D be a domain in some surface M. In fact, M could have any dimension. We would like to construct global quantum states on M which are localized in D and completely fill up D without much spillover outside D. We answer the question by making the quantum states eigenstates of a certain Hamiltonian defined so that D is the energy set H < E. The projection onto the corresponding eigenspaces defines a partial Bergman kernel and when normalized properly, it tends to 1 on D and zero outside D as Planck's constant tends to zero. There is a universal transition from 0 to 1.


                              





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