



Seminar on mathematical aspects of the quantum Hall effect
and related topics
Mathematical Institute, Institute for Theoretical Physics, Cologne, 20162017
Organizers: M. Hermanns (ITP), S. Klevtsov (ITP and MI) and G. Marinescu (MI)


Summer Semester 2017

11.07.2017 Samuel Monnier (University of Geneva, Switzerland) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: Spin ChernSimons theories for the quantum Hall effect
Abstract: We will review the effective description of the quantum Hall effect in terms of abelian ChernSimons theory, and show that it actually involves "spin" ChernSimons theory, whose action is half the standard ChernSimons action. We will then discuss two ways of defining the ChernSimons action for gauge fields of arbitrary topology and generalize them to the spin case. In particular, we will show that issues of invariance under large gauge transformations make a purely 3dimensional definition of the spin ChernSimons action less straightforward than one may think, requiring the concept of
generalized cohomology theory.

04.07.2017 Eldad Bettelheim (Hebrew University, Israel) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: Derivation of semiclassical kinetic theory in the presence of nonabelian Berry curvature
Abstract: In quantum mechanics it is often required to describe in a semiclassical approximation the motion of particles moving within a given energy band. Such a representation leads to the appearance of an analogues of fictitious forces, associated with the Berry curvature, in the semiclassical equations of motion. The purpose of this paper is to derive systematically the kinetic Boltzmann equations displaying these effects in the case that the band is degenerate, and as such the Berry curvature is generically nonAbelian. We use the formalism of phasespace quantum mechanics to derive the results.

27.06.2017 Julien Keller (Aix Marseille Université, France) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: About Jflow
Abstract: Given a complex projective manifold, it is very difficult to detect in the Kähler cone which classes admit a Kähler metric with constant scalar curvature. Fix a Kähler class $\alpha$ with such a special Kähler metric, one can ask if it is easier to describe explicitly a neighborhood V of $\alpha$, such that for all $\beta$ in V, the Kähler class $\beta$ admits a Kähler metric with constant scalar curvature. Of course, one can ask what is the largest possible neighborhood V? Under certain conditions, a partial answer to these questions can be given by studying Donaldson's Jflow that I will present. This flow also appears in mirror symmetry. I will explain that if the behavior of this flow is still unclear, its finite dimensional analogue can be well understood thanks to geometric quantization. This is joint works with R. Dervan and Y. Hashimoto.

20.06.2017 Jerome Dubail (Université de Lorraine, Nancy, France) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: Largescale approaches to correlations and dynamics of the inhomogeneous onedimensional Bose gas
Abstract: The onedimensional delta Bose gas, also known as the LiebLiniger model, is instrumental in the description of onedimensional ultracold quantum gases. In its homogeneous, translationinvariant version, a key property of the model is that it is solvable by Bethe Ansatz. However, in a trap potential, this property is lost.
I will give an introduction to the results that have accumulated recently
on correlation functions and on the dynamics of the inhomogeneous 1d delta Bose gas.
The results are obtained from largescale approaches that are valid in those situations where the external potential and the local thermodynamic quantities vary slowly at the inter particle distance.

13.06.2017 Benoit Estienne (LPTHE, Paris, France) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: [pdf] Trial states for the FQHE: from conformal field theory to matrix product states
Abstract: A large part of our understanding of the fractional quantum hall (FQH) physics comes from model wave functions. Some of the most famous examples are the Laughlin, MooreRead, ReadRezayi and Gaffnian states. A preferred method for obtaining bulk FQH model wave functions is to express them as correlation functions of some primary field V(z) in a Conformal Field Theory (CFT). After reviewing the CFT approach to the FQHE, I will derive an exact matrix product state (MPS) representations for these model states. I will conclude with some numerical results obtained with this MPS formalism concerning correlation lengths, topological entanglement entropy and quasihole braiding of various FQH model states.

30.05.2017 Vatsal Dwivedi (University of Cologne, Germany) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: [pdf] Transfer matrices, Riemann surfaces and topological phases of matter
Abstract: Transfer matrices naturally arise in discrete calculus as a representation of linear difference equations. A transfer matrix approach can be useful in analyzing noninteracting fermionic systems, esp noninteracting topological phases of matter, since it treats the "bulk" and "edge" states on an equal footing. In this talk, I will discuss a general construction of transfer matrices for such systems, starting from the single particle Hamiltonian.
Exploiting the algebraic nature of the eigenvalue problem for the transfer matrix and allowing the energy to be complex, one naturally arrives at an energy Riemann surface. The windings around the noncontractible loops of this Riemann surface then corresponds to the edge invariants of the system. Assuming symplecticity of the transfer matrix, these invariants can also be identified as a Maslov index on the symplectic group manifolds.

23.05.2017 Martin Zirnbauer (University of Cologne, Germany) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: [pdf] 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 twoparameter renormalization group flow conjectured from the nonlinear 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 nonlinear sigma model coupled to a maximally gauged WessZuminoWitten 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: CohnVossen 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 singleparticle observations alone. Indeed, it turns out that the eigenvalues of singleparticle states compatible with certain global entanglement classes form configurations of convex polytopes. Mathematically, these sets can be identified as nonabelian moment polytopes associated with the action of local symmetry groups on the global state vector. For lowdimensional 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: CohnVossen 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 ErdosRenyi 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 nonvanishing
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 20162017

31.01.2017 Eddy Ardonne (Stockholm University, Sweden) 10:15
Room: CohnVossen 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: CohnVossen Raum 313 des Mathematischen Instituts
Title: [pdf] Some recent estimates on nodal sets and nodal domains in the highenergy 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 highenergy 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: CohnVossen 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 HaldaneHalperin 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 HaldaneHalperin hierarchy to nonabelian states, and discuss their respective predictions and properties.

20.12.2016 Joaquim OrtegaCerdà (University of Barcelona, Spain) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: [pdf] Energy of determinantal point processes in the torus and the sphere
Abstract: We study expected Riesz senergies 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 twodimensional 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 (pseudomagnetic) fields [1] which result, in particular, in formation of pseudoLandau 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 valleypolarized 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 ParisSud, France) 10:15
Room: CohnVossen Raum 313 des Mathematischen Instituts
Title: [pdf] Random perturbations of nonselfadjoint differential operators, and the Gaussian analytic function
Abstract: It is wellknown 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 1dimensional 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 wellknown 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 BleherSchiffmanZelditch. 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 wellknown 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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