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

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: Spin Chern-Simons theories for the quantum Hall effect

Abstract: We will review the effective description of the quantum Hall effect in terms of abelian Chern-Simons theory, and show that it actually involves "spin" Chern-Simons theory, whose action is half the standard Chern-Simons action. We will then discuss two ways of defining the Chern-Simons 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 3-dimensional definition of the spin Chern-Simons action less straightforward than one may think, requiring the concept of generalized cohomology theory.

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: Derivation of semiclassical kinetic theory in the presence of non-abelian 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 non-Abelian. We use the formalism of phase-space quantum mechanics to derive the results.

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: About J-flow

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 J-flow 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.

Room: Cohn-Vossen Raum 313 des Mathematischen Instituts

Title: Large-scale approaches to correlations and dynamics of the inhomogeneous one-dimensional Bose gas

Abstract: The one-dimensional delta Bose gas, also known as the Lieb-Liniger model, is instrumental in the description of one-dimensional ultra-cold quantum gases. In its homogeneous, translation-invariant 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 large-scale approaches that are valid in those situations where the external potential and the local thermodynamic quantities vary slowly at the inter particle distance.

Room: Cohn-Vossen 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, Moore-Read, Read-Rezayi 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.

Room: Cohn-Vossen 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.

Room: Cohn-Vossen 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 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.

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.

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.

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.

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.

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.

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.

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).

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.