Department of Physics • University of Cologne

We will first review spectral flow and its application to bulk and boundary properties of the quantum Hall effect. We will then introduce the so-called KO-valued spectral flow, defined in joint work with Alan Carey, Matthias Lesch and Adam Rennie, and how it can be used to recover the classification of topological insulators and superconductors by Kennedy and Zirnbauer. Time permitting, we will then present recent joint work with Johannes Kellendonk and Adam Rennie, which uses the Cayley transform on Hilbert C*-modules to generalise many of these ideas and gives analogous results to recent work by Alldridge, Max and Zirnbauer.

In 2016 a localization dichotomy result for periodic Schrödinger operators has been proved. This result shows that the localization properties of the Wannier functions associated to a periodic Schrödinger operator are deeply connected to the topological properties of the quantum system. In this seminar I will start by discussing the generalization of Wannier functions and Chern number to non-periodic systems. After that, I will present some results regarding the generalization of the localization dichotomy to generic gapped quantum systems.

The late time behavior of disordered, wave bearing systems, acoustic, elastic or optical, has been of great interest in recent years. From a practical perspective, late time measurements correspond to large distances in a disordered system, and are thus relevant in many imaging or non-destructive evaluation applications. From a basic science perspective, late time observations can reveal interference behavior resulting from the approach to the Anderson transition, and offer the prospect of observing classical wave systems in a localized phase. While the Anderson transition has been theoretically predicted to occur in such systems for some time, clear evidence of a localized phase has unexpectedly been quite difficult to find. Our theoretical understanding of the current status is still incomplete, with continuing efforts employing direct numerical simulations, self-consistent, multiple scattering models, and field theoretic methods. Of these, direct numerical simulations and field theoretic methods enable a description of the localized phase. Direct numerical simulations, particularly those based on a monopole or dipole scattering approximation, have achieved some promising results but still describe highly idealized systems, and require substantial computational resources to describe systems substantially larger than a mean free path. Field theoretic methods on the other hand, while yielding semi-analytic results, have in the past been restricted to even more idealized systems. We discuss recent results in these areas, particularly with a view of moving towards a more realistic description of disordered, elastic systems. This research was funded by the Office of Naval Research.

The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group ow conjectured from Pruisken's non-linear sigma model. Yet, the conformal Field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. Here we carry out a detailed study of the current-current correlation function for the conductivity tensor, initially in the Chalker- Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We take the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n = 4. By proposing precise lattice expressions for the CFT primary Fields we predict the multifractal scaling exponents of critical wave intensities to be Δ_q = q(1-q)/4.

https://arxiv.org/abs/1509.04507

This group seminar talk is based on a part of the speaker's PhD thesis on applications of reflection positivity in strongly correlated lattice models. Some mathematical structures behind sign-problem-free quantum Monte Carlo simulations has been proposed by different research groups in the past a few years. The speaker is trying to unify all these theories with his limited mathematical knowledge. Comments and suggestions are welcome.

Edge reinforced random walk (ERRW) and vertex reinforced jump processes (VRJP) are history dependent stochastic processes, where the particle tends to come back more often on sites it has already visited in the past. For a particular scheme of reinforcement these processes are random walks in a random environment whose mixing measure can be related to a nonlinear sigma model introduced in the context of random matrix models for quantum diffusion. This relation allows to prove, in particular, transience for weak reinforcement of both processes, in d larger or equal to 3. I will give a review of the problem and some recent results.

We study the affine nilTemperley-Lieb algebra, an algebra given by generators and relations. We introduce its action on combinatorially defined particle configurations, and using this action, we describe explicitly several bases and the centre of the algebra.

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza–Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and, at least provisionally, the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub–Newman, Unti, Tamburino (Taub–NUT) manifold as a model for the electron, the Atiyah–Hitchin manifold as a model for the proton, CP2 with the Fubini– Study metric as a model for the neutron and S4 with its standard metric as a model for the neutrino.

In this talk we discuss the method of Majorana reflection positivity and its applications in the study of strongly correlated systems. The first application concerns the ground state degeneracy of interacting spinless fermions. Based on Majorana reflection positivity, we prove rigourously that the ground state of this model is either unique or doubly degenerate if the lattice size N is even, and is always doubly degenerate if N is odd. This proof holds in all dimensions with arbitrary lattice structures. The second application is about the sign problem in quantum Monte Carlo simulations. We examine this problem with a new perspective based on the Majorana reflection positivity. Two sufficient conditions are proven for the absence of the fermion sign problem. Our proof provides a unified description for nearly all the interacting lattice fermion models that are previously known to be free of the sign problem. It also allows us to identify a number of new sign-problem-free interacting fermion models.

Paper by Rahul Nandkishore and David A. Huse.

Paper by Paul Zinn-Justin.

Journal reference: Entropy 17 (2015), 772-789.

We discuss the recent paper "Observation of Gravitational Waves from a Binary Black Hole Merger" by B. P. Abbott et al. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We study transport via edge modes in a disordered 2D topological insulator allowing for the presence of non-protected diffusive channels in addition to the topologically protected edge channels. This scenario can be realized at the interface between two quantum Hall system (class A), in a Weyl semimetal in a magnetic field (Class A) or at the edge of a quantum spin Hall system (class AII). The edge transport is described by a one-dimensional field theory in the form of a supersymmetric non-linear sigma model with a topological term. The transfer-matrix formalism is employed to map the problem to the problem of finding the eigenfunctions of the Laplace operator on a symmetric superspace with an additional vector potential. The latter problem is solved exactly, enabling us to obtain the full counting statistics and mesoscopic conductance fluctuations in the system. Our main finding is that disorder is much more effective in localizing the diffusive (non-protected) channels in the presence of topologically protected ones. This manifests itself as a suppression of the shot noise at scales much shorter than the localization length and the appearance of a gap in the distribution function of transmission eigenvalues close to unit transmission.

We discuss the paper "Supergeometry in mathematics and physics" by Mikhail Kapranov. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the second and the third Lecture of Edward Witten's recent paper "Three Lectures On Topological Phases Of Matter". Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the first Lecture of Edward Witten's recent paper "Three Lectures On Topological Phases Of Matter". Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Copenhagen Quantum Mechanics" by Timothy J. Hollowood. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

By a result of Bochner and Montgomery the automorphism group of a compact complex manifold M carries the structure of a complex Lie group, whose Lie algebra is isomorphic to the Lie algebra of holomorphic vector fields on M. In this talk, the case of a compact complex supermanifold M is considered. In this setting, the Lie superalgebra Vec(M) of holomorphic super vector fields on M is finite-dimensional. We investigate how the automorphism group of a compact complex supermanifold M can be defined and prove that it carries the structure of a complex Lie supergroup with Lie superalgebra Vec(M). (This is joint work with Matthias Kalus.)

We discuss the paper "Nonlocal Hidden-Variable Theories and Quantum Mechanics, an Incompatibility Theorem" by A.J. Leggett. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "A theorem relative to the motion of a point pulled towards a fixed centre" by J.L.F. Bertrand. The result is nowadays known as "Bertrand's Theorem". Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "The symplectic egg in classical and quantum mechanics" by Maurice A. de Gosson. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "T-Duality of Topological Insulators" by V. Mathai and G.C. Thiang. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Undecidability of the Spectral Gap" by Cubitt, Perez-Garcia and Wolf. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss chapter 4 of the paper "The Noncommutative Geometry of the Quantum Hall Effect" by Bellissard, van Elst, and Schulz-Baldes. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "The Free Will Theorem" by John Conway and Simon Kochen. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "The non-commutative geometry of the complex classes of topological insulators" by Emil Prodan. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Symmetric functions and random partitions" by A. Okounkov. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss extracts from Zee's book "Quantum Field Theory in a Nutshell". Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss an extract of the paper "Spectral triples: examples and index theory" by Alan Carey, John Phillips and Adam Rennie. Please get in contact with Thomas Quella in order to find out the precise content of the seminar. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the papers in advance.

We discuss extracts of the two papers "A short introduction to noncommutative geometry" by Peter Bongaarts and "Spectral triples: examples and index theory" by Alan Carey, John Phillips and Adam Rennie. Please get in contact with Thomas Quella in order to find out the precise content of the seminar. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the papers in advance.

We discuss the paper "The Non-Commutative Geometry of the Quantum Hall Effect" by Bellissard, van Elst, Schulz-Baldes. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

A theorem of Helgason states that if an action of a Lie group on a manifold admits a slice, then any differential operator admits a `radial’ component on the slice. For actions of Lie supergroups, this is no longer true in general, as the naive slice condition is too weak. We introduce a strong notion of slice that allows for the generalisation of Helgason’s theorem, using the concept of `orbits through generalised points’. We study in particular the isotropy group action for a Riemannian symmetric superspace. In this setting, the strong slice condition holds if and only if the supermanifold is of `even type’ (i.e. the slice in the naive sense is an ordinary manifold). To remedy this situation, we consider a weak notion of slices. We prove in general that at least any differential operator locally invariant under the transporter supergroup of the weak slice admits a radial part on the quotient of the weak slice by the transporter supergroup. This quotient is (mildly) singular in general. For the example of Riemannian symmetric superspaces, the transporter supergroup action factors through the Weyl group, which is a reductive Lie group of positive dimension in the cases not of odd type. We illustrate our findings by examples. This work is joint with J. Hilgert and T. Wurzbacher, and with K. Coulembier, respectively.

We discuss the paper "Spin Ice, Fractionalization and Topological Order" by C. Castelnovo, R. Moessner, S.L. Sondhi. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Quantum information and computation" by Charles Bennett and David DiVincenzo. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Relating Field Theories via Stochastic Quantization" by Dijkgraaf, Orlando, Reffert. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the beginning of the chapter "Quantum Master Equations" from the book "The theory of open quantum systems" by Breuer and Petruccione. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Recovering low-rank matrices from few coefficients in any basis" by David Gross. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "On 2d TQFTs whose values are holomorphic symplectic varieties" by G. Moore and Y. Tachikawa. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "String-net condensation: A physical mechanism for topological phases" by M. Levin and X.-G. Wen. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "Geometry and Empirical Science" by C.G. Hempel. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

See http://arxiv.org/abs/1405.2971

We discuss the paper "Gauge symmetry and spontaneous breaking of gauge symmetry: philosophical approach" by Pascal Lederer. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We introduce the notion of a G-structure on a supermanifold M and indicate a proof that the automorphisms of such a structure constitute a Lie supergroup, provided it is of finite type. This relies on a representability criterion for subsheaves of certain representable sheaves on supermanifolds. Examples include the cases of even and odd Riemannian metrics as well as G-structures (G a suitable Poincare supergroup) on spinor supermanifolds as constructed by Cortes et al.

We discuss the paper "Consistent thermostatistics forbids negative absolute temperatures" by Dunkel and Hilbert. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

In physics, important information about a system sometimes can be obtained by studying its response to geometry. Recently, the gravitational response of the Quantum Hall effect on Riemann surfaces attracted considerable attention. I will begin by explaining that the problem can be formulated naturally using Kähler metrics, line bundles and projective embeddings. Then I will explain how to fully constrain the large N expansion of the QHE partition function on Riemann surfaces (and on Kähler manifolds) by relating it to the Bergman kernel, extending previous math (S. Donaldson, R. Berman) and physics (P. Wiegmann – A. Zabrodin) results. Then I will discuss the meaning of Liouville and Mabuchi functionals, appearing in the large N expansion. I will also discuss possible extension of these results to the case of fractional QHE.

In this informal talk, I will discuss the question whether it is possible to find a Tensor Product State which is the ground state of a gapped, local, Hamiltonian lying inside a chiral topological phase. Prominent examples of such chiral phases are quantum Hall states, Chern bands, or complex (e.g., p+ip) paired states. This question appears to have been mentioned by various people, but so far only a handful of papers have tackled it directly. I will start with a preprint from last year (JD-Read, 2013, see also Wahl-Tu-Schuch-Cirac, 2013 and Hastings, 2014) and elaborate on some subsequent developments (unpublished). Only basic knowledge of quantum Hall trial wave functions (the approach of Moore-Read, 1991) and of conformal field theory will be required.

We discuss the paper "The entropy concept for non-equilibrium states" by Lieb and Yngvason. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "The Excitation Spectrum for Bose Fluids with Weak Interactions" by Robert Seiringer. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We are going to have rehearsals for the DPG-Tagung. Speakers and titles: 1. Roberto Bondesan: "Critical theory of the spin quantum Hall transition" 2. Daniel Wieczorek: "Pure scaling operators at the integer quantum Hall plateau transition" 3. Thomas Quella: "The Cirac-Sierra construction and long range SU(N) spin models" 4. Jochen Peschutter: "The gl(1|1) Lie superalgebra as a basis for the Nielsen-Cirac-Sierra construction" 5. Thomas Quella: "Haldane phases of SU(N) spin chains and their realization in ultra-cold atom gases"

Projected entangled-pair states (PEPS), higher-dimensional generalizations of matrix-product states in one dimension, provide a useful representation for topological states. Many examples of nonchiral topological states, such as toric code and short-range resonating valence-bond states, allow exact PEPS representations. One of the challenging question in this field is to understand whether PEPS can describe chiral topological states with broken time-reversal symmetry. In this talk, I will present our recent results on constructing explicit examples of PEPS, which belong to Chern insulators and topological superconductors with nonzero Chern number. These provide a proof-of-principle result that PEPS can describe chiral topological states.

We discuss Lectures 2 and 3 of the lecture notes "Soluble models of turbulent transport" by Krzysztof Gawedzki. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance. (Contact me to get the paper.)

The difference between strong and weak topological insulators is rooted in the difference between homotopy classes of maps from a sphere as opposed to maps from a torus. I will give a simple proof that the former constitute a subset of the latter, which I use to give a definition of "strong". Another aspect I will address is the possibility of stacking lower dimensional systems into higher dimensions. I will give two examples that demonstrate that not all weak topological insulators come from stacking and - vice versa - that stacking insulators can give more topological phases than one might guess from considering only the lower dimension.

We read and discuss the paper "Geometry of physical dispersion relations" by Dennis Raetzel, Sergio Rivera and Frederic P. Schuller. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

We discuss the paper "How (not) to teach Lorentz covariance of the Dirac equation" by Hrvoje Nikolic. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

I will talk about the construction of the continuous series of sl(2,R) and its possible relation to modular double representations of quantum sl(2,R). The relations to physical models will be overviewed.

We discuss the paper "Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem" by Connes and Kreimer. Modus operandi: The dice will determine who will lead the discussion, so every seminar participant is kindly asked to prepare the paper in advance.

http://prl.aps.org/abstract/PRL/v90/i12/e120404

http://arxiv.org/pdf/1303.6917.pdf

http://www.maxschlosshauer.com/publications/decoherence_book.pdf http://www.maxschlosshauer.com/publications/decoherence_book.pdf http://www.maxschlosshauer.com/publications/decoherence_book.pdf http://www.maxschlosshauer.com/publications/decoherence_book.pdf http://www.maxschlosshauer.com/publications/decoherence_book.pdf

http://www.fme.upc.edu/arxius/butlleti-digital/riemann/071218_conferencia_atiyah-d_article.pdf

http://link.springer.com/article/10.1007%2Fs10955-009-9870-6

http://dx.doi.org/10.1016/0040-9383(84)90021-1

http://arxiv.org/abs/1207.5304

http://iopscience.iop.org/1367-2630/10/6/063016/

http://de.arxiv.org/abs/1206.3116

http://de.arxiv.org/abs/1203.4565

http://arxiv.org/abs/arXiv:1201.2714

http://arxiv.org/abs/hep-th/9502010

http://arxiv.org/abs/quant-ph/9508027

http://rmp.aps.org/abstract/RMP/v82/i3/p2213_1

http://arxiv.org/abs/arXiv:1201.2714

http://xxx.lanl.gov/abs/math.QA/0402057

http://arxiv.org/abs/hep-th/0506098

http://arxiv.org/abs/1112.2133

http://dx.doi.org/10.1016/0550-3213(84)90052-X

http://dx.doi.org/10.1088/0034-4885/70/6/R03

http://www.springerlink.com/content/365574223623w045/

http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf

http://link.aps.org/doi/10.1103/RevModPhys.80.1083

http://link.aps.org/doi/10.1103/PhysRevA.54.2759

http://dx.doi.org/10.1016/j.aop.2005.04.001

http://www.math.ubc.ca/~db5d/Seminars/PCMILectures/

http://arxiv.org/abs/math-ph/0404057

http://dx.doi.org/10.1088/0264-9381/26/22/224001

http://dx.doi.org/10.1016/j.jalgebra.2010.10.031

http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:1009.6032

http://de.arxiv.org/abs/1007.1844

http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:1001.0785

http://prb.aps.org/abstract/PRB/v82/i11/e115120

http://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01045-6/home.html