## Sponsored by ICMT Engineering Sciences Building 1101 West Springfield Avenue Urbana, IL 61801

There have been number of exciting recent developments both in the theory of non-abelian quantum Hall states and on the experimental detection of non-abelian quasiparticles in quantum Hall systems and (possibly) in time-reversal breaking superconductors. This motivated us to organized a small workshop in which we hope to bring together a small number of people who are working on these exciting problems. The launching of our new Institute provides a natural venue for a workshop of this type, as well as some amount of funding to organize an event at this limited scale.

Organizer: Prof. Eduardo Fradkin Co-Organizers: Prof. Mike Stone and Prof. Dale vanHarlingen
Local Contacts: Prof. Fradkin and Becky McDuffee

TO VIEW A TALK, CLICK ON THE SPEAKER'S NAME

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## Friday morning: Quantum Information and Topological Phases

Speaker: Nicholas Bonesteel, FSU and NHMFL
Title: Monte Carlo Simulations of Chains of Interacting Non-Abelian Particles

Abstract:In non-Abelian states of matter, quasiparticle excitations carry quantum numbers (topological charge) which characterize a degenerate Hilbert space whose dimensionality grows exponentially with the number of quasiparticles. When these quasiparticles are close enough together, the degeneracy of this Hilbert space is lifted and the quasiparticles are said to interact. Here we show that the valence-bond Monte Carlo method introduced by Sandvik to study ordinary spin-1/2 systems can be generalized to simulate 1D chains of interacting non-Abelian quasiparticles.  For uniform chains of SU(2)k particles (corresponding to exactly solvable q-state Potts models) our Monte Carlo results for the so-called valence-bond entanglement entropy agree with recent exact results of Jacobsen and Saleur. For random chains we confirm numerically that, as expected, the ground state freezes into a "random singlet phase" in which the entanglement entropy of a block of length L scales as S(L) ~ (ln d)/3 log L, where d is the quantum dimension of the particles.

Speaker: Parsa Bonderson, Microsoft Station Q, UCSB
Title: Measurement-Only Topological Quantum Computation

Abstract: The topological approach to quantum computing derives intrinsic fault-tolerance by encoding qubits in the non-local state spaces of non-Abelian anyons. The original prescription required topological charge measurement for qubit readout, and used braiding exchanges of anyons to execute computational gates. We present an anyonic analog of quantum state teleportation, and use it to show how a series of topological charge measurements may replace the physical transportation of computational anyons in the implementation of computational gates.

Speaker: Zhenghan Wang, Microsoft Station Q, UCSB
Title: Spin modular category and fermionic quantum Hall states

Abstract: Topological properties of bosonic quantum Hall states are well-modelled by unitary modular tensor categories. For fermionic quantum Hall states, the analogue of a modular tensor category is a unitary fusion category with double braidings and double twists covered by a modular category with a fermion. The unitary fusion category in general has neither braidings nor twists. This is a joint work with N. Read.

Speaker: Kirill Shtengel, UCR
Title: Local interactions, orthogonality and non-Abelian quantum loop gases

Abstract: Two-dimensional quantum loop gases provide elementary examples of states with Abelian or non-Abelian topological order. While Abelian loop gases appear as ground states of local, gapped Hamiltonians such as the toric code, it turns out that gapped non-Abelian loop gases require either non-local interactions or non-trivial inner products. In particular, perturbing a local, gapless Hamiltonian with an anticipated "non-Abelian" ground-state wavefunction immediately drives the system into the Abelian phase. This result is related to a curious failure of the "plasma analogy" which I will also discuss.

## Friday afternoon: Interferometers and non-abelian states

Speaker: Andrei Bernevig, Princeton.
Title: A Unified Description of Unitary and Nonunitary FQH states

Abstract: We present several first principle calculations of experimentally observable properties of abelian and non-abelian FQH states. We describe several general families of non-Abelian FQHE states. One of them, at filling ν = k/r, has polynomial wavefunctions Jλα (z1; ... ; zN) where Jλα is a symmetric Jack polynomial with negative (coprime) rational parameter α = -(k+1)/(r-1), and λ is a compressed partition. These polynomials are dominated by an occupation-number pattern maximally-obeying the generalized Pauili rule that no (consecutive) group of r orbitals contains more than k particles. This exclusion rule defines a space of polynomials characterized by how they vanish as clusters of particle coordinates contract to a point (pattern of zeroes). We also describe a new series of non-Jack non-abelian FQH states forming a hierarcy that ends in the Moore-Read state.

The edge of these FQHE states has a fractionally-quantized thermal Hall effect with ceff = k(r + 1)/(k + r), derived from the exclusion rule. The r = 2 family are the Laughlin, Moore-Read, and Read-Rezayi states, related to unitary conformal field theories (CFT). Their squeezing properties have been previously found by Haldane. The r > 2 families are related to non-unitary CFT, but (as polynomials) have well-defined (not obtainable from CFT) quasi-hole propagators; their fusion rules can be obtained from their root partions, as recently showed by Ardonne. We also present model wavefunctions for quasiparticle (as opposed to quasihole) excitations of the Zk parafermion sequence (Laughlin/Moore-Read/Read-Rezayi) of Fractional Quantum Hall states. For Abelian Fractional Quantum Hall states (k=1), our construction reproduces the Jain quasielectron wavefunction, and elucidates the difference between the Jain and Laughlin quasiparticle constructions. For two (or more) quasiparticles, our states differ from those constructed using Jain's method. By adding our quasiparticles to the Laughlin state, we obtain a hierarchy scheme which gives rise to a non-abelian non-unitary Gaffnian ν= 2⁄5 FQH state (same as the Jack state).

We perform first principle calculations of physical properties such as electron and quasihole propagators or edge thermal Hall coefficient and find that they reproduce the CFT conjectured results only for unitary FQH states. Non-unitary FQH states show several disagreements between first principle calculations and CFT predictions.

Title: Non-Abelian adiabatic transport of conformal block trial wavefunctions

Abstract: I will argue that when conformal blocks from a rational conformal field theory are used as trial wavefunctions as in the work of Moore and Read, the adiabatic statistics of quasiholes is given by the same braiding as the analytic continuation of the wavefunctions if the corresponding two-dimensional perturbed CFT is in a massive phase, and if not then the system is gapless. In particular, the use of non-unitary RCFTs that contain negative quantum dimensions does not lead to gapped (topological) phases. Quasihole spin, modular transformations, and Hall viscosity can be approached in the same way.

Speaker: Claudio Chamon, Boston University
Title: Topological order at finite temperatures

Abstract: We study topological order in a toric code in three spatial dimensions at finite temperature. We compute exactly the topological entropy in the system, and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature Tc, dropping to zero above Tc. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of strings and membranes, and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature, its quantum and classical aspects, and the possibility that certain quantum memories can be stable as long as the temperature is kept below a critical value.

Speaker: Shivaji Sondhi, Princeton
Title: Deconfinement Diagnostics, Symmetric Sponges and Finite Temperature Topological Phases

Abstract: Inspired by a mapping between symmetric sponges and deconfined phases, we construct a diagnostic for deconfinement. One version of this diagnostic is the Fredenhagen-Marcu order parameter known to lattice gauge theorists. Armed with this we establish the existence of finite temperature topological phases in d > 2 dimensions. We also show how to use a multiplet of Fredenhagen-Marcu order parameters to diagnose the phase structure of theories with U(1) gauge symmetry exhibiting condensation. We will also describe an exact reduction of Kitaev's model (toric code) at T>0 which makes its phase structure manifest in all dimensions.

Speaker: Sankar Das Sarma, University of Maryland
Title: Non-Abelian Fractional Quantum Hall Effect at Half-Filled Landau Levels

Abstract: I will discuss the current understanding of the nu =5/2 and 1/2 FQH states as non-Abelian states, using recent results from numerical, theoretical, and experimental works.

## Saturday morning: Non-abelian topological phases

Speaker: Chetan Nayak, Microsoft Station Q/UCSB
Title: Edge Excitations of non-Abelian States and Quasiparticle Tunneling

Abstract: I will discuss gapless edge excitations of non-Abelian topological states, especially in the quantum Hall regime. Inter-edge quasiparticle tunneling reflects some of the underlying topological structure of such states, and its relation to experiments will be discussed.

Speaker: Edward Rezayi, Cal State Los Angeles
Title: "Non-Abelian Hall states in high Landau levels and atomic Bose gases"

Abstract: Numerical studies appear to lend strong support to the notion that Non-Abelian Hall states could be materially realized in high Landau levels and rapidly-rotating Bose gases with dipolar interactions.  The focus of this talk will be on the Moore-Read paired Hall state and its generalization to states with k-particle grouping correlations for k=3 and 4.  The numerical studies of bulk and edge states and the conditions that stabilize them will be reviewed.

Speaker: Xiao-Gang Wen, MIT
Title: Characterize quantum Hall states through their pattern of zeros

Abstract: We show that different Abelian and non-Abelian fractional quantum Hall states can be characterized by patterns of zeros described by sequences of integers Sa. Using the data Sa we can calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types, their quantum numbers, and the fusion rules.

Speaker: Paul Fendley, Virginia.
Title: Classical and Quantum Duality

Abstract: I explain how the familiar Kramers-Wannier duality of two-dimensional classical lattice models and the "topological symmetry" of interacting anyonic chains are essentially the same symmetry. This allows this duality to found for essentially any lattice model with an integrable critical point. The idea is naturally generalized to two-dimensional quantum lattice models with ground states likely to possess topological order. Here the analogous quantum self-duality results in an easy way to construct relatively simple local Hamiltonians which have ground states comprised of superpositions of nets.

## Saturday afternoon: Non-abelian FQH states and superconductors

Speaker: Yiming Zhang, Harvard University (C. Marcus' Group)
Title: Sigantures for Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Perot interferometers

Abstract: Two different types of resistance oscillations are observed in two electronic Fabry-Perot interferometers of different sizes. Measuring these oscillations as a function of magnetic field, gate voltage, or both, we observe three signatures that distinguish the two types. The oscillations observed in a μ m2 device are understood to arise from Coulomb blockade, and those observed in an 18 μ m2 device from Aharonov-Bohm interference. As a function of dc bias and magnetic field, a checkerboard pattern is observed in the 18 μ m2 device, but not in the 2 μ m2 device. The energy scale for the checkerboard pattern is used to measure the edge state velocity, obtaining good agreement with simple models for both the low and high magnetic field limits.

Speaker: Dale Van Harlingen, Illinois
Title: Experimental Evidence for Complex Chiral Superconductivity in Sr2RuO4 and UPt3

Abstract: A leading candidate for topologically-protected quantum computing is the ruthenate superconductor Sr2RuO4 that is suspected to have a complex p-wave order parameter of the form px +ipy that breaks time-reversal symmetry. Josephson interferometry measurements give strong direct evidence for this order parameter and reveal the presence of a chiral domain structure that exhibits dynamics, coupling to external magnetic fields, and anomalous memory effects. However, several other experiments are not fully consistent with this picture, raising quest ions about the nature of the superconducting state. I will discuss experiments designed to verify the pairing symmetry of Sr2RuO4 and test for the nucleat ion of half-integer vortices that may be a signature of non-Abelian statistics.

A second candidate for complex superconductivity is the heavy ferm ion superconductor UPt3, one of the few materials that exhibits two distinct superconducting phases. I will present recent evidence for broken time-reversal symmetry in this material from Josephson interferometry measurements and describe ongoing experiments designed to explore the transition between the two superconducting states.

Speaker: Raffi Budakian Illinois
Title: Search for Chiral Domains in Mesoscopic Sr2RuO4 Samples Using Ultra-Sensitive Torque Magnetometry

Speaker: Eun-Ah Kim, Cornell
Title: In search of topological phases through half-quantum" vortices

Abstract: The existence of fractionalized excitations, together with the emergence of topological invariance in their long distance physics, characterize topological phases.  In this talk I will first give an overview of the connection between topology and fractionalized excitations highlighting common features between fractional quantum Hall states, the best established example of topological states, and Sr2 RuO4 as a possible example of a p+ip superconductor. I will then discuss our recent results and on-going efforts towards analyzing and proposing conditions for stabilizing and detecting half-quantum vortices in Sr2RuO4, Na-Cobaltates and fermions near a p-wave Feshbach resonance.

## Sunday morning: Topological Insulators

Speaker: Alexei Kitaev, Caltech
Title: A periodic table for topological insulators and superconductors

Abstract: Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class that corresponds to one of the 2 types of complex Clifford algebras and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z or Z2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided the electrons stay localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.

Speaker: Duncan Haldane, Princeton.
Title: "Gapless entanglement spectrum of Fractional Quantum Hall states and Topological Insulators"

Abstract: The Von Neumann bipartite "entanglement entropy" can formally be expressed as the entropy of a system of (dimensionless) "pseudo-energy levels" (the eigenvalues of the logarithm of a density matrix) at a "pseudo-temperature" T=1. This does not distinguish between gapless or gapped "entanglement spectra". If the ground state of a gapped infinite D-dimensional "topologically-ordered" system is Schmidt-decomposed into two semi-infinite halves along a translationally-invariant (D-1)-dimensional interface, it appears that the "entanglement spectrum" is general gapless, with the same topological character as the physical edge-state energy spectrum that would result from cutting the sample to make a physical edge.
In the case of fractional quantum Hall states, the gapless entanglement spectrum is equivalent to the conformal field theory that would describe a physical edge. In the case of topological insulators, the gapless entanglement spectrum, which is an intrinsic bulk ground-state property, may be regarded as the physical property that forces the appearance of gapless edge states when the system is physically cut.

Speaker: Charles Kane, University of Pennsylvania
Title: Majorana Fermions in Topological Insulators

Abstract: We will begin by reviewing recent experiments which have established the existence of topological insulators in two and three dimensions and probe their unique conducting edge and surface states. We will then argue theoretically that the interface between a topological insulator and a superconductor leads to a state that supports Majorana fermions, which are the basis for proposals for topological quantum computation. At the surface of a 3D topological insulator, the Majorana fermions occur at vortices, and can be created, manipulated and fused by engineering suitable superconducting junctions on the surface. At the edge of a 2D topological insulator, Majorana fermions occur at the interface between regions with superconducting and magnetic order. Due to the Majorana fermions, a Josephson junction mediated by the 2D topological insulator is predicted to exhibit a "fractional Josephson effect", in which the Josephson current has a 4 pi rather than a 2 pi periodicity.

Speaker: Joel Moore, UC Berkeley
Title: Magnetoelectric polarizability and axion electrodynamics in crystalline insulators

Abstract: Spin-orbit coupling can lead in two- and three-dimensional solids to time-reversal-invariant insulating phases that are topological'' in the same sense as the integer quantum Hall effect and similarly have protected edge or surface states. The three-dimensional topological insulator has a set of unusual magnetoelectric properties referred to as axion electrodynamics'': it supports an electromagnetic coupling LEM = (θ e2 / 2 π h) E \cdot B with θ=π, giving a half-integer surface Hall conductivity σ xy=(n + 1/2) e2 / h. We first compute a spatially resolved Chern number in a slab geometry to separate bulk and surface contributions to σ xy and confirm this behavior. An approach to θ in any three-dimensional crystal is then developed based on the Berry-phase theory of polarization: θ e2/ 2 π h is the bulk contribution to the orbital magnetoelectric polarizability (the polarization produced by an applied magnetic field). Although θ is no longer quantized once time-reversal is broken, it remains a universal bulk quantity for the same reasons as ordinary polarization, and the unit-cell-dependent ambiguity in bulk polarization is related to an e2/h ambiguity in magnetoelectric polarizability. Potentially measurable effects of the θ coupling include a switchable integer quantum Hall effect for θ = π and an electrical dressing of monopole excitations in frustrated magnets.