Session K51: Fractional QHE: Topological, Non-Abelian, and 5/2 States

Possibility of topological order in partially flat band systems

In this talk, we discuss the possibility of topological order in two-dimensional partially flat bands. We first consider a tight-binding model whose valence band is (nearly) flat only in some regions of the Brillouin zone, where the Berry curvature is mostly concentrated, and dispersive otherwise. We then investigate the ground-states of these systems when the flat regions are fractionally filled. We argue that under certain conditions spontaneous symmetry breaking orders such as charge density wave (CDW) can emerge. The resulting CDW enlarges the unit cell and separates the flat regions from the rest of the valence band by a finite energy gap. Interestingly, the broken symmetry state can exhibit fractional Chern insulator phase with quantized Hall conductivity. Finally, we present our numerical results for the phase diagram of such systems.   

Measurement of 2D topological invariants from anomalous edge spectral flow

A hallmark example of a TQFT is the 2$+$1 D Chern-Simons (CS) theory, which describes topological properties of both integer and fractional quantum Hall effects. The gauge invariant form of the CS theory with boundaries, encompassing both edge and bulk terms, provides an unambiguous way to relate bulk topological invariants to the edge dynamics. This bulk-edge correspondence is manifested as a gauge anomaly in the bulk and chiral anomaly at the edge, and provides a direct insight into the bulk topological order. In this work, we experimentally implement the integer quantum Hall model in a photonic system where the edge modes are described by the anomalous chiral conformal field theory. By selectively manipulating and probing the edge, we exploit the chiral anomaly of the edge theory, for the first time. The associated spectral edge flow associated to the chiral anomaly allows us to unambiguously measure topological invariants, i.e., the winding number of the edge states. This experiment provides a new approach for direct measurement of topological invariants, independent of the microscopic details, and thus could be extended to probe strongly-correlated topological orders.   

Bose condensation in topologically ordered quantum liquids

The condensation of bosons can induce transitions between topological quantum field theories (TQFTs). This as been previously investigated through the formalism of Frobenius algebras and with the use of Vertex lifting coefficients. We discuss an alternative, algebraic approach to boson condensation in TQFTs that is physically motivated and computationally efficient. With a minimal set of assumptions, such as commutativity of the condensation with the fusion of anyons, we can prove a number of theorems linking boson condensation in TQFTs with algebra extensions in conformal field theories and with the problem of factorization of completely positive matrices over the positive integers. We propose an algorithm for obtaining a condensed theory fusion algebra and its modular matrices. For example, this formalism can be used to build multi-layer TQFTs which could be a starting point to build three-dimensional topologically ordered phases. Using this formalism, we also give examples of bosons that cannot undergo a condensation transition due to topological obstructions.   

Fermion Parity Flips and Majorana Defects in Superconducting Fractional Topological Phases

We consider layered heterojunctions of s-wave superconductors and Abelian topologically ordered (TO) phases. We derive the emergent theories for a wide variety of fractional quantum Hall states promoted by a $\mathbb{Z}_2$ gauge theory. The theory always carries an anyonic symmetry (AS) which effects a fermion parity flip. The associated twist defects, which flip the parities of some types of orbiting quasiparticles, trap ordinary zero energy Majorana bound states (MBS), and can bind fractional charge. For example, an $h/2e$ flux vortex of the superconductor that circulates around the MBS undergoes a fermion parity flip and is accompanied by a level crossing in the vortex energy spectrum. We show numerical evidence of the level crossing in the simplest examples: a Chern insulator and a normal insulator/topological insulator/superconductor junction. Finally, we briefly describe the resulting twist liquid theory after gauging the AS where the twist defects become deconfined anyonic excitations.   

A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations

We introduce exactly solvable models of interacting (Majorana) fermions in $d \ge 3$ spatial dimensions that realize a new kind of topological quantum order, building on a model presented in ref. [1]. These models have extensive topological ground-state degeneracy and a hierarchy of point-like, topological excitations that are only free to move within sub-manifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically-ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a ``geometric" framework for the emergence of topological order. [1] S. Vijay, T. H. Hsieh and L. Fu, arXiv:1504.01724 (2015).   

Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators

Particle-vortex duality is a powerful theoretical tool that has been used to study systems of bosons. In arXiv:1505.05142, we propose an analogous duality for Dirac fermions in 2+1 dimensions. The physics of a single Dirac cone is proposed to be described by a dual theory, QED3 with a dual Dirac fermion coupled to a u(1) gauge field. This duality is established by considering two alternate descriptions of the 3d topological insulator (TI) surface. The first description is the usual Dirac cone surface state. The second description is accessed via an electric-magnetic duality of the bulk TI coupled to a gauge field, which maps it to a gauged topological superconductor. This alternate description ultimately leads to a new surface theory - dual QED3. The dual theory provides an explicit derivation of the T-Pfaffian state, a proposed surface topological order of the TI, which is simply the paired superfluid state of the dual fermions. The roles of time reversal and particle-hole symmetry are exchanged by the duality, which connects some of our results to a recent conjecture by Son on particle-hole symmetric quantum Hall states at $\nu = 1/2$.   

Half-filled topological flat bands, dualities and SPT surface states

The surface states of 3D symmetry protected topological phases are impossible to realize in pure 2D systems with a local implementation of symmetry. However, systems with non-local symmetries, e.g. associated with filling or emptying a topological flat band, can circumvent this limitation. A well studied recent example is that of the 1/2-filled Landau level, which effectively realizes the physics of a particular type of time-reversal invariant topological superconductor. In this talk, I will generalize these concepts to other classes of 2D topological flat bands with non-local symmetries, that are related to 3D SPT surface states with local implementations of the same symmetry. This generalization reveals new dualities between strongly interacting gapless phases. In addiition, physical implications for new gapless and topologically ordered states in multicomponent quantum Hall systems will be mentioned.   

Non-Abelian states in Fractional Quantum Hall effect in charge carrier hole systems

Quasiparticle excitations obeying non-Abelian statistics represent the key element of topological quantum computing. Crossing of levels and strong coupling between angular momentum and orbital motion, described by Luttinger Hamiltonian, make properties of charge carrier holes different from those of electrons. Peculiarities of hole spectrum in magnetic field provide an opportunity for controlling Landau level mixing in charge carier hole systems. In order to describe Fractional Quantum Hall effect for holes, we propose a method to map hole spectrum and wavefunctions using a spherical shell. We investigate the experimentally observed $\nu=1/2$ state in spherical geometry. Haldane pseudopotentials are computed and the effect of Landau level mixing is evaluated. Exact diagonalization of Coulomb interaction in systems with eight to fourteen holes is performed. We determine that the ground state superposition with Abelian 331 state is very small and the overlap with Moore-Read state is significant. The quasihole and quasielectron excitations are discussed.   

The Fractional Quantum Hall States at $\nu=13/5$ and $12/5$ and their Non-Abelian Nature

Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for their exotic fundamental physics and potential applications in universal quantum computing. The fractional quantum Hall (FQH) state at filling factor $\nu=12/5$ is such a promising candidate, however, its precise nature is still under debate and no consensus has been achieved so far. Here, we investigate the nature of the FQH $\nu=13/5$ state and its particle-hole conjugate state at $12/5$ with the Coulomb interaction, and address the issue of possible competing states. Based on a large-scale density-matrix renormalization group (DMRG) calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at $\nu=13/5$ and $12/5$ is captured by the $k=3$ parafermion Read-Rezayi state ($\text{RR}_3$), including a robust excitation gap and the topological fingerprint from entanglement spectrum and topological entanglement entropy. Furthermore, by considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a signature of the $\text{RR}_3$ state at $13/5$ and $12/5$ filling numbers.   

Interferometric measurements to test non-Abelian properties of e/4 charges in the fractional quantum Hall state at 5/2

The excitations of charge e/4 at 5/2 filling factor are proposed to obey non-Abelian statistics. To test this, interferometry at fractional quantum Hall states can be performed that controllably braids edge currents around localized charges. We have conducted these measurements in a large number of interferometers of different sizes, also using multiple designs of high quality 2D electron heterostructures. We observe properties of the interference measurements at 5/2 that are specifically consistent with non-Abelian e/4. In particular, magnetic field sweeps around 5/2 show interference oscillations with frequency spectra that are consistent in detail with non-Abelian e/4 properties. Four frequency spectra peaks are observed corresponding to both e/4 and e/2 charges; importantly a rapid non-Abelian e/4 component is seen that is split due to beating between the two e/4 braiding processes. We review these results and their observation in a range of interferometer dimensions and in different heterostructure designs.   

Abelian and non-Abelian states in $\nu=2/3$ bilayer fractional quantum Hall systems

There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\nu=n+2/3$, for integer $n$. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\nu=n+2/3$, including in particular the possibility of the non-Abelian $Z_4$ parafermion state. In $\nu=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the $Z_4$ state. On the other hand, in single-component systems at $\nu=8/3$, we find that the $Z_4$ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\nu=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.   

Particle-hole symmetry without particle-hole symmetry in the quantum Hall effect at $\nu=5/2$.

Numerical results suggest that the quantum Hall effect at $\nu=5/2$ is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau level mixing. In realistic samples both disorder and Landau level mixing are strong on the $5/2$ plateau. The experimental observation of the upstream neutral mode on the sample edge is incompatible with the Pfaffian state. Tunneling experiments give an upper bound on the universal exponent $g$ in the zero bias conductance $G\sim T^{2g-2}$. That bound is inconsistent with the anti-Pfaffian state. We show that a recent proposal of the PH-Pfaffian topological order by Son is compatible with the tunneling experiments and the observation of the upstream mode. The quasiparticle statistics of the PH-Pfaffian state is similar to the statistics in the Pfaffian and anti-Pfaffian states and its interferometric signatures are also similar to those of the Pfaffian and anti-Pfaffian topological orders. The absence of the particle-hole symmetry at $\nu=5/2$ is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.   

Electric Fields in the 5/2 fractional quantum Hall effect

The potential for non-Abelian quasiholes in the 5/2 fractional quantum Hall effect makes the state of interest theoretically and experimentally. The presence of such features in the ground state of the system would allow for the implementation of a topological quantum computation scheme. In order to probe the system for these features, a small measuring voltage, i.e. an electric field, is applied. In Corbino geometries, these electric fields are applied radially. This breaks the Galilean invariance, which in an infinite planar geometry allows us to transform to a moving frame of reference, eliminating the electric field. To study the effects of these fields, we carry out exact diagonalization calculations in a disk geometry. We find that application of small fields can lead to an improvement in the overlap with the Moore-Read Pfaffian long before the state is destroyed by the field. Additionally, we find that the coherence length of quasiholes travelling along the edge of the sample increases significantly when compared to the case with no applied field.   

Probing bulk physics in the 5/2 fractional quantum Hall effect using the Corbino geometry

We present two- and four-point Corbino geometry transport measurements in the second Landau level in GaAs/AlGaAs heterostructures. By avoiding edge transport, we are able to directly probe the physics of the bulk quasiparticles in fractional quantum Hall (FQH) states including 5/2. Our highest-quality sample shows stripe and bubble phases in high Landau levels, and most importantly well-resolved FQH minima in the second Landau level. We report Arrhenius-type fits to the activated conductance, and find that $\sigma_0$ agrees well with theory and existing Hall geometry data in the first Landau level, but not in the second Landau level. We will discuss the advantages the Corbino geometry could bring to various experiments designed to detect the non-Abelian entropy at 5/2, and our progress towards realizing those schemes. The results of these experiments could complement interferometry and other edge-based measurements by providing direct evidence for non-Abelian behaviour of the bulk quasiparticles. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL8500.   

Discovery of competing 5/2 fractional quantum Hall states

With an even denominator, $\backslash $nu $=$ 5/2 fractional quantum Hall state (FQH) is different from most of the other FQH states. Some of its proposed wave functions may exhibit novel non-Abelian statistics, which is related to topological quantum computation. We carried out tunneling measurements within a quantum point contact (QPC) at the 5/2 state and we were able to match the QPC's density to the two-dimensional electron gas bulk density. Such a density match guarantees the uniform filling factor inside and outside the QPC. The interaction parameter g and the effective charge e* can be extracted through the weak tunneling theory [1]. We found g and e* similar to what people believed to be the Abelian 331 state [2, 3]. By tuning the confinement, we observed another region where the experimental data agree well with the weak tunneling theory, which leads to e*$=$0.25 and g$=$0.52, implying non-Abelian wavefunctions such as anti-Pfaffian or U(1)×SU2(2). Our discovery suggests that there are competing 5/2 fractional quantum Hall ground states depending on the confinement. [1] Science 320, 899 (2008). [2] Phys. Rev. B 85, 165321 (2012). [3] Phys. Rev. B 90, 075403 (2014).