non-Abelian Quantum Hall States Research Articles

Anatomy of Abelian and Non-Abelian Fractional Quantum Hall States

We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and non-Abelian fractional quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow us to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read-Rezayi, and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures for these model Hamiltonians.

Non-Abelian Anyons: When Ising Meets Fibonacci

We consider an interface between two non-Abelian quantum Hall states: the Moore-Read state, supporting Ising anyons, and the k=2 non-Abelian spin-singlet state, supporting Fibonacci anyons. It is shown that the interface supports neutral excitations described by a (1+1)-dimensional conformal field theory with a central charge c=7/10. We discuss effects of the mismatch of the quantum statistical properties of the quasiholes between the two sides, as reflected by the interface theory.

Collective States of Interacting Anyons, Edge States, and the Nucleation of Topological Liquids

Quantum mechanical systems, whose degrees of freedom are so-called su(2)k anyons, form a bridge between ordinary SU(2) quantum magnets (of arbitrary spin-S) and systems of interacting non-Abelian anyons. Anyonic spin-1/2 chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit (k-->infinity) of the ordinary spin-1/2 Heisenberg chain. For anyonic spin-1 chains the phase diagram closely mirrors the one of the biquadratic SU(2) spin-1 chain. Our results describe, at the same time, nucleation of different 2D topological quantum fluids within a "parent" non-Abelian quantum Hall state, arising from a macroscopic occupation with localized, interacting anyons. The edge states between the "nucleated" and the parent liquids are neutral, and correspond precisely to the gapless modes of the anyonic chains.

Transition from Abelian to non-Abelian quantum liquids in the second Landau level

The search for non-Abelian quantum Hall states in the second Landau level is narrowed to the range of filling factors 7/3<\nu_e<8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at \nu_e=5/2 (Moore--Read) and 12/5 (Read--Rezayi). Outside of it, the four-flux noninteracting composite fermion model is validated. The borderline \nu_e=7/3 state is adiabatically connected to the Laughlin liquid, but its short-range correlations and charge excitations are different.

Experimental signatures of non-Abelian statistics in clustered quantum Hall states

We discuss transport experiments for various non-Abelian quantum Hall states, including the Read-Rezayi series and a paired spin-singlet state. We analyze the signatures of the unique characters of these states on Coulomb blockaded transport through large quantum dots. We show that the non-Abelian nature of the states manifests itself through modulations in the spacings between Coulomb blockade peaks as a function of the area of the dot. Even though the current flows only along the edge, these modulations vary with the number of quasiholes that are localized in the bulk of the dot. We discuss the effect of relaxation of edge states on the predicted Coulomb blockade patterns, and show that it may suppress the dependence on the number of bulk quasiholes. We predict the form of the lowest-order interference term in a Fabry-P\'erot interferometer for the spin-singlet state. The result indicates that this interference term is suppressed for certain values of the quantum numbers of the collective state of the bulk quasiholes, in agreement with previous findings for other clustered states belonging to the Read-Rezayi series.

Theory of Topological Edges and Domain Walls

We investigate domain walls between topologically ordered phases in two spatial dimensions. We present a method which allows for the determination of the superselection sectors of excitations of such walls and which leads to a unified description of the kinematics of a wall and the two phases to either side of it. This incorporates a description of scattering processes at domain walls which can be applied to questions of transport through walls. In addition to the general formalism, we give representative examples including domain walls between the Abelian and non-Abelian topological phases of Kitaev's honeycomb lattice model in a magnetic field, as well as recently proposed domain walls between spin polarized and unpolarized non-Abelian fractional quantum Hall states at different filling fractions.

BRAIDING AND ENTANGLEMENT IN NONABELIAN QUANTUM HALL STATES

Certain fractional quantum Hall states, including the experimentally observed ν = 5/2 state, and, possibly, the ν = 12/5 state, may have a sufficiently rich form of topological order (i.e. they may be nonabelian) to be useful for quantum information processing. For example, in some cases they may be used for topological quantum computation, an intrinsically fault tolerant form of quantum computation which is carried out by braiding the world lines of quasiparticle excitations in 2+1 dimensional space time. Here we briefly review the relevant properties of nonabelian quantum Hall states and discuss some of the methods we have found for finding specific braiding patterns which can be used to carry out universal quantum computation using them. Recent work on one-dimensional chains of interacting quasiparticles in nonabelian states is also reviewed.

Observable Bulk Signatures of Non-Abelian Quantum Hall States

We show that non-Abelian quantum Hall states can be identified by experimental measurements of the temperature dependence of either the electrochemical potential or the orbital magnetization. The predicted signals of non-Abelian statistics are within experimental resolution, and can be clearly distinguished from other contributions under realistic circumstances. The proposed measurement technique also has the potential to resolve spin-ordering transitions in low density electronic systems in the Wigner crystal and strongly interacting Luttinger liquid regimes.

Probing the Neutral Edge Modes in Transport across a Point Contact via Thermal Effects in the Read-Rezayi Non-Abelian Quantum Hall States

Non-Abelian quantum Hall states are characterized by the simultaneous appearance of charge and neutral gapless edge modes, with the structure of the latter being intricately related to the existence of bulk quasiparticle excitations obeying non-Abelian statistics. Here we propose a scenario for detecting the neutral modes by having two point contacts in series separated by a distance set by the thermal equilibration length of the charge mode. We show that by using the first point contact as a heating device, the excess charge noise measured at the second point contact carries a nontrivial signature of the presence of the neutral mode. We also obtain explicit expressions for the thermal conductance and corresponding Lorentz number for transport across a quantum point contact between two edges held at different temperatures and chemical potentials.

Properties of Non-Abelian Fractional Quantum Hall States at Fillingν=k/r

We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.

Magnetization and Spin Excitations of Non-Abelian Quantum Hall States

Significant insights into non-Abelian quantum Hall states are obtained from studying special multiparticle interaction Hamiltonians, whose unique ground states are the Moore-Read and Read-Rezayi states for the case of spinless electrons. We generalize this approach to include the electronic spin-1/2 degree of freedom. We demonstrate that in the absence of Zeeman splitting, the ground states of such Hamiltonians have large degeneracies and very rich spin structures. The spin structure of the ground states and low-energy excitations can be understood based on an emergent SU(3) symmetry for the case corresponding to the Moore-Read state. These states with different spin quantum numbers represent non-Abelian quantum Hall states with different magnetizations, whose quasihole properties are likely to be similar to those of their spin-polarized counterparts.

Multichannel Kondo Models in Non-Abelian Quantum Hall Droplets

We study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state which is spatially separated from it by an integer quantum Hall state. Near a resonance, the physics at energy scales below the level spacing of the edge states of the dot is governed by a k-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction nu=2+k/(k+2) or its particle-hole conjugate at nu=2+2/(k+2). The k-channel Kondo model is channel isotropic even without fine-tuning in the former state; in the latter, it is generically channel anisotropic. In the special case of k=2, our results provide a new venue, realized in a mesoscopic context, to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction nu=5/2.

Topological properties of Abelian and non-Abelian quantum Hall states classified using patterns of zeros

It has been shown that different Abelian and non-Abelian fractional quantum Hall states can be characterized by patterns of zeros described by sequences of integers ${{S}_{a}}$. In this paper, we will show how to use the data ${{S}_{a}}$ to calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types and their quantum numbers, as well as the actions of the quasiparticle tunneling and modular transformations on the degenerate ground states on torus.

Non-Abelian anyons and topological quantum computation

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Bridge Between Abelian and Non-Abelian Fractional Quantum Hall States

We propose a scheme to construct the most prominent Abelian and non-Abelian fractional quantum Hall states from K-component Halperin wave functions. In order to account for a one-component quantum Hall system, these SU(K) colors are distributed over all particles by an appropriate symmetrization. Numerical calculations corroborate the picture that K-component Halperin wave functions may be a common basis for both Abelian and non-Abelian trial wave functions in the study of one-component quantum Hall systems.

Model Fractional Quantum Hall States and Jack Polynomials

We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a "squeezing rule" that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.

Classification of symmetric polynomials of infinite variables: Construction of Abelian and non-Abelian quantum Hall states

The classification of complex wave functions of infinite variables is an important problem since it is related to the classification of possible quantum states of matter. In this paper, we propose a way to classify symmetric polynomials of infinite variables using the pattern of zeros of the polynomials. Such a classification leads to a construction of a class of simple non-Abelian quantum Hall states which are closely related to parafermion conformal field theories.

Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformalfield theory

We analyze the non-Abelian Read–Rezayi quantum Hall states on the torus, where it isnatural to employ a mapping of the many-body problem onto a one-dimensionallattice model. On the thin torus—the Tao–Thouless (TT) limit—the interactingmany-body problem is exactly solvable. The Read–Rezayi states at fillingν = k/(kM+2) are known to be exact ground states of a local repulsivek+1-bodyinteraction, and in the TT limit this is manifested in that all states in the ground state manifold have exactlyk particleson any kM+2 consecutive sites. For the two-body correlations of these states also imply that there is no more than one particle onM adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domainwalls between the ground states, and we show that the number of distinct domain wallpatterns gives rise to the nontrivial degeneracies, required by the non-Abelianstatistics of these states. In the second part of the paper we consider the quasiholedegeneracies from a conformal field theory (CFT) perspective, and show that thecounting of the domain wall patterns maps one to one on the CFT counting via thefusion rules. Moreover we extend the CFT analysis to topologies of higher genus.

Effect of Landau Level Mixing on Braiding Statistics

We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of Abelian and non-Abelian quantum Hall states. While path dependent geometric phases can perturb the Abelian part of the statistics, we find that the non-Abelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read–Rezayi state whose effective theory is the SU(2) K Chern–Simons theory. As a generalization of the Pfaffian ( K = 2) and the Fibonacci ( K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.