Monte Carlo evaluation of non-Abelian statistics

We develop a general framework to (numerically) study adiabatic braiding of quasiholes in fractional quantum Hall systems. Specifically, we investigate the Moore-Read (MR) state at nu=1/2 filling factor, a known candidate for non-Abelian statistics, which appears to actually occur in nature. The non-Abelian statistics of MR quasiholes is demonstrated explicitly for the first time, confirming the results predicted by conformal field theories.     

A plasma analogy and Berry matrices for non-abelian quantum Hall states

We present an approach to the computation of the non-abelian statistics of quasiholes in quantum Hall states, such as the Pfaffian state, whose wavefunctions are related to the conformal blocks of minimal model conformal field theories. We use the Coulomb gas construction of these conformal field theories to formulate a plasma analogy for the quantum Hall states. A number of properties of the Pfaffian state follow immediately, including the Berry phases, which demonstrate the quasiholes' fractional charge, the abelian statistics of the two-quasihole state, and equal-time ground state correlation functions. The non-abelian statistics of multi-quasihole states follows from an additional assumption.     

Topological entanglement in Abelian and non-Abelian excitation eigenstates

Entanglement in topological phases of matter has so far been investigated through the perspective of their ground-state wave functions. In contrast, we demonstrate that the excitations of fractional quantum Hall (FQH) systems also contain information to identify the system's topological order. Entanglement spectrum of the FQH quasihole (QH) excitations is shown to differentiate between the conformal field theory (CFT) sectors, based on the relative position of the QH with respect to the entanglement cut. For Read-Rezayi model states, as well as Coulomb interaction eigenstates, the counting of the QH entanglement levels in the thermodynamic limit matches exactly the CFT counting, and sector changes occur as non-Abelian quasiholes successively cross the entanglement cut.     

Nonabelions in the fractional quantum hall effect

Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.     

Nature of excitations of the 5/2 fractional quantum Hall effect

It is shown, with the help of exact diagonalization studies on systems with up to 16 electrons, in the presence of up to two delta function impurities, that the Pfaffian model is not accurate for the actual quasiholes and quasiparticles of the 5/2 fractional quantum Hall effect. Implications for non-Abelian statistics are discussed.     

Qubit Approach to the Topological Degeneracy of 4 and 6-Quasihole Pfaffian Paired Hall State

By virtue of the two-layer picture of Pfaffian pair Hall state, a qubit representation of topological degeneracy for quasiholes excitation is displayed. The non-Abelian feature of states can be manifested readily by the new wave functions. The virtue of this approach is that one does not need to find the equalities of Pfaffians, which is a so tedious task as exemplified for the case of six quasiholes. Then the braiding matrices are also constructed readily by just permutating single-qubit states, which are unitary and hermite.     

Topological invariants for fractional quantum hall states

We calculate a topological invariant, whose value would coincide with the Chern number in the case of integer quantum Hall effect, for fractional quantum Hall states. In the case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In the case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.     

Wavefunctions for topological quantum registers

We present explicit wavefunctions for quasi-hole excitations over a variety of non-abelian quantum Hall states: the Read-Rezayi states with k ? 3 clustering properties and a paired spin-singlet quantum Hall state. Quasi-holes over these states constitute a topological quantum register, which can be addressed by braiding quasi-holes. We obtain the braid properties by direct inspection of the quasi-hole wavefunctions. We establish that the braid properties for the paired spin-singlet state are those of ‘Fibonacci anyons’, and thus suitable for universal quantum computation. Our derivations in this paper rely on explicit computations in the parafermionic conformal field theories that underly these particular quantum Hall states.     

Non-abelian spin-singlet quantum Hall states: wave functions and quasihole state counting

We investigate a class of non-abelian spin-singlet (NASS) quantum Hall phases, proposed previously. The trial ground and quasihole excited states are exact eigenstates of certain (k+1)-body interaction Hamiltonians. The k=1 cases are the familiar Halperin abelian spin-singlet states. We present closed-form expressions for the many-body wave functions of the ground states, which for k>1 were previously defined only in terms of correlators in specific conformal field theories. The states contain clusters of k electrons, each cluster having either all spins up, or all spins down. The ground states are non-degenerate, while the quasihole excitations over these states show characteristic degeneracies, which give rise to non-abelian braid statistics. Using conformal field theory methods, we derive counting rules that determine the degeneracies in a spherical geometry. The results are checked against explicit numerical diagonalization studies for small numbers of particles on the sphere.     

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.     

Fractional quantum Hall regime of a gas of ultracold atoms

We study the physics of a rapidly rotating gas of ultracold bosonic atoms. In the limit of very rapid rotation of the trap the system exhibits a fractional quantum Hall regime analogous to that of electrons in the fractional quantum Hall effect. We show that the ground state of the system is a 1/2-Laughlin liquid, a highly correlated atomic liquid. Exotic excitations consisting of localized quasiholes of 1/2 of an atom can be created by focusing lasers at the desired positions. We show how to manipulate these quasiholes in order to probe directly their 1/2-statistics.     

Detecting non-Abelian statistics of Majorana fermions in quantum nanowire networks

A scheme in semiconducting quantum nanowire structure has been proposed to demonstrate the non-Abelian statistics for Majorana fermions in terms of braid group. The Majorana fermions are localized at the endpoints of semiconducting wires, which are deposited on an s-wave superconductor. The non-Abelian nature of Majorana fermion is manifested by the fact that the output of the different applied orders of two operations, constructed by the braid group elements, are different. In particular, the difference can be unambiguously imprinted on the quantum states of a superconducting flux qubit.     

Non-Abelian braiding of lattice bosons

We report on a numerical experiment in which we use time-dependent potentials to braid non-Abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where ν, the ratio of particles to flux quanta, is near 1/2, 1, or 3/2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for ν near 1 and 3/2, with Berry matrices, respectively, consistent with Ising and Fibonacci anyons. Near ν=1/2, the braids commute.     

Rectangular amplitudes,conformal blocks,and applications to loop models

In this paper we continue the investigation of partition functions of critical systems on a rectangle initiated in [R. Bondesan, et al., Nucl. Phys. B 862 (2012) 553–575]. Here we develop a general formalism of rectangle boundary states using conformal field theory, adapted to describe geometries supporting different boundary conditions. We discuss the computation of rectangular amplitudes and their modular properties, presenting explicit results for the case of free theories. In a second part of the paper we focus on applications to loop models, discussing in details lattice discretizations using both numerical and analytical calculations. These results allow to interpret geometrically conformal blocks, and as an application we derive new probability formulas for self-avoiding walks.     

Fusion and braiding in W-algebra extended conformal theories

We define the chiral conformal blocks of integer-spin extended (W-algebra) conformal theories by the fusion of elementary ones. The braid group representation matrices which realize the exchange algebra are computed. They are shown to coincide with the Boltzmann weights — in a certain limit of the spectral parameter — of the critical face models of Jimbo et al. In the unitary cases, where the extended conformal theories can be realized as cosets , we relate the braiding matrices of the former to those of the WZW models. In this article we restrict ourselves to the case corresponding to symmetric tensor representations of A_n.     

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.     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.