Properties of Non-Abelian Fractional Quantum Hall States at Filling ν=kr\nu=\frac{k}{r}

We compute the physical properties of non-Abelian Fractional Quantum Hall (FQH) states described by Jack polynomials at general filling $\nu=\frac{k}{r}$. For $r=2$, these states are identical to the $Z_k$ 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 show that the non-Abelian quasihole has a well-defined propagator falling off with the distance. The clustering properties of the Jack polynomials, provide a strong indication that the states with $r>2$ can be obtained as correlators of fields of \emph{non-unitary} conformal field theories, but the CFT-FQH connection fails when invoked to compute physical properties such as thermal Hall coefficient or, more importantly, the quasihole propagator. The quasihole wavefuntion, when written as a coherent state representation of Jack polynomials, has an identical structure for \emph{all} non-Abelian states at filling $\nu=\frac{k}{r}$.Comment: 2 figure

Spinon-Holon Attraction in the Supersymmetric t-J Model with 1/r^2-Interaction

We derive the coordinate representation of the one-spinon one-holon wavefunction for the supersymmetric $t-J$ model with $1/r^2$-interaction. This result allows us to show that spinon and holon attract each other at short distance. The attraction gets stronger as the size of the system is increased and, in the thermodynamic limit, it is responsible for the square root singularity in the hole spectral function.Comment: 4 pages, 1 .eps figur

The 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 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.Comment: modify comment in Ref. 1

Central Charge and Quasihole Scaling Dimensions From Model Wavefunctions: Towards Relating Jack Wavefunctions to W-algebras

We present a general method to obtain the central charge and quasihole scaling dimension directly from groundstate and quasihole wavefunctions. Our method applies to wavefunctions satisfying specific clustering properties. We then use our method to examine the relation between Jack symmetric functions and certain W-algebras. We add substantially to the evidence that the (k,r) admissible Jack functions correspond to correlators of the conformal field theory W_k(k+1,k+r), by calculating the central charge and scaling dimensions of some of the fields in both cases and showing that they match. For the Jacks described by unitary W-models, the central charge and quasihole exponents match the ones previously obtained from analyzing the physics of the edge excitations. For the Jacks described by non-unitary W-models the central charge and quasihole scaling dimensions obtained from the wavefunctions differ from the ones obtained from the edge physics, which instead agree with the "effective" central charge of the corresponding W-model.Comment: 22 pages, no figure

Highest weight Macdonald and Jack Polynomials

Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant under the rotation group. Excited quasiparticle/quasihole states are member of multiplets under the rotation group and generically there is a nontrivial highest weight member of the multiplet from which all states can be constructed. Some of the trial states proposed in the literature belong to classical families of symmetric polynomials. In this paper we study Macdonald and Jack polynomials that are highest weight states. For Macdonald polynomials it is a (q,t)-deformation of the raising angular momentum operator that defines the highest weight condition. By specialization of the parameters we obtain a classification of the highest weight Jack polynomials. Our results are valid in the case of staircase and rectangular partition indexing the polynomials.Comment: 17 pages, published versio

Hierarchical structure in the orbital entanglement spectrum in Fractional Quantum Hall systems

We investigate the non-universal part of the orbital entanglement spectrum (OES) of the nu = 1/3 fractional quantum Hall effect (FQH) ground-state with Coulomb interactions. The non-universal part of the spectrum is the part that is missing in the Laughlin model state OES whose level counting is completely determined by its topological order. We find that the OES levels of the Coulomb interaction ground-state are organized in a hierarchical structure that mimic the excitation-energy structure of the model pseudopotential Hamiltonian which has a Laughlin ground state. These structures can be accurately modeled using Jain's "composite fermion" quasihole-quasiparticle excitation wavefunctions. To emphasize the connection between the entanglement spectrum and the energy spectrum, we also consider the thermodynamical OES of the model pseudopotential Hamiltonian at finite temperature. The observed good match between the thermodynamical OES and the Coulomb OES suggests a relation between the entanglement gap and the true energy gap.Comment: 16 pages, 19 figure

Decomposition of fractional quantum Hall states: New symmetries and approximations

We provide a detailed description of a new symmetry structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall states first obtained in Ref. 1, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The symmetry rules in Ref. 1 as well as the ones we obtain for the spin singlet states allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size). We show that these overlaps reach unity in the thermodynamic limit even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state which can be written as an expectation value of parafermionic operators.Comment: 22 pages, 8 figure

Scaling and non-Abelian signature in fractional quantum Hall quasiparticle tunneling amplitude

We study the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is consistent with the scaling anaysis; this allows for the extraction of the conformal dimensions of the quasiparticles. We analyze the scaling behavior of both Abelian and non-Abelian quasiparticles in the Read-Rezayi Z_k-parafermion states. Interestingly, the non-Abelian quasiparticle tunneling amplitudes exhibit nontrivial k-dependent corrections to the scaling exponent.Comment: 16 pages, 4 figure

Entanglement entropy of integer Quantum Hall states in polygonal domains

The entanglement entropy of the integer Quantum Hall states satisfies the area law for smooth domains with a vanishing topological term. In this paper we consider polygonal domains for which the area law acquires a constant term that only depends on the angles of the vertices and we give a general expression for it. We study also the dependence of the entanglement spectrum on the geometry and give it a simple physical interpretation.Comment: 8 pages, 6 figure

Supersymmetric Quantum Hall Liquid with a Deformed Supersymmetry

We construct a supersymmetric quantum Hall liquid with a deformed supersymmetry. One parameter is introduced in the supersymmetric Laughlin wavefunction to realize the original Laughlin wavefunction and the Moore-Read wavefunction in two extremal limits of the parameter. The introduced parameter corresponds to the coherence factor in the BCS theory. It is pointed out that the parameter-dependent supersymmetric Laughlin wavefunction enjoys a deformed supersymmetry. Based on the deformed supersymmetry, we construct a pseudo-potential Hamiltonian whose groundstate is exactly the parameter-dependent supersymmetric Laughlin wavefunction. Though the SUSY pseudo-potential Hamiltonian is parameter-dependent and non-Hermitian, its eigenvalues are parameter-independent and real.Comment: 14 pages, contribution to the proceedings of the Group 27 conference, Yerevan, Armenia, August 13-19, 2008, published versio