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arXiv:cond-mat/0601678 · Physical Review B

Wavefunctions and counting formulas for quasiholes of clustered quantum Hall states on a sphere

N. Read

Published Jun 27, 2006• 40 citations
arXivHTMLPDF

Abstract

The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree $k$ of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a $k+1$-body interaction, and so are symmetric polynomials that vanish when any $k+1$ particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each angular momentum. An interesting recursive structure emerges in the states that relates those for $k$ to those for $k\ensuremath{-}1$. It is pointed out that the same numbers of zero-energy states can be proved to occur in certain one-dimensional models that have recently been obtained as limits of the two-dimensional $k+1$-body interaction Hamiltonians, using results from the combinatorial literature.

DOI
10.1103/physrevb.73.245334

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The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree kkk of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a k+1k+1k+1-body interaction, and so are symmetric polynomials that vanish when any k+1k+1k+1 particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each angular momentum. An interesting recursive structure emerges in the states that relates those for kkk to those for k\ensuremath−1k\ensuremath{-}1k\ensuremath−1. It is pointed out that the same numbers of zero-energy states can be proved to occur in certain one-dimensional models that have recently been obtained as limits of the two-dimensional k+1k+1k+1-body interaction Hamiltonians, using results from the combinatorial literature.
We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the conformal field theory (CFT) that describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the Fractional Quantum Hall plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the three-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and ``squeezing.''
Number of zeros seen by a particle around small clusters of other particles is encoded in the root partition, and partly characterizes the correlations in fractional quantum Hall trial wavefunctions. We explore a generalization wherein we consider the counting of zeros seen by a cluster of particles on another cluster. Numbers of such zeros between clusters in the Laughlin wavefunctions are fully determined by the root partition. However, such a counting is unclear for general Jain states where a polynomial expansion is difficult. Here we consider the simplest state beyond the Laughlin wavefunction, namely a state containing a single quasiparticle of the Laughlin state. We show numerically and analytically that in the trial wavefunction for the quasiparticle of the Laughlin state, counting of zeros seen by a cluster on another cluster depends on the relative dimensions of the two clusters. We further ask if the patterns in the counting of zeros extend, in at least an approximate sense, to wavefunctions beyond the trial states. Using numerical computations in systems up to N=9N=9N=9, we present results for the statistical distribution of zeros around particle clusters at the center of an FQH droplet in the ground state of a Hamiltonian that is perturbed away from the V1V_1V1​ interaction (short-range repulsion). Evolution of this distribution with the strength of the perturbation shows that the counting of zeros is altered by even a weak perturbation away from the parent Hamiltonian, though the perturbations do not change the phase of the system.
Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire ZkZ_kZk​ Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.
We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the CFT which describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the FQH plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the 3-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and 'squeezing'.
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We continue the program started by N. Read and one of us and explain the statistics of the excitations for the generalizations of the paired states in the quantum Hall effect in terms of the parafermion statistics. We show that these excitations behave as combinations of bosons and parafermions. That generalizes the prior treatment of the paired (Moore-Read) state where the excitations behave as combinations of bosons and fermions. We explain what it means, from a quantum-mechanical point of view, for a particle to be a ``parafermion'' rather than a boson or a fermion and work through several explicit examples. The resulting multiplets coincide exactly with the angular momentum multiplets found numerically for the k+1k+1k+1-particle interaction Hamiltonian on a sphere. We present proof that the wave functions found previously are indeed the correlation functions of the parafermion conformal field theory. We also obtain the tunneling characteristics into the parafermion edge, which can be used to distinguish the parafermion states from other known states at the same filling.