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Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level

N. Read, E. Rezayi

TL;DR

The paper extends Pfaffian physics to a family of parafermion quantum Hall states by constructing exact zero-energy ground states of $k+1$-body delta-function Hamiltonians using ${\mathbb Z}_k$ parafermion CFT and OPEs. It provides explicit wavefunctions built from clustered $k$-particle groups, analyzes the quasihole sectors with nonabelian statistics, and counts the degeneracies (e.g., Fibonacci for $k=3$). For the first excited Landau level, it shows large overlaps (up to $~97\%$) between the Coulomb ground state at $\nu=2+3/5$ and the parafermion states, with robustness against short-range perturbations. The results suggest parafermion liquids as viable descriptions of incompressible states in higher Landau levels and offer a framework for understanding nonabelian anyons beyond the Pfaffian case.

Abstract

The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k+1-body interactions, for all integers k > 0. The remarkably simple wavefunctions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2 case is the Pfaffian. For k > 1, the quasiparticle excitations of these systems are expected to possess nonabelian statistics, like those of the Pfaffian. For k=3, these ground states have large overlaps with the ground states of the (2-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors ν=2+3/5, 2+2/5.

Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level

TL;DR

The paper extends Pfaffian physics to a family of parafermion quantum Hall states by constructing exact zero-energy ground states of -body delta-function Hamiltonians using parafermion CFT and OPEs. It provides explicit wavefunctions built from clustered -particle groups, analyzes the quasihole sectors with nonabelian statistics, and counts the degeneracies (e.g., Fibonacci for ). For the first excited Landau level, it shows large overlaps (up to ) between the Coulomb ground state at and the parafermion states, with robustness against short-range perturbations. The results suggest parafermion liquids as viable descriptions of incompressible states in higher Landau levels and offer a framework for understanding nonabelian anyons beyond the Pfaffian case.

Abstract

The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k+1-body interactions, for all integers k > 0. The remarkably simple wavefunctions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2 case is the Pfaffian. For k > 1, the quasiparticle excitations of these systems are expected to possess nonabelian statistics, like those of the Pfaffian. For k=3, these ground states have large overlaps with the ground states of the (2-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors ν=2+3/5, 2+2/5.

Paper Structure

This paper contains 8 sections, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Solution of special $k+1$-body interaction Hamiltonians
  3. Notation and Hamiltonians
  4. Ope's and the Pfaffian state
  5. Parafermion states
  6. Quasihole states and nonabelian statistics
  7. Coulomb interaction in the first excited Landau level
  8. Conclusion

Figures (4)

  • Figure 1: The low-lying spectrum for the four-body Hamiltonian for $N=15$ electrons. The ground state is at $L=0$, $\Delta E=0$.
  • Figure 2: Pair correlation function of the $\nu=3/5$ parafermion state for $N=18$ electrons on the sphere. $r$ is the great-circle distance.
  • Figure 3: Overlap-squared of the two reference states, the $\nu=3/5$ parafermion state ($N=15$) and the $\nu=3/5$ hierarchy state ($N=12$), with the state obtained by diagonalizing the ${\cal N}=1$ Coulomb potential with an added $\delta V_1$ component.
  • Figure 4: Same as Fig. \ref{['fig:specM']} but for the ${\cal N}=1$ Coulomb potential. The ground-state energy has been subtracted in the spectrum.