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arXiv:cond-mat/9809384 · Physical review. B, Condensed matter

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

N. Read, Edward H. Rezayi

Published Mar 15, 1999• 943 citations
arXivHTMLPDF

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 $k>~1.$ The remarkably simple wave functions 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 $k>~2,$ the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For $k=3,$ these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors $\ensuremath{\nu}=2+3/5,2+2/5.$

DOI
10.1103/physrevb.59.8084

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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+1k+1k+1-body interactions, for all integers k> 1.k>~1.k> 1. The remarkably simple wave functions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2k=2k=2 case is the Pfaffian. For k> 2,k>~2,k> 2, the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For k=3,k=3,k=3, these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors \ensuremathν=2+3/5,2+2/5.\ensuremath{\nu}=2+3/5,2+2/5.\ensuremathν=2+3/5,2+2/5.
The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a family of incompressible paired parton FQH states at filling $\ensuremath{\nu}=\frac{1}{6}$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\ensuremath{\nu}=\frac{1}{3}$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-type anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principal existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimensions.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, 4/54/54/5, and 4/34/34/3, and partially spin-polarized states at filling factors 3/43/43/4 and 3/23/23/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, 4/54/54/5, and 3/43/43/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.
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.
In this review, the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ([Formula: see text]-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the particle–hole (PH) Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern–Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.
Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level.We consider filling fractions ν = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed.Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential.Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ν = 1/2 ("Pfaffian") and ν = 2/5 ("parafermion" state).
It is demonstrated that an understanding of the 5/2 fractional quantum Hall effect can be achieved within the composite fermion theory without appealing to the Pfaffian wave function. The residual interaction between composite fermions plays a crucial role in establishing incompressibility at this filling factor. This approach has the advantage of being amenable to systematic perturbative improvements, and produces ground as well as excited states. It, however, does not relate to non-Abelian statistics in any obvious manner.