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arXiv:1811.02646 · Physical review. B./Physical review. B

Particle-hole-symmetric model for a paired fractional quantum Hall state in a half-filled Landau level

William Hutzel, John J. McCord, Peter Raum, Ben Stern, Hao Wang, V. W. Scarola, Michael R. Peterson

Published Jan 14, 2019• 10 citations
arXivHTMLPDF

Abstract

The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.

DOI
10.1103/physrevb.99.045126

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The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.
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.
The Moore-Read Pfaffian and anti-Pfaffian states have been under scrupulous review as candidates which describe the fractional quantum Hall effect at filling factor 5/2. Quantum states in the universality class of the Moore-Read Pfaffian/anti-Pfaffian have non-trivial intrinsic topological order and support low-energy non-Abelian excitations that have applications in fault-tolerant topological quantum computing schemes. Both states are exact ground states of three-body Hamiltonians that explicitly break particle-hole symmetry. We study the topological order of a competing ansatz state &PSgr;2 that is the exact ground state of a two-body Hamiltonian that preserves particle-hole symmetry. In particular, we calculate the bipartite entanglement entropy and spectra in the lowest Landau level in the spherical geometry for &PSgr; 2. We perform such calculations for a finite number of electrons up to 14. We then extrapolate to the thermodynamic limit the topological entanglement entropy γ as a measure of the topological order of the ansatz and compare to the known value of the Moore-Read Pfaffian/anti-Pfaffian state. We also study the orbital entanglement spectra for &PSgr;2 and compare with the Moore-Read Pfaffian and two-body Coulomb ground states. We show that our extrapolation of γ lies within the uncertainty of the known value of γ for the Moore-Read Pfaffian state, and that the orbital entanglement spectra of &PSgr;2 assumes a similar structure to that of the two-body Coulomb interaction.
We propose a derivative operator formed as a function of derivatives of the electron coordinates. When the derivative operator is applied to the Laughlin wave function, two new wave functions in the lowest Landau level at filling factor 1/2 are generated. For systems of 4, 6, and 8 electrons in spherical geometry, it is shown that the first wave function has nearly unity overlap with the particle-hole conjugate of the Moore-Read Pfaffian wave function, therefore together with the Moore-Read Pfaffian state forms a particle-hole conjugate pair. The second wave function has essentially perfect particle-hole symmetry itself, with a positive parity when the number of electron pairs N/2 is an even integer and and a negative parity when N/2 is an odd integer. An equivalent form suggests the first wave function forms a f-wave pairing of composite fermions, and the second wave function forms a p-wave pairing. The corresponding Non-Abelian statistics quasiparticle wave functions are also proposed.
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.
The fractional quantum Hall effect (FQHE) at the filling factor with an even denominator 52\frac{5}{2}25​ occurs despite the expectation, due to the electron statistics, that the denominator must be an odd number. It is believed that the Cooper pairing of underlying quasiparticles, composite fermions (CFs), leads to the explanation of this effect. Such a state should have a Pfaffian form of the BCS wave function (due to the fully polarized spin) and non-Abelian statistics of possible vortexlike excitations (due to the ppp-wave nature of the pairing). Here we expose the origin of pairing by using the effective dipole representation of the problem and show that pairing is encoded in a Hamiltonian that describes the interaction of the charge density with dipoles, i.e., the current of CFs. The necessary condition for the paired state to exist is the effective dipole physics at the Fermi level as a consequence of the nontrivial topology of the ideal band in which electrons live, a Landau level (LL); the paired state is a resolution of the unstable, critical behavior characterized by the distancing of correlation hole with respect to electron (and thus dipole) at the Fermi level due to the topology. We describe analytically this deconfined critical point, at which deconfinement of Majorana neutral fermions takes place. In the presence of large, short-range repulsive interaction inside a LL, the critical behavior may be stabilized into a regularized Fermi-liquid-like state, like the one that characterizes the physics in the lowest LL, but in general, for an interaction with slowly decaying pseudopotentials, the system is prone to pairing.
In this work we propose a parton state as a candidate state to describe the fractional quantum Hall effect in the half-filled second Landau level. The wave function for this parton state is ${\mathcal{P}}_{\mathrm{LLL}}{\mathrm{\ensuremath{\Phi}}}_{1}^{3}{[{\mathrm{\ensuremath{\Phi}}}_{2}^{*}]}^{2}\ensuremath{\sim}{\mathrm{\ensuremath{\Psi}}}_{2/3}^{2}/{\mathrm{\ensuremath{\Phi}}}_{1}$ and in the spherical geometry it occurs at the same flux as the anti-Pfaffian state. This state has a good overlap with the anti-Pfaffian state and with the ground state obtained by exact diagonalization, using the second Landau level Coulomb interaction pseudopotentials for an ordinary semiconductor such as GaAs. By calculating the entanglement spectrum we show that this state lies in the same phase as the anti-Pfaffian state. A major advantage of this parton state is that its wave function can be evaluated for large systems, which makes it amenable to variational calculations. In the Appendix of this work, we have numerically assessed the validity of another candidate state at filling factor $\ensuremath{\nu}=\frac{5}{2}$, namely, the particle-hole-symmetric Pfaffian (PH-Pfaffian) state. We find that the proposed candidate wave function for the PH-Pfaffian state is particle-hole symmetric to a high degree but it does not appear to arise as the ground state of any simple Hamiltonian with two-body interactions.