Parton construction of a wave function in the anti-Pfaffian phase

Ajit C. Balram, Maissam Barkeshli, and Mark S. Rudner

Phys. Rev. B 98, 035127 – Published 18 July 2018

Abstract

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 $P_{LLL} Φ_{1}^{3} {[Φ_{2}^{*}]}^{2} \sim Ψ_{2 / 3}^{2} / Φ_{1}$ 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 $ν = \frac{5}{2}$ , 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.

Received 2 April 2018
Revised 6 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.035127

Physics Subject Headings (PhySH)

Fractional quantum Hall effect

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ajit C. Balram¹, Maissam Barkeshli², and Mark S. Rudner¹

¹Niels Bohr International Academy and the Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
²Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20472 USA

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Vol. 98, Iss. 3 — 15 July 2018

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Images

Figure 1
Orbital entanglement spectrum of the $Ψ_{1 / 2}^{\bar{2} \bar{2} 111}$ state for $N = 10$ electrons at a flux $2 Q = 21$ on the sphere. The two subsystems $A$ and $B$ with respect to which the entanglement spectrum is calculated each have $N_{A} = N_{B} = 5$ electrons and $l_{A} = l_{B} = 11$ orbitals (top panels) and $l_{A} = 12$ and $l_{B} = 10$ orbitals (bottom panels). The entanglement levels are labeled by the $z$ component of the total orbital angular momentum of the $A$ subsystem, $L_{z}^{A}$ . For comparison, we also show the corresponding entanglement spectra for the anti-Pfaffian state, $Ψ_{1 / 2}^{aPf}$ (right panels). The counting of low-lying levels (blue dashes) in the top panels, from $L_{z}^{A} = 24$ (going from left to right) goes as $1, 1, 3, 5, \dots$ , while in the bottom panels, from $L_{z}^{A} = 25.5$ (going from left to right) goes as $1, 2, 4, \dots$ , and is identical for the states within each row.
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Figure 2
Pair-correlation function $g (r)$ , where $r$ is the chord distance between electrons on the sphere, and its Fourier transform, the structure factor $S (q)$ (inset), in the state $Ψ_{1 / 2}^{\bar{2} \bar{2} 111}$ . The calculation is performed for $N = 100$ electrons on the sphere, with flux $2 Q = 201$ . The pair-correlation function shows a “shoulder”-like feature at intermediate distances, which is typical of non-Abelian states.
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Figure 3
Ground-state orbital angular momentum, $L$ , obtained in the spherical geometry using exact diagonalization at the particle-hole-symmetric Pfaffian flux for a model interaction (see Appendix pp1-s1a for a description of the interaction). Regions of parameter space where the ground state is uniform (i.e., has $L = 0)$ , and therefore compatible with maintaining a finite-energy gap in the thermodynamic limit, are indicated in dark blue shade. In the right panels of the top two rows we show the overlaps of the particle-hole-symmetric Pfaffian trial state $P_{LLL} {[Pf]}^{*} J^{2} = P_{LLL} {[Pf ({{(z_{i} - z_{j})}^{- 1}})]}^{*} \prod_{i < j} {(z_{i} - z_{j})}^{2}$ with the ground state of the model interaction, for $N = 8$ and $N = 10$ particles.
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Figure 4
Ground-state orbital angular momentum, $L$ , obtained in the spherical geometry using exact diagonalization at the particle-hole-symmetric Pfaffian flux for a model interaction (see Appendix pp1-s1b for a description of the interaction). Regions of parameter space where the ground state is uniform (i.e., has $L = 0)$ , and is therefore compatible with maintaining a finite-energy gap in the thermodynamic limit, are indicated in dark blue shade. By perturbing $V_{1}$ and $V_{3}$ around the second Landau level Coulomb point (white dot in the center), we seek conditions where the PH-Pf state is stabilized. In the right panel of the top two rows we show the overlap of the particle-hole-symmetric Pfaffian trial state $P_{LLL} {[Pf]}^{*} J^{2} = P_{LLL} {[Pf ({{(z_{i} - z_{j})}^{- 1}})]}^{*} \prod_{i < j} {(z_{i} - z_{j})}^{2}$ with the ground state of the model interaction. In all cases, the PH-Pf does not appear to describe the ground state well.
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Figure 5
Ground-state orbital angular momentum, $L$ , obtained in the spherical geometry using exact diagonalization at the particle-hole-symmetric Pfaffian flux for a model interaction (see Appendix pp1-s1b for a description of the interaction). Regions of parameter space where the ground state is uniform (i.e., has $L = 0)$ , and is therefore compatible with maintaining a finite-energy gap in the thermodynamic limit, are indicated in dark blue shade. By perturbing $V_{1}$ and $V_{3}$ around the lowest Landau level Coulomb point (white dot in the center), we seek conditions where the PH-Pf state is stabilized. In the right panel of the top two rows we show the overlap of the particle-hole-symmetric Pfaffian trial state $P_{LLL} {[Pf]}^{*} J^{2} = P_{LLL} {[Pf ({{(z_{i} - z_{j})}^{- 1}})]}^{*} \prod_{i < j} {(z_{i} - z_{j})}^{2}$ with the ground state of the model interaction. In all cases, the PH-Pf does not appear to describe the ground state well.
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