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arXiv:2103.12026 · Physical review. B./Physical review. B

Stability of the particle-hole Pfaffian state and the 52 -fractional quantum Hall effect

Edward H. Rezayi, Kiryl Pakrouski, F. D. M. Haldane

Published Aug 23, 2021• 23 citations
arXivHTMLPDF

Abstract

After decades since its discovery, a complete understanding of the topological phase of matter at $\frac{5}{2}$ filling appears to be lacking. Previous numerical studies have narrowed the candidates to the Moore-Read Pfaffian or the anti-Pfaffian. Recent thermal transport measurements point to another variant, the particle-hole Pfaffian, which has a nonholomorphic pairing part that complicates comparison with other competing states. Here, the authors have constructed the exact holomorphic particle-hole Pfaffian wave function with full particle-hole symmetry and compared its energetics with the Moore-Read Pfaffian and the anti-Pfaffian.

DOI
10.1103/physrevb.104.l081407

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After decades since its discovery, a complete understanding of the topological phase of matter at 52\frac{5}{2}25​ filling appears to be lacking. Previous numerical studies have narrowed the candidates to the Moore-Read Pfaffian or the anti-Pfaffian. Recent thermal transport measurements point to another variant, the particle-hole Pfaffian, which has a nonholomorphic pairing part that complicates comparison with other competing states. Here, the authors have constructed the exact holomorphic particle-hole Pfaffian wave function with full particle-hole symmetry and compared its energetics with the Moore-Read Pfaffian and the anti-Pfaffian.
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 report on the study of the fractional quantum Hall effect at the filling factor 5/2 using exact diagonalization method with torus geometry. The particle-hole symmetry breaking effect is considered using an additional three-body interaction. Both Pfaffian and anti-Pfaffian states can be the ground state depending on the sign of the three-body interaction. The results of the low-energy spectrum, the wave function overlap, and the particle-hole parity evolution, have shown clear evidence of a direct sharp transition (possibly first order) from the Pfaffian to the anti-Pfaffian state at the Coulomb point. A quantum phase diagram is established, where one finds further transitions from the Pfaffian or anti-Pfaffian state to the nearby compressible phases induced by a change of the pseudopotential.
We numerically assess model wave functions for the recently proposed particle-hole-symmetric Pfaffian (``PH-Pfaffian'') topological order, a phase consistent with the recently reported thermal Hall conductance [M. Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $\ensuremath{\nu}=5/2$ quantum Hall plateau. We find that the most natural Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $\ensuremath{\nu}=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak ppp-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.
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.
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.