Numerical exploration of trial wave functions for the particle-hole-symmetric Pfaffian

TL;DR

This study numerically evaluates the PH-Pfaffian wave functions in the quantum Hall effect, revealing their similarities to composite Fermi liquids and questioning their suitability as ground state models at filling factor 5/2.

Contribution

It provides the first detailed numerical assessment of the PH-Pfaffian wave function's properties and its relation to known phases in the quantum Hall regime.

Findings

PH-Pfaffian trial state resembles composite Fermi liquid at accessible sizes.

It performs reasonably well energetically but may not represent the true ground state.

The wave function may encode weak pairing or lack a gapped phase.

Abstract

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 [Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $\nu=5/2$ 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 $\nu=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak $p$-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.