Josephson Coupled Moore-Read States

TL;DR

This paper investigates a bilayer quantum Hall system of bosons at total filling factor 1, revealing how pair tunneling induces a topologically ordered phase with unique edge properties, modeled as coupled composite fermion superconductors.

Contribution

Introduces two exactly solvable Hamiltonians demonstrating how pair tunneling induces topological order in a bosonic quantum Hall bilayer system, with a detailed analysis of the resulting phases and edge spectra.

Findings

Pair tunneling gaps out Goldstone mode and induces topological order.

The resulting phase has Halperin 220 topological order in a symmetric/antisymmetric basis.

Edge spectrum exhibits a $U(1)_4 imes U(1)$ structure.

Abstract

We study a quantum Hall bilayer system of bosons at total filling factor $\nu = 1$, and study the phase that results from short ranged pair-tunneling combined with short ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys.~Rev.~Lett.~{\bf 108}, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase while the other is gapless. However, on introduction of a pair tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with $U(1)_2\times U(1)_2$ topological order, but it is realized in the symmetric/antisymmetric-basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected $U(1)_4 \times U(1)$ structure.