The Broken Symmetry of Two-Component $\nu=1/2$ Quantum Hall States

TL;DR

This paper proposes that the $ u=1/2$ quantum Hall states in bilayer systems are triplet p-wave pairing states of composite fermions, showing a continuous deformation from the (331) state to the Pfaffian state, with implications for electron vortex dynamics.

Contribution

It introduces a triplet p-wave pairing model for $ u=1/2$ quantum Hall states, linking them to $^{3}$He superfluids and explaining their behavior during the two- to one-component crossover.

Findings

$ u=1/2$ states are triplet p-wave pairing states.

Continuous deformation from (331) to Pfaffian state.

Persistence of $ u=5/2$ state explained by triplet pairing.

Abstract

We show that the recently discovered $\nu=1/2$ quantum Hall states in bilayer systems are triplet p-wave pairing states of composite Fermions, of exactly the same form as $^{3}$He superfluids. The observed persistence (though weakening) of the $\nu=1/2$ state in the two- to one-component crossover region corresponds to a continuous deformation of the so-called (331) state towards the ``Pfaffian" state, identical to the well known A to A$_{1}$ transition in $^{3}$He. This deformation also demonstrates the remarkable fact that electrons can release and capture ``vortices" in a continuous and incompressible manner through spin rotations. The broken symmetry of the triplet pairing state is a ``pairing" vector ${\bf d}$. It also implies a (pseudo-spin) magnetization $\propto i{\bf d}\times {\bf d}^{\ast}$. In the presence of layer tunneling, the (331) state ({\bf d} real) is unstable against other states with a magnetization ({\bf d} complex). The recently observed persistence of the $\nu=5/2$ state in single layer systems in the two- to one-component crossover region is also consistent with triplet pairing interpretation.