The gauge theory dual of the bilayer XY model with second order Josephson coupling

TL;DR

This paper develops a duality transformation for a bilayer XY model with second order Josephson coupling, revealing a phase transition driven by Z2 domain wall loops and providing a new gauge theory perspective on vortex confinement.

Contribution

It introduces a dual gauge theory formulation for the bilayer XY model with second order Josephson coupling, highlighting the role of Z2 domain walls in phase transitions.

Findings

The phase transition is an Ising transition driven by Z2 domain wall loops.

Second order Josephson coupling induces vortex confinement.

Dual gauge theory provides an intuitive understanding of the model's physics.

Abstract

We formulate a duality transformation for a bilayer XY model where the layers are coupled by second order Josephson effect, which favors inter-layer phase difference of either $0$ or $\pi$. The model may represent a bilayer superconductor or a spin-1 ferromagnetic Bose gas in the easy-plane limit. The second order Josephson term is mapped to a U(1) gauge field, known to be trivially confining in two dimensions, and we argue that a Coulomb-gas analysis is not applicable to the dual theory. Instead, we appeal to the vast knowledge of gauge theory and infer that the only phase transition out of low-temperature ordered phase is an Ising transition driven by condensation of $\mathbb{Z}_2$ domain wall loops. The domain wall loops can be seen as a surviving vestige of single-layer vortex-anti-vortex pair, heavily deformed by the second order Josephson coupling. A theoretical or computational method that concentrates on point defects would most likely miss out on these excitations and reach erroneous results. Our dual theory offers a clear, intuitive picture of how the second order Josephson coupling induces confinement of vortices and drastically changes the physics.