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

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 $π$. 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.