Lattice duality for the compact Kardar-Parisi-Zhang equation

Abstract

Duality transformations have a long history in physics, and recently saw a surge of renewed interest in the context of strongly correlated fermion systems. A prime example for such a transformation was established to describe the phase transition in two-dimensional superfluids in thermal equilibrium. Here, Kosterlitz and Thouless developed a dual representation of the superfluid, which maps the vortices in the latter to charges in a Coulomb gas. In this framework, the dissociation of vortex-antivortex pairs at the critical temperature corresponds to the formation of a plasma of free charges. How can such a framework be leveraged over to nonequilibrium situations, relevant to the understanding of driven open fluids of light such as exciton-polariton systems? In this work, the authors make a crucial step in this direction by deriving a transformation that maps the stochastic equation of motion for the phase of a driven open condensate --- the compact Kardar-Parisi-Zhang (cKPZ) equation --- to a dual electrodynamic theory. This results in modified, and in particular, nonlinear Maxwell equations for the electromagnetic fields, and a diffusion equation for the charges representing vortices in the cKPZ equation. In a companion paper, the authors apply this theoretical framework to the study of nonequilibrium vortex unbinding.