Lattice duality for the compact Kardar-Parisi-Zhang equation

TL;DR

This paper develops a duality transformation that maps the compact KPZ equation, describing driven-dissipative condensate phases, to an electrodynamic theory, enhancing understanding of vortex dynamics in non-equilibrium systems.

Contribution

It introduces a novel duality mapping from the compact KPZ equation to an electrodynamic framework, extending vortex unbinding concepts to driven-dissipative systems.

Findings

Derived a transformation linking the compact KPZ equation to a dual electrodynamic theory.

Formulated modified Maxwell equations for vortex dynamics in driven-dissipative condensates.

Utilized an adapted Villain approximation within a functional integral approach.

Abstract

A comprehensive theory of the Kosterlitz-Thouless transition in two-dimensional superfluids in thermal equilibrium can be developed within a dual representation which maps vortices in the superfluid 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. The physics of vortex unbinding in driven-dissipative systems such as fluids of light, on the other hand, is much less understood. Here we make a crucial step to fill this gap by deriving a transformation that maps the compact Kardar-Parisi-Zhang (KPZ) equation, which describes the dynamics of the phase of a driven-dissipative condensate, to a dual electrodynamic theory. The latter is formulated in terms of modified Maxwell equations for the electromagnetic fields and a diffusion equation for the charges representing vortices in the KPZ equation. This mapping utilizes an adaption of the Villain approximation to a generalized Martin-Siggia-Rose functional integral representation of the compact KPZ equation on a lattice.