Nonequilibrium Phase Transition in a Two-Dimensional Driven Open Quantum System

The Berezinskii-Kosterlitz-Thouless mechanism, inwhich a phase transition is mediated by the proliferationof topological defects, governs the critical behaviour of awide range of equilibrium two-dimensional systems with acontinuous symmetry, ranging from superconducting thinfilms to two-dimensional Bose fluids, such as liquid heliumand ultracold atoms. We show here that this phenomenonis not restricted to thermal equilibrium, rather it survivesmore generally in a dissipative highly non-equilibrium sys-tem driven into a steady-state. By considering a light-matter superfluid of polaritons, in the so-called opticalparametric oscillator regime, we demonstrate that it in-deed undergoes a vortex binding-unbinding phase tran-sition. Yet, the exponent of the power-law decay of thefirst order correlation function in the (algebraically) or-dered phase can exceed the equilibrium upper limit – asurprising occurrence, which has also been observed in arecent experiment [1]. Thus we demonstrate that the or-dered phase is somehow more robust against the quantumfluctuations of driven systems than thermal ones in equi-librium.The Hohenberg-Mermin-Wagner theorem prohibits spon-taneous symmetry breaking of continuous symmetries and as-sociated off-diagonal long-range order for systems with short-range interactions at thermal equilibrium in two (or fewer) di-mensions [2]. This is because long-range fluctuations due tothe soft Goldstone mode are so strong as to be able to “shakeapart” any possible long-ranged order. The Berezinskii-Kosterlitz-Thouless (BKT) mechanism (for an overview seeRefs. [3, 4]) provides a loophole to the Hohenberg-Mermin-Wagner theorem: Two-dimensional (2D) systems can still ex-hibit a phase transition between a quasi-long-range orderedphase below a critical temperature, where correlations decayalgebraically and topological defects are bound together, anda disordered phase above such a temperature, where defectsunbind and proliferate, causing exponential decay of correla-tions. Further, it can be shown [5] that the algebraic decayexponent in the ordered phase cannot exceed the upper boundvalue of 1=4.The BKT transition is relevant for a wide class of systems,perhaps the most celebrated examples are those in the contextof 2D superfluids, as in