The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is mediated by the proliferation of topological defects, governs the critical behavior of a wide range of equilibrium twodimensional systems with a continuous symmetry, ranging from spin systems to superconducting thin films and two-dimensional Bose fluids, such as liquid helium and ultracold atoms. We show here that this phenomenon is not restricted to thermal equilibrium, rather it survives more generally in a dissipative highly nonequilibrium system driven into a steady state. By considering a quantum fluid of polaritons of an experimentally relevant size, in the so-called optical parametric oscillator regime, we demonstrate that it indeed undergoes a phase transition associated with a vortex binding-unbinding mechanism. Yet, the exponent of the power-law decay of the first-order correlation function in the (algebraically) ordered phase can exceed the equilibrium upper limit: this shows that the ordered phase of driven-dissipative systems can sustain a higher level of collective excitations before the order is destroyed by topological defects. Our work suggests that the macroscopic coherence phenomena, observed recently in interacting twodimensional light-matter systems, result from a nonequilibrium phase transition of the Berezinskii-Kosterlitz-Thouless rather than the Bose-Einstein condensation type.

In a recent paper [1] Cariñena et al analyzed a non-polynomial onedimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schrödinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator. We then use this observation to generate an infinite set of potentials which can be exactly solved in a similar manner.

Two-dimensional fluid of polaritons sheds light on quantum vortex dynamics.