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.

The Berezinskii-Kosterlitz-Thouless phase transition from a disordered to a quasi-ordered state, mediated by the proliferation of topological defects in two dimensions, governs seemingly remote physical systems ranging from liquid helium, ultracold atoms and superconducting thin films to ensembles of spins. Here we observe such a transition in a short-lived gas of exciton-polaritons, bosonic light-matter particles in semiconductor microcavities. The observed quasi-ordered phase, characteristic for an equilibrium two-dimensional bosonic gas, with a decay of coherence in both spatial and temporal domains with the same algebraic exponent, is reproduced with numerical solutions of stochastic dynamics, proving that the mechanism of pairing of the topological defects (vortices) is responsible for the transition to the algebraic order. This is made possible thanks to long polariton lifetimes in high-quality samples and in a reservoir-free region. Our results show that the joint measurement of coherence both in space and time is required to characterize driven-dissipative phase transitions and enable the investigation of topological ordering in open systems.

We study the spin vortices and skyrmions coherently imprinted into an exciton-polariton condensate on a planar semiconductor microcavity. We demonstrate that the presence of a polarization anisotropy can induce a complex dynamics of these structured topologies, leading to the twist of their circuitation on the Poincaré sphere of polarizations. The theoretical description of the results carries the concept of generalized quantum vortices in two-component superfluids, which are conformal with polarization loops around an arbitrary axis in the pseudospin space.quantum vortices | topology | Poincaré space | condensates | polaritons T opological defects represent a wide class of objects relevant to different fields of physics from condensed matter to cosmology. The universality of monopoles, vortices, skyrmions, domain walls, and of their formation processes in different systems, has largely motivated their study in the condensed matter context. In particular, the interplay between the symmetry breaking in phase transitions and the formation of topological defects has been the focus of intensive research in the last century. In high-energy physics, the existence of an isolated point source intrigued a great number of physicists (1). Dirac was "surprised if Nature had made no use of it" and postulated the possibility of the magnetic monopoles linked to the quantization of electric charge (2). However, the elusiveness of their observation in free space has motivated an extensive study of monopole analogues in the form of quasiparticles in many-body systems (3) such as the exotic spin ices (4, 5), liquid crystals (6), exciton-polariton (7), and rubidium Bose-Einstein condensates (BECs) (8, 9), as well as other systems (10, 11). In a 2D multicomponent BEC, an equivalent topological structure to the monopole is given by the hedgehog polarization vortex (12), which together with the hyperspin vortex (13) belongs to the class of spin vortices that when combined lead to a well-defined polarization pattern. Such topological states are characterized by a linear polarization vector which rotates an integer number of times (spin winding number) around a singular central point, in a way analogous to what the magnetization does in the spin vortices of a ferromagnetic spinor BEC (14).In this work we excite complex vortex states in an excitonpolariton superfluid and study their amplitude, phase, and polarization dynamics. We demonstrate that temporal evolution of such topologies leads, in general, to a twist of the polarization plane in the Poincaré space. The observed features of the vortex behavior are explained within the concept of generalized spin vortices, where the rotation of polarization at large distances occurs around an arbitrary axis on the Poincaré sphere.Polaritons emerge in planar semiconductor microcavities as eigenmodes of the strong coupling regime between the exciton resonance and the photon cavity mode, combining the properties of light and matter. The photons confer on polaritons a very small effective mass (10...