The concept of universality pertains to order and classifies a great variety of different physical systems in terms of their common collective behaviour in the long-wavelength limit. While dynamical critical phenomena in equilibrium are by now quite well understood, the extension to non-equilibrium is a relatively new field. At the same time, due to unprecedented experimental progress on a range of light-matter realisations in recent years, particularly in microcavity-polariton systems, there is a particular interest in the collective behaviour of driven-dissipative quantum systems. In these systems energy is not conserved, the detailed balance condition is broken and fluctuation-dissipation relations are not satisfied. Particularly, in two dimensional systems it has been shown that the driven-dissipative nature of the system has a more profound qualitative effect, completely destroying the analogous equilibrium order in some regimes, bringing the system to a different, genuinely non-equilibrium, universality class, the Kardar-Parisi-Zhang (KPZ) class.

Numerical simulations and experiments find many similarities between the transition from the disordered to quasi-ordered phase of 2D driven-dissipative systems (e.g. microcavity polaritons) and the Berezinskii-Kosterlitz-Thouless (BKT) transition in 2D equilibrium systems. Like in the equilibrium BKT mechanism, the quasi-ordered phase is reached by the binding of topological defects (i.e. forming vortex-antivortex pairs). Free vortices destroy order resulting in an exponential decay of first order spacial correlations, while bound vortex-antivortex pairs do not affect the order at long range, giving algebraic decay.

> Binding of vortex-antivortex pairs across phase transition to optical parametric oscillator regime in polaritons (from ref 2.)

The Kardar-Parisi-Zhang (KPZ) equation defines a universality class to which many non-equilibrium systems, from growing bacterial colonies to burning paper, belong. Coherently pumped microcavity polaritons in the optical parametric oscillator regime also represent a paradigmatical example of a system showing a genuine non-equilibrium universality class. Additionally, this system can be driven to different universalities simply by changing the strength of the pumping mechanism in an appropriate parameter range. Thus the system can be driven between (i) the classic algebraically ordered superfluid below the BKT transition, as in equilibrium; (ii) the non-equilibrium KPZ phase; and the two associated topological defect dominated disordered phases caused by proliferation of (iii) entropic BKT vortex-antivortex pairs or (iv) repelling vortices in the KPZ. Some of these different transitions may be observed in current experimental setups, but there are also many cases where, at the length scales of typical experiments, the KPZ phase is indistinguishable from an equilibrium-like algebraically ordered phase.

Microcavity-polariton systems can reveal universal properties in their dynamics when the system is quenched from the highly disordered phase (no polariton condensate) to the quasi-ordered phase (non-zero polariton condensate) by an infinitely rapid quench of one of the driving parameters. In this quench scenario, both incoherently and coherently pumped polariton systems, despite their driven-dissipative nature, fulfil the dynamical scaling hypothesis, exhibiting self-similar patterns for the two-point correlator at late times of the phase ordering. Numerical simulations of the polariton system under both pumping schemes show that, for polaritons, this phase ordering process is characterised by the dynamical critical exponent z≈2, suggestive of the equilibrium-like XY (BKT) universality. Topological defects play a fundamental role in the dynamics, giving logarithmic corrections both to the power-law decay of the number of vortices and to the associated growth of the characteristic length-scale.