Universality class and effective field theory of the motility-induced critical point

In this talk I will present the results of large-scale simulations of a two-dimensional model of active particles close to the motility-induced critical point. I will show how finite-size scaling analysis provides exhaustive evidence that the critical behaviour of this active system belongs to the Ising universality class. In addition to the scaling observables that are also typical of passive systems, in the active system we find a kinetic temperature difference emerging between the two cohexisting phases. This quantity, which is always zero in equilibrium, is instead singular in the active system and it is well described by the same exponent of the order parameter in agreement with mean-field theory. In the seond part of the talk I will focus on the critical non-equilibrium dynamics of this model, which is studied by analyzing the breakdown of the Fluctuation-Dissipation Theorem (FDT). The FDT-violation manifests in the short time and wavelength regime, where the response function has a much larger amplitude than the fluctuation spectrum. Conversely, at larger spatiotemporal scales, the FDT is restored and the critical slowing-down is compatible with the dynamic Ising universality class. Building on these results, we develop a novel field-theoretical description employing a space-time correlated noise which qualitatively captures the numerical results already at the Gaussian level. A one-loop renormalization group analysis of such a field theory shows that the correlated noise does not change the critical exponents with respect to the equilibrium. I will discuss how these results demonstrate that a correlated noise field is a fundamental ingredient to capture the features of critical active matter at the coarse-grained level.