## Periodic Report Summary 2 - CORPHO (Theory of strongly correlated photonic systems)

Driven-dissipative quantum systems are attracting a considerable interest for the richness of their fundamental physics and also for possible applications in quantum technologies. In the case of lattice systems, the theoretical challenge is represented by the quickly prohibitive dimension of the Hilbert space, which increases exponentially with the number of lattice sites. Moreover, the non-equilibrium nature of the steady-state phases can add to the complexity. The project CORPHO is developing and applying original theoretical methods to describe such systems, with a particular focus on the physics of strongly correlated photonic systems. The project CORPHO has led to the development of the corner-space renormalization method, an approach which can be applied to determine the steady-state density matrix both for one and two-dimensional lattices. Such a method is based on an iterative spatial renormalization procedure where a relatively small subspace of the Hilbert space, the so-called 'corner', is selected by maximizing joint probabilities obtained from the density matrices of smaller blocks. Solving the master equation in the corner space can provide an accurate approximation of the physical observables. The method is controlled, as the convergence of the results can be checked by increasing the size of the corner space. A second approach which has been developed is a truncated correlation hierarchy scheme for driven-dissipative multimode quantum systems. Moreover, exact analytical results have been obtained for a few models using the formalism of the complex-P representation. The original theoretical methods developed in the project have been applied to explore the physics of complex photonic lattices with geometric frustration such as the Lieb lattice: it has been shown that an inhomogeneous driving, where only a fraction of the sites is pumped, can lead to strong correlations even when the value of the on-site interaction is nominally small. Currently, CORPHO is focusing on the physics of dissipative phase transitions. Via the corner-space renormalization method, we have explored a ferromagnetic dissipative transition in the anisotropic Heisenberg model in two-dimensional lattices: we have determined the first estimate of the critical exponents, the behavior of entropy across the critical point and characterized entanglement via the quantum Fisher information. Interesting results have been obtained concerning the power-law properties of dynamical optical hysteresis in Kerr quantum nonlinear systems and the closing of the Liouvillian gap in a well defined thermodynamical limit. Current studies are aimed at further exploring the collective critical phenomena in quantum lattice systems and the active control of such phases.