We explore critical properties of two-dimensional lattices of spins interacting via an anisotropic Heisenberg Hamiltonian and subject to incoherent spin flips. We determine the steady-state solution of the master equation for the density matrix via the corner-space renormalization method. We investigate the finite-size scaling and critical exponent of the magnetic linear susceptibility associated to a dissipative ferromagnetic transition. We show that the Von Neumann entropy increases across the critical point, revealing a strongly mixed character of the ferromagnetic phase. Entanglement is witnessed by the quantum Fisher information which exhibits a critical behavior at the transition point, showing that quantum correlations play a crucial role in the transition.

We study an array of coupled optical cavities in presence of two-photon driving and dissipation. The system displays a critical behavior similar to that of a quantum Ising model at finite temperature. Using the corner-space renormalization method, we compute the steady-state properties of finite lattices of varying size, both in one-and two-dimensions. From a finite-size scaling of the average of the photon number parity, we highlight the emergence of a critical point in regimes of small dissipations, belonging to the quantum Ising universality class. For increasing photon loss rates, a departure from this universal behavior signals the onset of a quantum critical regime, where classical fluctuations induced by losses compete with long-range quantum correlations.The emergence of critical phenomena in the nonequilibrium steady state (NESS) of open quantum systems [1-3], arising from the competition between the incoherent and coherent dynamics, is a topic that has gathered increasing attention in recent years, especially in view of the possible experimental realization of model systems using circuit QED [4][5][6], lattices of ultracold atoms [7-10], or other advanced quantum platforms [11,12]. Several theoretical studies have highlighted the possibility of dissipative phase transitions in various many-body systems [1][2][3][4][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], possibly displaying novel universal properties [33]. In this regard, the question about the role played by quantum fluctuations in these critical phenomena is still a matter of debate.Bosonic systems on a lattice have been the object of several studies, motivated by the analogy with the Hamiltonian Bose-Hubbard model and by the possibility to realize experimental studies using arrays of optical cavities with a third-order optical nonlinearity [26][27][28][29][30][34][35][36][37][38][39]. In recent years, the driven-dissipative Bose-Hubbard model in presence of a two-photon -i.e. quadratic in the field -driving term has been studied both theoretically [19,25,[40][41][42][43][44][45][46][47][48][49] and experimentally [50,51]. This quadratically driven scheme has been in particular proposed as a possible realization of a noise-resilient quantum code, where photonic Schrödinger's cat states with even and odd photon number parity behave as interacting spin degrees of freedom [42][43][44][45]50]. This finding is suggestive of a possible scheme for realizing a photonic simulator of quantum spin models [52] and allows for a completely novel approach to the study of dissipative phase transitions. Indeed, while widely studied one-photon driving breaks the U (1) symmetry of the Hamiltonian, twophoton driving preserves a Z 2 symmetry which can be spontaneously broken [25,49], giving rise to a second order phase transition that is similar to that of the quantum transverse Ising model [53].The analogy to a quantum spin model leads to expect that, in the limit of low losses, this model should dis-play a quantum critical point ...