Validity of mean-field theory in a dissipative critical system: Liouvillian gap, PT -symmetric antigap, and permutational symmetry in the XYZ model

Abstract

We study the all-to-all connected $\mathit{XYZ}$ (anisotropic-Heisenberg) spin model with local and collective dissipations, comparing the results of mean-field (MF) theory with the solution of the Lindblad master equation. Exploiting the weak $\mathcal{PT}$ symmetry of the model (referred to as Liouvillian $\mathbb{PT}$ symmetry), we efficiently calculate the Liouvillian gap, introducing the idea of an antigap, and we demonstrate the presence of a paramagnetic-to-ferromagnetic phase transition. Leveraging the permutational symmetry of the model [N. Shammah et al., Phys Rev. A 98, 063815 (2018)], we characterize criticality, finding exactly (up to numerical precision) the steady state for $N$ up to $N=95$ spins. We demonstrate that the MF theory correctly predicts the results in the thermodynamic limit in all regimes of parameters, and quantitatively describes the finite-size behavior in the small anisotropy regime. However, for an intermediate number of spins and for large anisotropy, we find a significant discrepancy between the results of the MF theory and those of the full quantum simulation. We also study other more experimentally accessible witnesses of the transition, which can be used for finite-size studies, namely, the bimodality coefficient and the angular-averaged susceptibility. In contrast to the bimodality coefficient, the angular-averaged susceptibility fails to capture the onset of the transition, in striking difference with respect to lower-dimensional studies. We also analyze the competition between local dissipative processes (which disentangle the spin system) and collective dissipative ones (generating entanglement). The nature of the phase transition is almost unaffected by the presence of these terms. Our results mark a stark difference with the common intuition that an all-to-all connected system should fall onto the mean-field solution also for intermediate number of spins.