Mean-field validity in a dissipative critical system: Liouvillian gap, $\mathbb{PT}$-symmetric antigap, and permutational symmetry in the $XYZ$ model

TL;DR

This paper investigates the validity of mean-field theory in a dissipative all-to-all connected XYZ spin model, analyzing phase transitions, symmetries, and dissipative effects through exact solutions and numerical methods up to 95 spins.

Contribution

It provides the first exact solutions for large spin systems with permutational symmetry, introduces the concept of an antigap, and compares mean-field predictions with full quantum solutions in a dissipative setting.

Findings

Mean-field theory agrees with exact results for small anisotropy.

Significant deviations occur at large anisotropy, indicating limits of mean-field approximation.

The phase transition is robust against local and collective dissipation effects.

Abstract

We study the all-to-all connected $XYZ$ (anisotropic-Heisenberg) spin model with local and collective dissipations, comparing the results of mean field theory with the solution of the Lindblad quantum evolution. Leveraging the permutational symmetry of the model [N. Shammah et al., Phys Rev. A 98, 063815 (2018)], we find exactly (up to numerical precision) the steady state up to $N=95$ spins. We characterize criticality, studying, as a function of the number of spins $N$, the spin structure factor, the magnetization, the Liouvillian gap and the Von Neumann entropy of the steady state. Exploiting the weak $\mathcal{PT}$-symmetry of the model, we efficiently calculate the Liouvillian gap, introducing the idea of an antigap. For small anisotropy, we find a paramagnetic-to-ferromagnetic phase transition in agreement with the mean-field theory. For large anisotropy, instead, we find a significant discrepancy from the scaling of the low-anisotropy ferromagnetic phase. 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.