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We study the all-to-all connected $XYZ$ (anisotropic-Heisenberg) spin modelwith local and collective dissipations, comparing the results of mean fieldtheory with the solution of the Lindblad quantum evolution. Leveraging thepermutational 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 thenumber of spins $N$, the spin structure factor, the magnetization, theLiouvillian gap and the Von Neumann entropy of the steady state. Exploiting theweak $\mathcal{PT}$-symmetry of the model, we efficiently calculate theLiouvillian gap, introducing the idea of an antigap. For small anisotropy, wefind a paramagnetic-to-ferromagnetic phase transition in agreement with themean-field theory. For large anisotropy, instead, we find a significantdiscrepancy from the scaling of the low-anisotropy ferromagnetic phase. We alsostudy other more experimentally-accessible witnesses of the transition, whichcan be used for finite-size studies, namely the bimodality coefficient and theangular averaged susceptibility. In contrast to the bimodality coefficient, theangular averaged susceptibility fails to capture the onset of the transition,in striking difference with respect to lower-dimensional studies. We alsoanalyze the competition between local dissipative processes (which disentanglethe spin system) and collective dissipative ones (generating entanglement). Thenature of the phase transition is almost unaffected by the presence of theseterms. Our results mark a stark difference with the common intuition that anall-to-all connected system should fall onto the mean-field solution also forintermediate number of spins.

physical review. b./physical review. b

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