Resolving mean-field solutions of dissipative phase transitions using permutational symmetry

Abstract

Phase transitions in dissipative quantum systems have been investigated using various analytical approaches, particularly in the mean-field (MF) limit. However, analytical results often depend on specific methodologies. For instance, the Keldysh formalism shows that the dissipative transverse Ising (DTI) model exhibits a discontinuous transition at the upper critical dimension, dc=3, whereas the fluctuationless MF approach predicts a continuous transition in infinite dimensions (d∞). These two solutions cannot be reconciled, because the MF solutions above dc should be the same. Thus numerical check is required. However, numerical studies in large systems may not be feasible owing to the exponential increase in computational complexity as O(22N) with system size N. Here, we notice that because spins can be regarded as being fully connected at d∞, the spin indices can be permutation invariant, and then the number of quantum states can be considerably contracted as O(N3). The Lindblad equation is transformed into the dynamic equation based on the contracted states. Applying the Runge–Kutta algorithm to the dynamic equation, we obtain all critical exponents including the dynamic exponent as z≈0.5. Moreover, using the property that the DTI model has Z2 symmetry and thus the hyperscaling relation has the form 2β+γ=ν(dc+z), we obtain the relation dc+z=4 in the MF limit. Hence, dc≈3.5 and thus the discontinuous transition in d=3 cannot be treated as an MF solution. We conclude that the permutation invariance at d∞ can be used effectively for checking the validity of an analytic MF solution in quantum phase transitions.