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, Keldysh formalism shows that the dissipative transverse Ising (DTI) model exhibits a discontinuous transition at the upper critical dimension, d_c= 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 d_c should be identical. This necessitates a numerical verification. However, numerical studies on large systems may not be feasible because of the exponential increase in computational complexity as O(2^2N) with system size N. Here, we note that because spins can be regarded as being fully connected at d_∞, the spin indices can be permutation invariant, and the number of quantum states can be considerably contracted with the computational complexity O(N³). The Lindblad equation is transformed into a dynamic equation based on the contracted states. Applying the Runge--Kutta algorithm to the dynamic equation, we obtain all the critical exponents, including the dynamic exponent z≈ 0.5. Moreover, since the DTI model has Z₂ symmetry, the hyperscaling relation has the form 2β+γ=ν(d+z), we obtain the relation d_c+z=4 in the MF limit. Hence, d_c≈ 3.5; thus, the discontinuous transition at d=3 cannot be treated as an MF solution. We conclude that the permutation invariance at d_∞ can be used effectively to check the validity of an analytic MF solution in quantum phase transitions.