  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 ($d_\infty$). 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 $\mathcal{O}(2^{2N})$ with system size $N$. Here,
we note that because spins can be regarded as being fully connected at
$d_\infty$, the spin indices can be permutation invariant, and the number of
quantum states can be considerably contracted with the computational complexity
$\mathcal{O}(N^3)$. 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 $z\approx 0.5$. Moreover, since the DTI model has
$\mathbb{Z}_2$ symmetry, the hyperscaling relation has the form
$2\beta+\gamma=\nu(d+z)$, we obtain the relation $d_c+z=4$ in the MF limit.
Hence, $d_c\approx 3.5$; thus, the discontinuous transition at $d=3$ cannot be
treated as an MF solution. We conclude that the permutation invariance at
$d_\infty$ can be used effectively to check the validity of an analytic MF
solution in quantum phase transitions.
