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

• URL: http://arxiv.org/abs/2110.09435v2
• Date: Sun, 27 Mar 2022 13:07:34 GMT
• Title: Resolving mean-field solutions of dissipative phase transitions using permutational symmetry
• Authors: Minjae Jo, Bukyoung Jhun, and B. Kahng
• Abstract summary: Phase transitions in dissipative quantum systems have been investigated using various analytical approaches, particularly in the mean-field (MF) limit. These two solutions cannot be reconciled because the MF solutions above $d_c$ should be identical. numerical studies on large systems may not be feasible because of the exponential increase in computational complexity.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 ($d_\infty$). 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.

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