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, 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.

12 pages, 8 figures

References in corpus (12)

• Quantum fluids of light
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Dynamical phase transition in the open Dicke model
• Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems
• Signatures of a dissipative phase transition in photon correlation measurements
• Implementation of the Dicke lattice model in hybrid quantum system arrays
• Tricritical directed percolation
• Non-equilibrium effective field theory for absorbing state phase transitions in driven open quantum spin systems
• Nonequilibrium phase transition in an open quantum spin system with long-range interaction
• Tricritical directed percolation with long-range interaction in one and two dimensions
• Upper critical dimension of the 3-state Potts model