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.

arXiv comment12 pages, 8 figures

References in corpus (56)

• Efficient classical simulation of slightly entangled quantum computations
• Quantum fluids of light
• The Dicke Quantum Phase Transition with a Superfluid Gas in an Optical Cavity
• The Quantum Jump Approach to Dissipative Dynamics in Quantum Optics
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Observation of discrete time-crystalline order in a disordered dipolar many-body system
• Preparation of Entangled States by Quantum Markov Processes
• Quantum phase transitions
• Keldysh Field Theory for Driven Open Quantum Systems
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Exploring Symmetry Breaking at the Dicke Quantum Phase Transition
• Dynamical phase transition in the open Dicke model
• A Dissipatively Stabilized Mott Insulator of Photons
• Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems
• Dynamical Critical Phenomena in Driven-Dissipative Systems
• Quantum simulations and many-body physics with light
• Open quantum systems with local and collective incoherent processes: Efficient numerical simulation using permutational invariance
• Observation of the Mott Insulator to Superfluid Crossover of a Driven-Dissipative Bose-Hubbard System
• Cooperative non-equilibrium phase transition in a dilute Rydberg ensemble
• Steady-state phases and tunneling-induced instabilities in the driven-dissipative Bose-Hubbard model
• Unconventional magnetism via optical pumping of interacting spin systems
• Cluster mean-field approach to the steady-state phase diagram of dissipative spin systems
• A positive tensor network approach for simulating open quantum many-body systems
• Quantum phase transitions in transverse field spin models: from statistical physics to quantum information
• Signatures of a dissipative phase transition in photon correlation measurements
• Dynamical phases and intermittency of the dissipative quantum Ising model
• Quantum critical states and phase transitions in the presence of non equilibrium noise
• Full counting statistics and phase diagram of a dissipative Rydberg gas
• Nonequilibrium many-body steady states via Keldysh formalism
• Discrete time crystals in the absence of manifest symmetries or disorder in open quantum systems
• Dissipative Floquet Dynamics: from Steady State to Measurement Induced Criticality in Trapped-ion Chains
• A simple tensor network algorithm for two-dimensional steady states
• Non-equilibrium Functional Renormalization for Driven-Dissipative Bose-Einstein Condensation
• Driven Markovian Quantum Criticality
• Implementation of the Dicke lattice model in hybrid quantum system arrays
• Signatures of Self-Organised Criticality in an Ultracold Atomic Gas
• Absorbing state phase transition with competing quantum and classical fluctuations
• Dynamics and universality in noise driven dissipative systems
• Perturbative Field-Theoretical Renormalization Group Approach to Driven-Dissipative Bose-Einstein Criticality
• Multicritical behavior in dissipative Ising models
• Experimental signatures of an absorbing-state phase transition in an open driven many-body quantum system
• The Open-System Dicke-Model Quantum Phase Transition with a Sub-Ohmic Bath
• Phase diagram of the dissipative quantum Ising model on a square lattice
• Quantum contact process
• Tricritical directed percolation
• Spatial correlations of one dimensional driven-dissipative systems of Rydberg atoms
• Non-equilibrium effective field theory for absorbing state phase transitions in driven open quantum spin systems
• Synchronization in a semiclassical Kuramoto model
• Tricritical directed percolation in 2+1 dimensions
• Non-equilibrium phase transitions in $(1+1)$-dimensional quantum cellular automata with controllable quantum correlations
• Absorbing phase transition with a continuously varying exponent in a quantum contact process: a neural network approach
• Numerical Simulation of Critical Dissipative Non-Equilibrium Quantum Systems with an Absorbing State
• Epidemic dynamics in 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