Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems

Abstract

We show that short-range correlations have a dramatic impact on the steady-state phase diagram of quantum driven-dissipative systems. This effect, never observed in equilibrium, follows from the fact that ordering in the steady state is of dynamical origin, and is established only at very long times, whereas in thermodynamic equilibrium it arises from the properties of the (free) energy. To this end, by combining the cluster methods extensively used in equilibrium phase transitions to quantum trajectories and tensor-network techniques, we extend them to nonequilibrium phase transitions in dissipative many-body systems. We analyze in detail a model of spin-1=2 on a lattice interacting through an XYZ Hamiltonian, each of them coupled to an independent environment that induces incoherent spin flips. In the steady-state phase diagram derived from our cluster approach, the location of the phase boundaries and even its topology radically change, introducing reentrance of the paramagnetic phase as compared to the single-site mean field where correlations are neglected. Furthermore, a stability analysis of the cluster mean field indicates a susceptibility towards a possible incommensurate ordering, not present if short-range correlations are ignored.

Linked cluster expansions for open quantum systems on a lattice

“…However, the value of the critical point we found is in agreement with the results reported in Ref. 53 and Ref. 39, so far.…”

“…Here we see NLCE at work in an interacting twodimensional spin-1/2 model with incoherent spin relaxation 32 , which is believed to exhibit a rich phase diagram, and represents a testing ground for strongly correlated open quantum systems 39,53,58 . We will test our method both far from critical points, and in the proximity of a phase transition: in the first case NLCE allows us to accurately compute the value of the magnetization, while in the latter we are able to estimate the critical point as well as the critical exponent γ for the divergent susceptibility.…”

“…The phase transition point for the same choice of parameters of Eq. (12) has been estimated to be α c = 0.1 32 , 0.14 ± 0.01 53 and 0.17 ± 0.02 39 .…”

“…Numerical strategies specifically suited for this purpose have been recently put forward, including cluster mean-field 53 , correlated variational Ansätze 54,55 , truncated correlation hierarchy schemes 56 , corner-space renormalization methods 57 , and even two-dimensional tensor-network structures 58 . The nonequilibrium extension of the dynamical mean-field theory (which works directly in the thermodynamic limit) has been also proved to be very effective in a wide class of lattice systems [59][60][61] .…”

Critical behavior of dissipative two-dimensional spin lattices

“…If a quantum system is coupled to an external reservoir, dissipative phase transitions may take place. They appear due to the competition between the coherent Hamiltonian dynamics and dissipation processes [2][3][4][5][6] . In contrast to equilibrium critical phenomena at zero temperature, the physical properties do not depend on the Hamiltonian ground state, but instead on the steady-state density matrix of a master equation accounting for the dissipation.…”