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arXiv:2105.12747 · SciPost Physics

Time-reversal symmetry breaking and emergence in driven-dissipative Ising models

Daniel A. Paz, Mohammad F. Maghrebi

Published Feb 17, 2022• 4 citations
arXivHTMLPDF

Abstract

Fluctuation-dissipation relations (FDRs) and time-reversal symmetry (TRS), two pillars of statistical mechanics, are both broken in generic driven-dissipative systems. These systems rather lead to non-equilibrium steady states far from thermal equilibrium. Driven-dissipative Ising-type models, however, are widely believed to exhibit effective thermal critical behavior near their phase transitions. Contrary to this picture, we show that both the FDR and TRS are broken even macroscopically at, or near, criticality. This is shown by inspecting different observables, both even and odd operators under time-reversal transformation, that overlap with the order parameter. Remarkably, however, a modified form of the FDR as well as TRS still holds, but with drastic consequences for the correlation and response functions as well as the Onsager reciprocity relations. Finally, we find that, at criticality, TRS remains broken even in the weakly-dissipative limit.

DOI
10.21468/scipostphys.12.2.066

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Fluctuation-dissipation relations (FDRs) and time-reversal symmetry (TRS), two pillars of statistical mechanics, are both broken in generic driven-dissipative systems. These systems rather lead to non-equilibrium steady states far from thermal equilibrium. Driven-dissipative Ising-type models, however, are widely believed to exhibit effective thermal critical behavior near their phase transitions. Contrary to this picture, we show that both the FDR and TRS are broken even macroscopically at, or near, criticality. This is shown by inspecting different observables, both even and odd operators under time-reversal transformation, that overlap with the order parameter. Remarkably, however, a modified form of the FDR as well as TRS still holds, but with drastic consequences for the correlation and response functions as well as the Onsager reciprocity relations. Finally, we find that, at criticality, TRS remains broken even in the weakly-dissipative limit.
Fluctuation-dissipation relations (FDRs) and time-reversal symmetry (TRS), two pillars of statistical mechanics, are both broken in generic driven-dissipative systems. These systems rather lead to non-equilibrium steady states far from thermal equilibrium. Driven-dissipative Ising-type models, however, are widely believed to exhibit effective thermal critical behavior near their phase transitions. Contrary to this picture, we show that both the FDR and TRS are broken even macroscopically at, or near, criticality. This is shown by inspecting different observables, both even and odd operators under time-reversal transformation, that overlap with the order parameter. Remarkably, however, a modified form of the FDR as well as TRS still holds, but with drastic consequences for the correlation and response functions as well as the Onsager reciprocity relations. Finally, we find that, at criticality, TRS remains broken even in the weakly-dissipative limit.
By considering a solvable driven-dissipative quantum model, we demonstrate that continuous phase transitions in dissipative systems may occur without an accompanying symmetry breaking. As such, the underlying mechanism for this type of transition is qualitatively different from that of continuous equilibrium phase transitions. In our model, the transition is solely a result of the interplay between Hamiltonian and dissipative dynamics and manifests as a nonanalyticity in the steady state ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{ss}}$ in the thermodynamic limit. Based on knowledge from critical classical models we suggest that this behavior derives from a rounding of a first-order phase transition into a continuous one due to large environment-induced fluctuations. Despite being conceptually different from the traditional continuous transitions, we show that expectations of local observables can still be characterized by a set of critical exponents.
Driven-dissipative quantum systems generically do not satisfy simple notions of detailed balance based on the time symmetry of correlation functions.We show that such systems can nonetheless exhibit a hidden time-reversal symmetry, which most directly manifests itself in a doubled version of the original system prepared in an appropriate entangled thermofield double state.This hidden time-reversal symmetry has a direct operational utility: it provides a general method for finding exact solutions of nontrivial steady states.Special cases of this approach include the coherent quantum absorber and complex-P function methods from quantum optics.We also show that hidden time-reversal symmetry has observable consequences even in single-system experiments, and can be broken by the nontrivial combination of nonlinearity, thermal fluctuations, and driving.To illustrate our ideas, we analyze concrete examples of driven qubits and nonlinear cavities.These systems exhibit hidden time-reversal symmetry but not conventional detailed balance.
Driven-dissipative quantum systems generically do not satisfy simple notions of detailed balance based on the time symmetry of correlation functions. We show that such systems can nonetheless exhibit a hidden time-reversal symmetry, which most directly manifests itself in a doubled version of the original system prepared in an appropriate entangled thermofield double state. This hidden time-reversal symmetry has a direct operational utility: it provides a general method for finding exact solutions of nontrivial steady states. Special cases of this approach include the coherent quantum absorber and complex-P function methods from quantum optics. We also show that hidden time-reversal symmetry has observable consequences even in single-system experiments, and can be broken by the nontrivial combination of nonlinearity, thermal fluctuations, and driving. To illustrate our ideas, we analyze concrete examples of driven qubits and nonlinear cavities. These systems exhibit hidden time-reversal symmetry but not conventional detailed balance.
Driven-dissipative systems are expected to give rise to nonequilibrium phenomena that are absent in their equilibrium counterparts.However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their nonequilibrium origin.In this paper, we show that multicritical points in such systems lead to a rich and genuinely nonequilibrium behavior.Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high-to a low-density phase-reminiscent of a liquid-gas transition-and another to an antiferromagnetic phase.Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) Z 2 symmetry.However, they coalesce at a multicritical point, giving rise to a nonequilibrium model of coupled Ising-like order parameters described by a Z 2 × Z 2 symmetry.Using a dynamical renormalization-group approach, we show that a pair of nonequilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system.We elucidate various exotic features of these NEFPs.In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance.This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition.As direct evidence of the nonequilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths.Finally, we argue that this nonequilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.
Driven-dissipative systems define a broad class of nonequilibrium systems where an external drive (e.g., laser) competes with a dissipative environment. The steady state of dynamics is generically distinct from a thermal state characteristic of equilibrium. As a representative example, a driven-dissipative system with a continuous symmetry is generically disordered in two dimensions in contrast with the well-known algebraic order in equilibrium XY phases. In this paper, we study a two-dimensional driven-dissipative model of weakly interacting bosons with a continuous U(1)U(1)U(1) symmetry. Our aim is twofold: First, we show that an effectively equilibrium XY phase emerges despite the driven nature of the model, and that it is protected by a natural Z2{\mathbb{Z}}_{2}Z2​ symmetry of the dynamics. Second, we argue that this phase is unstable against symmetry-breaking perturbations as well as static disorder, whose mechanism in most cases has no analog in equilibrium. In the language of renormalization group theory, we find that, outside equilibrium, there are more relevant directions away from the XY phase.