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arXiv:1712.01560

Dynamics of non-equilibrium steady state quantum phase transitions

Patrik Hedvall, Jonas Larson

Published Dec 5, 2017• 4 citations
arXivHTMLPDF

Abstract

In this paper we address the question how the Kibble-Zurek mechanism, which describes the formation of topological defects in quantum systems subjected to a quench across a critical point, is generalized to the same scenario but for driven-dissipative quantum critical systems. In these out of equilibrium systems, the critical behavior is manifested in the steady states rather than in the ground states as in quantum critical models of closed systems. To give the generalization we need to establish what is meant by adiabaticity in open quantum systems. Another most crucial concept that we clarify is the characterization of non-adiabatic excitations. It is clear what these are for a Hamiltonian systems, but question is more subtle for driven-dissipative systems. In particular, the important observation is that the instantaneous steady states serve as reference states. From these the excitations can then be extracted by comparing the distance from the evolved state to the instantaneous steady state. With these issues resolved we demonstrate the applicability of the generalized Kibble-Zurek mechanism to an open Landau-Zener problem and show universal Kibble-Zurek scaling for an open transverse Ising model. Thus, our results support any assumption that non-equilibrium quantum critical behavior can be understood from universal features.

DOI
10.48550/arxiv.1712.01560

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In this paper we address the question how the Kibble-Zurek mechanism, which describes the formation of topological defects in quantum systems subjected to a quench across a critical point, is generalized to the same scenario but for driven-dissipative quantum critical systems. In these out of equilibrium systems, the critical behavior is manifested in the steady states rather than in the ground states as in quantum critical models of closed systems. To give the generalization we need to establish what is meant by adiabaticity in open quantum systems. Another most crucial concept that we clarify is the characterization of non-adiabatic excitations. It is clear what these are for a Hamiltonian systems, but question is more subtle for driven-dissipative systems. In particular, the important observation is that the instantaneous steady states serve as reference states. From these the excitations can then be extracted by comparing the distance from the evolved state to the instantaneous steady state. With these issues resolved we demonstrate the applicability of the generalized Kibble-Zurek mechanism to an open Landau-Zener problem and show universal Kibble-Zurek scaling for an open transverse Ising model. Thus, our results support any assumption that non-equilibrium quantum critical behavior can be understood from universal features.
When a system is swept through a quantum critical point (QCP), the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp time scale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a non-critical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality.
When a system is swept through a quantum critical point, the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp timescale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a noncritical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality. © 2026 authors. Published by the American Physical Society.
The Kibble-Zurek mechanism provides a universal framework for predicting defect formation in non-equilibrium phase transitions. While Markovian dissipation typically degrades universal scaling, the impact of non-Markovian memory remains largely unexplored. We demonstrate that an Ohmic bath induces a Berezinskii-Kosterlitz-Thouless transition in the open quantum Rabi model. Using simulations based on Matrix Product States, we show that the excitation energy strictly follows universal Kibble-Zurek power-law scaling when evaluated at the freeze-out time. Crucially, we find that since the environment defines the universality class, dissipation does not inherently compete with adiabatic dynamics, in stark contrast to Markovian regimes. Our results establish the Kibble- Zurek mechanism as a robust witness of universality in open quantum systems, revealing that non-Markovian memory preserves the integrity of non-equilibrium scaling.
We analyze mechanisms for universal out-of-equilibrium dynamics near criticality by exploring the effect of randomized quantum resetting (QR) under a finite-time quench across a quantum phase transition. Using the transverse-field Ising chain as a generic model and exploiting its exact solution, QR is found to cause a crossover of the scaling of the topological defect density with the time scale τττ of the quench, from Kibble-Zurek to anti-Kibble-Zurek scaling as τττ increases. This reflects a competition between non-adiabatic quench-driven excitations and QR, giving rise to local minima of the defect densities at optimal annealing times. These times and the corresponding local minima are shown to scale as universal power laws with the rate of QR. Additional results for the scaling of the mean excess energy suggest that a system driven across a quantum critical point exhibits the same scaling behavior under a linear quench with QR as with uncorrelated noise.
Driven dynamics across a quantum critical point is usually described by the Kibble-Zurek scaling. Although the original Kibble-Zurek scaling requires an adiabatic initial state, it has been shown that scaling behaviors exist even when the driven dynamics is triggered from a thermal equilibrium state exactly at the critical point, in spite of the breakdown of the initial adiabaticity. In this paper, we show that the existence of the scaling behavior can be generalized to the case of the initial state being a thermal equilibrium state near the critical point. We propose a scaling theory in which the initial parameters are included as additional scaling variables due to the breakdown of the initial adiabaticity. In particular, we demonstrate that for the driven critical dynamics in a closed system, the nontrivial thermal effects are closely related to the initial distance to the critical point. We numerically confirm the scaling theory by simulating the real-time dynamics of the one-dimensional quantum Ising model at both zero and finite temperatures.
We investigate the quench dynamics of an open quantum system involving a quantum phase transition. In the isolated case, the quench dynamics involving the phase transition exhibits a number of scaling relations with the quench rate as predicted by the celebrated Kibble-Zurek mechanism. In contact with an environment however, these scaling laws break down and one may observe an anti-Kibble-Zurek behavior: slower ramps lead to less adiabatic dynamics, increasing thus nonadiabatic effects with the quench time. In contrast to previous works, we show here that such anti-Kibble-Zurek scaling can acquire a universal form in the sense that it is determined by the equilibrium critical exponents of the phase transition, provided the excited states of the system exhibit singular behavior, as observed in fully connected models. This demonstrates novel universal scaling laws granted by a system-environment interaction in a critical system. We illustrate these findings in two fully connected models, namely, the quantum Rabi and the Lipkin-Meshkov-Glick models. In addition, we discuss the impact of nonlinear ramps and finite-size systems.