Nonreciprocity can profoundly alter the spectra and dynamics of open quantum systems, yet its impact on the long-time steady-state phases of matter has remained largely unexplored. Here we show that the interplay of nonreciprocity, symmetry defects, and spatial boundaries can generate phases beyond the standard spontaneous-symmetry-breaking paradigm. We demonstrate this mechanism by showing that sufficiently strong nonreciprocity turns boundaries into sources and drains of symmetry defects, while simultaneously endowing these defects with chiral dynamics in the bulk. As a result, the conventional uniform symmetry-broken state gives way to a domain-wall traveling-wave phase, in which symmetry defects form a persistent chiral wave. We showcase this mechanism in a bosonic model with \(Z_{2}\) symmetry, where periodic boundary conditions support only the conventional symmetric and symmetry-broken phases, whereas open boundary conditions allow the traveling-wave phase. We further show that even in the absence of symmetry breaking, the steady state can exhibit anomalous chiral relaxation: owing to the non-Hermitian skin effect in the stability matrix, local fluctuations are chirally amplified as they approach a boundary, where they eventually decay. Combining mean-field theory with truncated Wigner simulations, we characterize these phases, analyze the order parameter and Goldstone-mode fluctuations of the traveling-wave phase, and confirm its existence in three spatial dimensions.
Nonreciprocity can profoundly alter the spectra and dynamics of open quantum systems, yet its impact on the long-time steady-state phases of matter has remained largely unexplored. Here we show that the interplay of nonreciprocity, symmetry defects, and spatial boundaries can generate phases beyond the standard spontaneous-symmetry-breaking paradigm. We demonstrate this mechanism by showing that sufficiently strong nonreciprocity turns boundaries into sources and drains of symmetry defects, while simultaneously endowing these defects with chiral dynamics in the bulk. As a result, the conventional uniform symmetry-broken state gives way to a domain-wall traveling-wave phase, in which symmetry defects form a persistent chiral wave. We showcase this mechanism in a bosonic model with (Z_{2}) symmetry, where periodic boundary conditions support only the conventional symmetric and symmetry-broken phases, whereas open boundary conditions allow the traveling-wave phase. We further show that even in the absence of symmetry breaking, the steady state can exhibit anomalous chiral relaxation: owing to the non-Hermitian skin effect in the stability matrix, local fluctuations are chirally amplified as they approach a boundary, where they eventually decay. Combining mean-field theory with truncated Wigner simulations, we characterize these phases, analyze the order parameter and Goldstone-mode fluctuations of the traveling-wave phase, and confirm its existence in three spatial dimensions.
Active agents are capable of exerting nonreciprocal forces upon one another. For instance, one agent, say A , may attract another agent B while B repels A . These antagonistic nonreciprocal interactions have been extensively studied in classical systems, revealing a wealth of exciting phenomena such as novel phase transitions and traveling-wave states. Whether these phenomena can originate in quantum many-body systems is an open issue, and proposals for their realization are lacking. In this work, we present a model of two species of quantum spins that interact in an antagonistic nonreciprocal way of the attraction-repulsion type. We propose an implementation based on two atomic ensembles coupled via chiral waveguides featuring both braided and nonbraided geometries. The spins are active due to the presence of local gain, which allows them to synchronize. In the thermodynamic limit, we show that nonreciprocal interactions result in a nonreciprocal phase transition to time-crystalline traveling-wave states, associated with spontaneous breaking of parity-time symmetry. We establish how this symmetry emerges from the microscopic quantum model. For a finite number of spins, signatures of the time-crystal phase can still be identified by inspecting equal-time or two-time correlation functions. Remarkably, continuous monitoring of the output field of the waveguides induces a quantum traveling-wave state: a time-crystalline state of a finite-size quantum system, in which parity-time symmetry is spontaneously broken. Our work lays the foundation to explore nonreciprocal interactions in active quantum matter.
Boundary time crystals are a class of exotic dissipative quantum phases that spontaneously break continuous time-translation symmetry in the thermodynamic limit of open quantum systems. In finite-size systems, the long-time evolution of boundary time crystals exhibits decaying oscillations that cannot be captured by widely used mean-field theory. To address this issue, we develop an effective approach called the stroboscopic rotating wave approximation, which provides a well-approximated state for the long-time evolution of boundary time crystals in the strongly driven regime. In this approach, the order parameter exhibits both a long-time decaying envelope governed by an effective Lindblad superoperator and short-time oscillations dominated by a reduced quantum dynamical semigroup. Our results reveal that the competition among dephasing processes along three directions induces persistent oscillations, marking the emergence of the boundary-time-crystal phase. We obtain the analytical expressions for the steady-state density operator, the oscillation period, and the decay rate of the order parameter when the coherent energy splitting exceeds the dissipation rate. Our work provides a beyond-mean-field tool for studying the dynamics of periodically driven open quantum systems and understanding the formation of time crystals.
One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely, that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wave front. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.
Continuous time-translation symmetry is often spontaneously broken in open quantum systems, and the condition for their emergence has been actively investigated. However, there are only a few cases in which its condition for appearance has been fully elucidated. In this Letter, we show that a Lindbladian parity-time (PT) symmetry can generically produce persistent periodic oscillations in a wide class of systems. This includes one-collective spin models, which have been studied thoroughly in the context of dissipative continuous time crystals, and spatially extended bipartite bosonic systems with a conserved particle number. By making an analogy to nonreciprocal phase transitions, we demonstrate that a transition point from the dynamical phase is associated with spontaneous PT symmetry breaking that typically corresponds to a critical exceptional point. Interestingly, the periodic orbits in the PT symmetric phase are found to be center type, implying an initial state dependence. These results are established by proving that the Lindbladian PT symmetry at the microscopic level implies a nonlinear PT symmetry and by performing a linear stability analysis near the transition point. This research will further our understanding of novel nonequilibrium phases of matter and phase transitions with spontaneous antiunitary symmetry breaking.
Non-Hermiticity and synthetic gauge fields play two fundamental roles in engineering exotic phases and dynamics in artificial quantum systems. Here, we explore the mean-field dynamics of interacting bosons in a two-leg ladder with synthetic magnetic flux and nonreciprocal hopping under the open boundary condition. In the Hermitian limit, we showcase the breakdown of the flux-driven chiral dynamics due to the nonlinear self-trapping effect. We further find that the nonreciprocity can drive the transition between chiral dynamics and antichiral dynamics. The antichiral motion is manifested as the non-Hermitian skin dynamics along the same direction on two legs that are not suppressed by the magnetic flux, while the chiral-antichiral transition is flux tunable. We also reveal the trap-skin dynamics with the coexistence of the self-tapping and skin dynamics in the ladder. Dynamical phase diagrams with respect to the chiral-antichiral dynamics, skin dynamics, self-trapping dynamics, and trap-skin dynamics are presented. Our results shed light on intriguing dynamical phenomena under the interplay among non-Hermiticity, nonlinearity, and artificial gauge fields.
In this paper, we develop a systematic approach to characterize the ′ t Hooft anomaly in open quantum systems. Owing to nontrivial couplings to the environment, symmetries in such systems manifest as either strong or weak type. By representing their symmetry transformation through superoperators, we incorporate them in a unified framework that enables a direct calculation of their anomalies. In the case where the full symmetry group Γ arises as a central extension of the strong-symmetry group K with the weak-symmetry group G , we find that anomalies of bosonic systems are classified by H d + 2 ( Γ , U ( 1 ) ) / H d + 2 ( G , U ( 1 ) ) in d spatial dimensions. To illustrate the power of anomalies in open quantum systems, we generally prove that anomaly must lead to nontrivial mixed-state quantum phases as long as the weak symmetry is imposed. Analogous to the “anomaly matching” condition ensuring nontrivial low-energy physics in closed systems, anomaly also guarantees nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. Notably, we identify a novel ( 1 + 1 ) D mixed-state quantum phase that has no counterpart in closed systems, where the steady state shows no nontrivial correlation function in the bulk, but displays long-range correlation on the boundary, which is enforced by anomalies. We further establish the general relations between mixed-state anomalies and such unconventional boundary correlation. Moreover, we explore the generalization of the “anomaly inflow” mechanism in open quantum systems. We construct ( 1 + 1 ) D and ( 2 + 1 ) D Lindbladians whose steady states have mixed-state symmetry-protected-topological order in the bulk, with corresponding edge theories characterized by nontrivial anomalies. Finally, we generalize our results to the cases including lattice translation symmetries, which serves as the Lieb-Schulz-Mattis (LSM) theorem for mixed states.