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.
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.
A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.
Spontaneous symmetry breaking plays a pivotal role in physics ranging from the emergence of elementary particles to the phase transitions of matter. The spontaneous breaking of continuous time translation symmetry leads to a novel state of matter named continuous time crystal (CTC). It exhibits periodic oscillation without the need for periodic driving, and the relative phases for repetitively realized oscillations are random. However, the mechanism behind the spontaneous symmetry breaking in CTCs, particularly the random phases, remains elusive. Here we propose and experimentally realize two types of CTCs based on distinct mechanisms: manifold topology and near-chaotic motion. We observe both types of CTCs in thermal atomic ensembles by artificially synthesizing spin-spin nonlinear interactions through a measurement-feedback scheme. Our work provides general recipes for the realization of CTCs, and paves the way for exploring CTCs in various systems.
Time crystals are a nonequilibrium phase of matter that extend fundamental spontaneous symmetry breaking into the temporal dimension, typically requiring external driving for their realization. Here, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system without external coherent or incoherent driving. Through numerical analysis of spin dynamics, we identify nonstationary steady states, some of which are limit cycles with persistent periodic oscillations, while others exhibit chaotic, aperiodic behavior. The emergence of limit cycle steady states breaks the continuous time-translation symmetry of this time-independent many-body system, classifying them as continuous time crystals. We further validate these oscillatory behaviors by testing their stability against local perturbations and assess the robustness of the emergent continuous time crystals by introducing isotropic Gaussian white noise. This work provides insights into the intricate interplay between the dissipation and spin interaction, and opens possibilities for realizing dissipation-induced, heating-immune time crystals. Time crystals represent a nonequilibrium phase of matter without an equilibrium counterpart. The authors investigate the nonequilibrium steadystate phase diagram of a two-dimensional dissipative Heisenberg model in the absence of driving, and identify a continuous time crystalline phase exhibiting persistent oscillations in time.
Continuous time crystals, i.e., nonequilibrium phases with a spontaneously broken continuous time-translational symmetry, have been studied and recently observed in the long time dynamics of open quantum systems. Here, we investigate a lattice of interacting three-level particles and find two distinct time-crystal phases that cannot be described within mean-field theory. Remarkably, one of them emerges only in the presence of correlations, upon accounting for beyond-mean-field effects. Our findings extend explorations of continuous time-translational symmetry breaking in dissipative systems beyond the classical phenomenology of periodic orbits in a low-dimensional nonlinear system. The proposed model applies directly to the laser-driven dynamics of interacting Rydberg states in neutral-atom arrays and suggests that the predicted time-crystal phases are observable in such experiments.
Dissipative time crystals can appear in spin systems, when the Z2 symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator S2 is conserved. In this paper, we relax the latter condition and show that time-translation-symmetry-breaking collective oscillations persist, in the thermodynamic limit, even in the absence of spin symmetry. We engineer an ad hoc Lindbladian using power-law-decaying spin operators and show that time-translation-symmetry breaking appears when the decay exponent obeys $0<\ensuremath{\eta}\ensuremath{\le}1$. This model shows a surprisingly rich phase diagram, including the time-crystal phase as well as first-order, second-order, and continuous transitions of the fixed points. We study the phase diagram and the magnetization dynamics in the mean-field approximation. We prove that this approximation is quantitatively accurate, when $0<\ensuremath{\eta}<1$ and the thermodynamic limit is taken, because the system does not develop sizable quantum fluctuations, if the Gaussian approximation is considered.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].