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.
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.
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 Z2 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.
We investigate the physics of an open two-component Dicke model, where the light field mediates nonreciprocal interactions between two spin species. We show that the model, which we dub nonreciprocal Dicke model, exhibits a discrete parity-time (PT) symmetry and we characterize the emergence of a nonstationary phase, so far explained in terms of dissipation-induced instability, as spontaneous breaking of PT symmetry. We further show that such PT symmetry breaking embodies an instance of a nonreciprocal phase transition, a concept recently introduced by Fruchart et al. [Nature (London) 592, 363 (2021)NATUAS0028-083610.1038/s41586-021-03375-9]. Remarkably, the phase transition in our model does not necessitate the presence of any underlying broken symmetry or exceptional points in the spectrum, both believed to be essential requirements for nonreciprocal phase transitions. Our results establish driven-dissipative light-matter systems as a new avenue for exploring nonreciprocal phase transitions and contribute to the theory of nonreciprocal collective phenomena.
Nonreciprocal interactions often create conflicting dynamical objectives that cannot be simultaneously satisfied, leading to nonreciprocal frustration. On the other hand, geometric frustration arises when conflicting static objectives in energy minimization cannot be satisfied. In this work, we show that nonreciprocal interaction among three collective quantum spins, mediated by a damped cavity, induces not only nonreciprocal frustration, intrinsic to nonreciprocity, but also geometric frustration with a remarkable robustness against disorder. It therefore ensures that the accidental degeneracy for steady states remains intact even when the system is perturbed away from a fine-tuned point of enhanced symmetry, in sharp contrast to the equilibrium case. Leveraging this finding, we identify a nonreciprocal phase transition driven by both geometric and nonreciprocal frustration. It gives rise to a time-dependent state, which shows a chiral dynamics along a geometry shaped by the geometric frustration and dynamically restores the broken discrete symmetries. Moreover, it constitutes a time-crystalline order, with multiple harmonics set by an emergent time scale that exhibits critical slowing down. Our predictions have important physical implications for a three-component spinor BEC-cavity system, which manifest as a geometric frustration in the structural phase transition and chiral dynamics of the frustrated self-organized BECs. We demonstrate the feasibility of experimental observation despite the presence of disorder in the spin-cavity coupling strengths.
Nonequilibrium many-body systems can spontaneously break symmetry in time, as in time crystals, or in space, through self-organized domains and interfaces. Whether these two forms of symmetry breaking can intertwine so that an emergent interface alone hosts time-crystalline order remains unknown. In this work, by introducing the Rabi-Hatano-Nelson model, we unveil the existence and mechanism of a self-partitioned interfacial time crystal (SPITC), where a homogeneous system generates its own internal and tunable interface, at which the time-translation symmetry is also spontaneously broken. Such a SPITC phase is intrinsically induced by nonreciprocity and open boundary conditions, without external pumping or long-range interaction. The periodic and open boundary phase diagrams of the system are both mapped out; vacuum and Dicke-like superradiance with static or active orders are identified, with analytical phase boundaries in the weak coupling limit. The frequency of the SPITC is found to scale quadratically with the spin-photon coupling strength, as we derive analytically for the slow dynamics of the spins. The position of the SPITC boundary scales with a critical exponent of −1 as a function of the degree of nonreciprocity, in stark contrast to −1/2 for an otherwise stationary boundary. Our construction of SPITC establishes a route to spatiotemporal order in non-Hermitian many-body systems.
We present a theoretical study of the interaction between an atom characterized by a degenerate ground state and a reciprocal environment, such as a semiconductor nanoparticle, without the presence of external bias. Our analysis reveals that the combined influence of the electron's intrinsic spin magnetic moment on the environment and the chiral atomic dipolar transitions may lead to either the spontaneous breaking of time-reversal symmetry or the emergence of time-crystal-like states with remarkably long relaxation times. The different behavior is ruled by the handedness of the precession motion of the atom's spin vector, which is induced by virtual chiral-dipolar transitions. Specifically, when the relative orientation of the precession angular velocity and the electron spin vector is as in a spinning top, the system manifests time-crystal-like states. Conversely, with the opposite relative orientation, the system experiences spontaneous symmetry breaking of time-reversal symmetry. Our findings introduce a novel mechanism for the spontaneous breaking of time-reversal symmetry in atomic systems, and unveil an exciting opportunity to engineer a nonreciprocal response at the nanoscale, exclusively driven by the quantum vacuum fluctuations.
Nonreciprocal interactions in many-body systems lead to time-dependent states, commonly observed in biological, chemical, and ecological systems. The stability of these states in the thermodynamic limit and the critical behavior of the phase transition from static to time-dependent states are not yet fully understood. To address these questions, we study a minimalistic system endowed with nonreciprocal interactions: an Ising model with two spin species having opposing goals. The mean-field equation predicts three stable phases: disorder, static order, and a time-dependent swap phase. Large-scale numerical simulations support the following: (i) in two dimensions, the swap phase is destabilized by defects; (ii) in three dimensions, the swap phase is stable and has the properties of a time crystal; (iii) the transition from disorder to swap in three dimensions is characterized by the critical exponents of the 3D XY model and corresponds to the breaking of a continuous symmetry, time translation invariance; (iv) when the two species have fully antisymmetric couplings, the static-order phase is unstable in any finite dimension due to droplet growth; and (v) in the general case of asymmetric couplings, static order can be restored by a droplet-capture mechanism preventing the droplets from growing indefinitely. We provide details on the full phase diagram, which includes first- and second-order-like phase transitions, and study how the system coarsens into swap and static-order states.