Abstract We explore the emergence of a discrete time crystalline (DTC) order and its stability against thermal fluctuations in a driven kinetic Ising model on a two-dimensional square lattice using the drive protocol invented in a recent work (Gambetta et al 2019 Phys. Rev. E 100 060105(R)). The DTC order is found to be quite robust in the presence of thermal fluctuations. We construct the resulting three-dimensional phase diagram for the DTC order, which manifests a striking resemblance to the jamming phase diagram proposed by Liu and Nagel. This finding may suggest a new way to view the DTC order as a new type of nonequilibrium soft matter. The quench dynamics exhibits a unique feature due to the nature of the employed drive protocol, namely, breakdown of the inverse relationship between the domain growth and defect relaxation, which holds in the usual quench dynamics of the kinetic Ising model.
Abstract We explore the emergence of a discrete time crystalline (DTC) order and its stability against thermal fluctuations in a driven kinetic Ising model on a two-dimensional square lattice using the drive protocol invented in a recent work (Gambetta et al 2019 Phys. Rev. E 100 060105(R)). The DTC order is found to be quite robust in the presence of thermal fluctuations. We construct the resulting three-dimensional phase diagram for the DTC order, which manifests a striking resemblance to the jamming phase diagram proposed by Liu and Nagel. This finding may suggest a new way to view the DTC order as a new type of nonequilibrium soft matter. The quench dynamics exhibits a unique feature due to the nature of the employed drive protocol, namely, breakdown of the inverse relationship between the domain growth and defect relaxation, which holds in the usual quench dynamics of the kinetic Ising model.
We describe a general and simple paradigm for discrete time crystals (DTCs), systems with a stable subharmonic response to an external driving field, in a classical thermal setting. We consider, specifically, an Ising model in two dimensions, as a prototypical system with a phase transition into stable phases distinguished by a local order parameter, driven by thermal dynamics and periodically kicked with a noisy protocol. By means of extensive numerical simulations for large sizes-allowed by the classical nature of our model-we show that the system features a true disorder-DTC order phase transition as a function of the noise strength, with a robust DTC phase extending over a wide parameter range. We demonstrate that, when the dynamics is observed stroboscopically, the phase transition to the DTC state appears to be in the equilibrium two-dimensional Ising universality class. However, we explicitly show that the DTC is a genuine nonequilibrium state. More generally, we speculate that systems with thermal phase transitions to multiple competing phases can give rise to DTCs when appropriately driven.
The stability of a discrete time crystal against thermal fluctuations has been studied numerically by solving a stochastic Landau-Lifshitz-Gilbert equation of a periodically driven classical system composed of interacting spins, each of which couples to a thermal bath. It is shown that in the thermodynamic limit, even though the long-range temporal crystalline order is stable at low temperature, it is melting above a critical temperature, at which the system experiences a nonequilibrium phase transition. The critical behaviors of the continuous phase transition have been systematically investigated, and it is shown that despite the genuine nonequilibrium feature of such a periodically driven system, its critical properties fall into the three-dimensional Ising universality class with a dynamical exponent (z=2) identical to that in the critical dynamics of a kinetic Ising model without driving.
A discrete time crystal (DTC) is an out-of-equilibrium phase of matter that spontaneously breaks discrete time-translation symmetry. Previous studies have been limited to a set of models with Ising-like couplings - and mostly only in one dimension - thus precluding our understanding of the existence (or not) of DTCs in models with more realistic interactions. In this work, by combining the latest generation of IBM quantum processors with state-of-the-art tensor network methods, we demonstrate the existence of a DTC in a two-dimensional system governed by anisotropic Heisenberg interactions. We uncover a rich phase diagram encompassing spin-glass, ergodic, and time-crystalline phases, and identify the interplay of initialization, interaction anisotropy, and driving protocols in stabilizing the DTC phase. By extending the study of Floquet matter beyond simplified models, we lay the groundwork for exploring how driven systems bridge the gap between quantum coherence and emergent non-equilibrium thermodynamics.
We analyze the quantum dynamics of periodically driven, disordered systems in the presence of long-range interactions. Focusing on the stability of discrete time crystalline (DTC) order in such systems, we use a perturbative procedure to evaluate its lifetime. For 3D systems with dipolar interactions, we show that the corresponding decay is parametrically slow, implying that robust, long-lived DTC order can be obtained. We further predict a sharp crossover from the stable DTC regime into a regime where DTC order is lost, reminiscent of a phase transition. These results are in good agreement with the recent experiments utilizing a dense, dipolar spin ensemble in diamond [Nature (London) 543, 221 (2017)NATUAS0028-083610.1038/nature21426]. They demonstrate the existence of a novel, critical DTC regime that is stabilized not by many-body localization but rather by slow, critical dynamics. Our analysis shows that the DTC response can be used as a sensitive probe of nonequilibrium quantum matter.
Discrete time crystals (DTC) have emerged as a significant phase of matter for out-of-equilibrium many-body systems. In this work, we study how random long-range interactions contribute to the stability of the DTC phase. Generally, a stable DTC phase is believed to be realized in disordered systems with short-range interactions. Here, we explore a periodically driven quantum Sherrington-Kirkpatrick (SK) model of Ising spin-glass in which all spins are coupled to each other randomly. We investigate the possibilities of the DTC phase in the SK model within three different driving protocols and found that the quantum SK model exhibits a robust DTC phase, although systems with uniform long-range interactions can exhibit only prethermal DTC phase. Further, we demonstrate that disorder in the local XY term or transverse field is crucial for stabilizing a broad DTC phase, despite random couplings in the SK model. Our analysis shows that the stability of the DTC phase is determined by the non-ergodic nature of the Floquet eigenstates.
We consider a dissipative quantum Ising model periodically driven by a train of $\ensuremath{\pi}$ pulses and investigate dissipative discrete time crystals (DTCs) in solids. In this model, the interaction between the spins spontaneously breaks the discrete time translation symmetry, giving rise to a dissipative DTC, where two ferromagnetic states are switched alternately by each pulse. We microscopically describe the generic dissipation due to thermal contact to an equilibrium heat bath using the Bloch-Redfield equation. In contrast to other DTC studies, this dissipation stabilizes, rather than destroys, the DTC order without fine-tuning as long as the temperature is low enough. Invoking the time-dependent mean-field theory and solving self-consistently the periodic drive, dissipation, and DTC order parameter, we investigate the nonequilibrium DTC phase transition and determine the critical exponents, including a dynamical one. We also find phase transitions without an equilibrium counterpart: a nontrivial interplay of the periodic drive and dissipation gives rise to reentrant DTC transition when changing the pulse interval at a fixed temperature. Also, to demonstrate the rigidity of the DTC, we consider imperfect $\ensuremath{\pi}$ pulses, showing that the DTC is robust against the small imperfections and finding that discrete time quasicrystals can appear for the larger imperfections. Together with experimental proposals in magnetic materials, our results pave the way for realizing the DTC and for uncovering nonequilibrium critical phenomena in real solid-state materials.