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arXiv:2007.08348 · Physical Review Letters

Dynamical Transitions and Critical Behavior between Discrete Time Crystal Phases

Xiaoqin Yang, Zi Cai

Published Jan 12, 2021• 34 citations
arXivHTMLPDF

Abstract

In equilibrium physics, spontaneous symmetry breaking and elementary excitation are two concepts closely related with each other: the symmetry and its spontaneous breaking not only control the dynamics and spectrum of elementary excitations, but also determine their underlying structures. In this Letter, based on an exactly solvable model, we propose a phase ramping protocol to study an excitationlike behavior of a nonequilibrium quantum matter: a discrete time crystal phase with spontaneous temporal translational symmetry breaking. It is shown that slow ramping could induce a dynamical transition between two Z_{2} symmetry breaking time crystal phases in time domain, which can be considered as a temporal analog of the soliton excitation spatially sandwiched by two degenerate charge density wave states in polyacetylene. By tuning the ramping rate, we observe a critical value at which point the transition duration diverges, resembling the critical slowing down phenomenon in nonequilibrium statistic physics. We also discuss the effect of stochastic sequences of such phase ramping processes and its implication to the stability of the discrete time crystal phase against noisy perturbations.

DOI
10.1103/physrevlett.126.020602

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In equilibrium physics, spontaneous symmetry breaking and elementary excitation are two concepts closely related with each other: the symmetry and its spontaneous breaking not only control the dynamics and spectrum of elementary excitations, but also determine their underlying structures. In this Letter, based on an exactly solvable model, we propose a phase ramping protocol to study an excitationlike behavior of a nonequilibrium quantum matter: a discrete time crystal phase with spontaneous temporal translational symmetry breaking. It is shown that slow ramping could induce a dynamical transition between two Z_{2} symmetry breaking time crystal phases in time domain, which can be considered as a temporal analog of the soliton excitation spatially sandwiched by two degenerate charge density wave states in polyacetylene. By tuning the ramping rate, we observe a critical value at which point the transition duration diverges, resembling the critical slowing down phenomenon in nonequilibrium statistic physics. We also discuss the effect of stochastic sequences of such phase ramping processes and its implication to the stability of the discrete time crystal phase against noisy perturbations.
Spontaneous symmetry breaking is one of the central organizing principles in physics. Time crystals have emerged as an exotic phase of matter, spontaneously breaking the time translational symmetry, and are mainly categorized as discrete or continuous. While these distinct types of time crystals have been extensively explored as standalone systems, intriguing effects can arise from their mutual interaction. Here, we demonstrate that a time-independent coupled system of discrete and continuous time crystals induces a simultaneous twofold temporal symmetry breaking, resulting in a hierarchical time crystal phase. Interestingly, one of the subsystems breaks a discrete temporal symmetry that is absent from the time-independent Liouvillian generating the dynamics, but instead emerges dynamically, giving rise to a convoluted non-equilibrium phase. We demonstrate that hierarchical time crystals are robust, emerging for fundamentally different coupling schemes and persisting across wide ranges of system parameters.
Spontaneous symmetry breaking is a ubiquitous concept and is well described in many textbooks of physics. However, direct observation of spontaneous symmetry breaking is lacking. Here we present the observation and analysis of a spontaneously broken discrete time-translation symmetry in our driven system. We experimentally find a 50-50 split between two stable and temporal-distinct solutions, indicative of the breaking of a Z2{\mathbb{Z}}_{2}Z2​ (Ising-like) time-translation symmetry. The experiment allows for further exploration of the symmetry breaking in our discrete time crystal and for engineering excitations in space and time in the quantum domain.
Discrete time crystals are related to nonequilibrium dynamics of periodically driven quantum many-body systems where the discrete time-translation symmetry of the Hamiltonian is spontaneously broken into another discrete symmetry. Recently, the concept of phase transitions has been extended to nonequilibrium dynamics of time-independent systems induced by a quantum quench, i.e., a sudden change of some parameter of the Hamiltonian. There, the return probability of a system to the ground state reveals singularities in time which are dubbed dynamical quantum phase transitions. We show that the quantum quench in a discrete time crystal leads to dynamical quantum phase transitions where the return probability of a periodically driven system to a Floquet eigenstate before the quench reveals singularities in time. It indicates that dynamical quantum phase transitions are not restricted to time-independent systems and can be also observed in systems that are periodically driven. We discuss how the phenomenon can be observed in ultracold atomic gases.
Spontaneous breaking of continuous time translation symmetry into a discrete one is related to time crystal formation. While the phenomenon is not possible in the ground state of a time-independent many-body system, it can occur in an excited eigenstate. Here, we concentrate on bosons on a ring with attractive contact interactions and analyze a quantum quench from the time crystal regime to the noninteracting regime. We show that dynamical quantum phase transitions can be observed where the return probability of the system to the initial state before the quench reveals a nonanalytical behavior in time. The problem we consider constitutes an example of the dynamical quantum phase transitions in a system where both time and space continuous translation symmetries are broken.
Nonequilibrium driving systems provide fertile ground for exploring intriguing spontaneous symmetry breaking phenomena. In this paper, we report on the intertwined discrete spacetime translational symmetry breaking in a driven-dissipative spin system without pure spatial translational symmetry, resulting in the emergence of an out-of-equilibrium nonmagnetic phase termed the ``discrete spacetime crystal.'' In contrast to the previously widely studied discrete time crystal, the spins establish a distinctive long-range crystalline order not only in time, but also in the intertwined spacetime direction. We further demonstrate the exponentially long lifetime of the spatial-temporal order against generic weak environmental noise fluctuations. Finally, possible experimental realizations of our model are discussed. Our findings shed light on the existence of exotic phases of matter characterized by intertwined spacetime symmetry breaking, paving the way for future investigations into the properties and potential applications of these exotic states.
A discrete time crystal is a phase unique to nonequilibrium systems, where discrete time translation symmetry is spontaneously broken. Most conventional time crystals proposed so far rely on the spontaneous breaking of on-site symmetries and their corresponding on-site symmetry operations. In this Letter, we propose a new time crystal dubbed the "spatial-translation-induced discrete time crystal," which is realized by spatial translation and its symmetry breaking. Owing to the properties of spatial translation, in this new time crystal, various time crystal orders can only emerge by changing the filling but not changing the driving protocol. We demonstrate that the local transport of charges or spins shows a nontrivial oscillation, enabling detection and applications of time crystal orders, and also provide promising platforms including quantum circuits. Our proposal opens up a new avenue of realizing time crystal orders by spatial translation in various quantum simulators.