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arXiv:2406.01686

Prethermal Time-Crystalline Corner Modes

Si Wen Jiang, Yuan Dong, Jiang, Wenjie, Dong-Ling Deng, Francisco Machado

Published Jun 3, 2024• 0 citations
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Abstract

We demonstrate the existence of prethermal discrete time crystals whose sub-harmonic response is entirely localized to zero-dimensional corner modes. Within the exponentially long prethermal regime, we show that the robustness of these corner modes arises from two related, yet distinct mechanisms: the presence of a higher-order symmetry-protected topological phase in the effective Hamiltonian, or the emergence of a dynamical constraint that prevents the decay of the corner mode. While the first mechanism ensures the stability of the sub-harmonic response throughout the entirety of the prethermal regime, it is restricted to initial states in the ground state manifold of the effective Hamiltonian. By contrast, the second mechanism enables the observation of the prethermal time-crystalline order for arbitrary initial states, albeit with a time scale that is not only determined by the frequency of the drive, but also the relative energy scale across the system's sublattices. We characterize these two mechanisms by simulating the dynamics of a periodically driven two-dimensional spin model, and discuss natural extensions of our model to all other dimensions.

DOI
10.48550/arxiv.2406.01686

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We demonstrate the existence of prethermal discrete time crystals whose sub-harmonic response is entirely localized to zero-dimensional corner modes. Within the exponentially long prethermal regime, we show that the robustness of these corner modes arises from two related, yet distinct mechanisms: the presence of a higher-order symmetry-protected topological phase in the effective Hamiltonian, or the emergence of a dynamical constraint that prevents the decay of the corner mode. While the first mechanism ensures the stability of the sub-harmonic response throughout the entirety of the prethermal regime, it is restricted to initial states in the ground state manifold of the effective Hamiltonian. By contrast, the second mechanism enables the observation of the prethermal time-crystalline order for arbitrary initial states, albeit with a time scale that is not only determined by the frequency of the drive, but also the relative energy scale across the system's sublattices. We characterize these two mechanisms by simulating the dynamics of a periodically driven two-dimensional spin model, and discuss natural extensions of our model to all other dimensions.
We demonstrate the existence of prethermal discrete time crystals whose subharmonic response is entirely localized to zero-dimensional corner modes. Within the exponentially long prethermal regime, we show that the robustness of these corner modes arises from two related, yet distinct, mechanisms: the presence of a higher-order symmetry-protected topological phase in the effective Hamiltonian, or the emergence of a dynamical constraint that prevents the decay of the corner mode. While the first mechanism ensures the stability of the subharmonic response throughout the entirety of the prethermal regime, it is restricted to initial states in the ground state manifold of the effective Hamiltonian. By contrast, the second mechanism enables the observation of the prethermal time-crystalline order for arbitrary initial states, albeit with a timescale that is not only determined by the frequency of the drive, but also the relative energy scale across the system's sublattices. We characterize these two mechanisms by simulating the dynamics of a periodically driven two-dimensional spin model, and discuss natural extensions of our model to all other dimensions.
We demonstrate that the prethermal regime of periodically driven (Floquet), classical many-body systems can host nonequilibrium phases of matter. In particular, we show that there exists an effective Hamiltonian that captures the dynamics of ensembles of classical trajectories despite the breakdown of this description at the single trajectory level. In addition, we prove that the effective Hamiltonian can host emergent symmetries protected by the discrete time-translation symmetry of the drive. The spontaneous breaking of such an emergent symmetry leads to a subharmonic response, characteristic of time crystalline order, that survives to exponentially late times in the frequency of the drive. To this end, we numerically demonstrate the existence of classical prethermal time crystals in systems with different dimensionalities and ranges of interaction. Extensions to higher order and fractional time crystals are also discussed.
While a clean, driven system generically absorbs energy until it reaches "infinite temperature," it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for nontrivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethernial regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localized cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al. [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.
A time crystal is a state of periodically driven matter that breaks discrete time-translation symmetry.Time crystals have been demonstrated experimentally in various programmable quantum simulators, and they exemplify how nonequilibrium, driven quantum systems can exhibit intriguing and robust properties absent in systems at equilibrium.These robust driven states need to be stabilized by some mechanism, with the preeminent candidates being many-body localization and prethermalization.This introduces additional constraints that make it challenging to experimentally observe time crystallinity in naturally occurring systems.Recent theoretical work has developed the notion of prethermalization without temperature, expanding the class of time-crystal systems to explain time-crystalline observations at (or near) infinite temperature.In this work, we conclusively observe the emergence of a prethermal Uð1Þ time-crystalline state at quasi-infinite temperature in a solid-state NMR quantum emulator by verifying the requisites of prethermalization without temperature.In addition to observing the signature period-doubling behavior, we show the existence of a long-lived prethermal regime whose lifetime is significantly enhanced by strengthening an emergent Uð1Þ conservation law.Not only do we measure this enhancement through the global magnetization, but we also exploit on-site disorder to measure local observables, ruling out the possibility of many-body localization and confirming the emergence of long-range correlations.
Prethermalization, where quasisteady states are realized in the intermediate long-time regime (prethermal regime), in periodically driven (Floquet) systems is an important phenomenon since it provides a platform for nontrivial Floquet many-body physics. In this Rapid Communication, we consider Floquet systems with dual energy scales: The Hamiltonian consists of two different terms whose amplitude is either comparable to or much smaller than the frequency. As a result, when the larger-amplitude drive induces a ZN{\mathbb{Z}}_{N}ZN​ symmetry operation, we obtain the effective static Hamiltonian respecting the emergent ZN{\mathbb{Z}}_{N}ZN​ symmetry in high-frequency expansions, which describes the dynamics of such Floquet systems in the prethermal regime. As an application of our formulation, our formalism gives a general way to analyze time crystals in prethermal regimes in terms of the static effective Hamiltonian. We also provide an application to Floquet engineering, with which we can perform the simultaneous control of phases and symmetries of the systems. For example, this enables us to control symmetry protected topological phases even when the original system does not respect the symmetry.
We study prethermal time-crystalline order in periodically driven quantum Ising models on disorder-free decorated lattices. Using a tensor network ansatz for the state which reflects the geometry of a unit cell of the lattice, we show through finite entanglement scaling that the system has an exponentially long-lived subharmonic response in the thermodynamic limit, which decays nonperturbatively in deviations from a perfect periodic drive. The resulting prethermal discrete time crystal is not only stable to imperfections in the transverse field, but also exhibits a bipartite rigidity to generic perturbations in the longitudinal field. We call this state a bipartite discrete time crystal and reveal a rich prethermal phase diagram, including multiple regions of bipartite time-crystalline order, uniform time-crystalline order, and thermalization, with boundaries depending delicately on the topology of the decorated lattice. Our results thus uncover a variety of time crystals which may be realized on current digital quantum processors and analog quantum simulators.