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Two-dimensional discrete time crystals protected by multiple Majorana modes

Raditya Weda Bomantara, Sen Mu, Jiangbin Gong

Published Dec 7, 2020• 0 citations
OpenAlex

Abstract

Discrete time crystals (DTCs) are exotic phases of matter characterized by robust and persistent subharmonic response of observables to a time-periodic driving. We uncover an important and rather general mechanism, made possible by the so-called Majorana $\pi$ modes (MPMs), in generating and stabilizing the DTCs in periodically driven spin-1/2 systems in a two-dimensional setting. Of particular significance is that we can increase the number of MPMs linearly with the number of spins and consequently strengthen the performance of the DTCs. Our findings not only provide a topological insight into DTCs, but also pave the way towards studies of DTC phases in higher dimension with sharper subharmonic response, by searching for a lattice configuration that maximizes the number of MPMs.

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Physical Review Letters•Jun 30, 2025•1 citation
Discrete time crystals (DTCs) are exotic phases of matter characterized by robust and persistent subharmonic response of observables to a time-periodic driving. We uncover an important and rather general mechanism, made possible by the so-called Majorana π\piπ modes (MPMs), in generating and stabilizing the DTCs in periodically driven spin-1/2 systems in a two-dimensional setting. Of particular significance is that we can increase the number of MPMs linearly with the number of spins and consequently strengthen the performance of the DTCs. Our findings not only provide a topological insight into DTCs, but also pave the way towards studies of DTC phases in higher dimension with sharper subharmonic response, by searching for a lattice configuration that maximizes the number of MPMs.
Studies of periodically driven one-dimensional many-body systems have advanced our understanding of complex systems and stimulated promising developments in quantum simulation. It is hence of interest to go one step further, by investigating the topological and dynamical aspects of periodically driven spin ladders as clean quasi-one-dimensional systems with spin-spin interaction in the rung direction. Specifically, we find that such systems display subharmonic magnetization dynamics reminiscent to that of discrete time crystals (DTCs) at finite system sizes. Through the use of generalized Jordan-Wigner transformation, this feature can be attributed to the presence of corner Majorana $\ensuremath{\pi}$ modes (MPMs), which are of topological origin, in the systems' equivalent Majorana lattice. Special emphasis is placed on how the coupling in the rung direction of the ladder prevents degeneracy from occurring between states differing by a single spin excitation, thus preserving the MPM-induced $\ensuremath{\pi}/T$ quasienergy spacing of the Floquet eigenstates in the presence of parameter imperfection. This feature, which is absent in their strict one-dimensional counterparts, may yield fascinating consequences in future studies of higher dimensional Floquet many-body systems.
Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a type of DTC in a periodically driven surface code, the subharmonic signature of which is only observable with respect to some nonlocal order parameter. By modifying the commonly used metrics for characterizing ordinary DTCs, i.e., the spectral functions, spin-glass order parameters, and two-point correlators, we further numerically demonstrate the rich phases of the proposed system. Specifically, depending on the system parameters and boundary conditions, we find that the system may fall into either a ``surface code'' phase, trivial paramagnet phase, nonlocal period-doubling DTC phase, or nonlocal period-quadrupling DTC phase. Our work thus demonstrates the prospect of exploring topological codes for discovering novel phases of matter.
Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple πππ-pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.
Discrete time crystals (DTCs) have recently attracted increasing attention, but most DTC models and their properties are only revealed after disorder average. In this Letter, we propose a simple disorder-free periodically driven model that exhibits nontrivial DTC order stabilized by Stark many-body localization (MBL). We demonstrate the existence of the DTC phase by analytical analysis from perturbation theory and convincing numerical evidence from observable dynamics. The new DTC model paves a new promising way for further experiments and deepens our understanding of DTCs. Since the DTC order does not require special quantum state preparation and the strong disorder average, it can be naturally realized on the noisy intermediate-scale quantum hardware with much fewer resources and repetitions. Moreover, in addition to the robust subharmonic response, there are other novel robust beating oscillations in the Stark-MBL DTC phase that are absent in random or quasiperiodic MBL DTCs.
We propose a Floquet time crystal model that responds in arbitrary multiples of the driving period. Such an nnn-tuple discrete time crystal is theoretically constructed by permutations in a disordered spin chain and is well suited for experimental implementations. Transitions between these time crystals with different periods give rise to a novel phase of matter that we call subspace-thermal discrete time crystals, where states within subspaces of definite charges are fully thermalized at an early time. However, the whole system still robustly responds to the periodic driving subharmonically, with a period being the greatest common divisor of the original two periods. Existing theoretical analysis from many-body localization cannot be used to understand the rigidity of such subspace-thermal time crystal phases. To resolve this, we develop a theoretical framework for the robustness of DTCs from the perspective of the robust $2\ensuremath{\pi}/n$ quasi-energy gap. Its robustness is rigorously proved if the system satisfies a certain condition where the mixing length, defined by the Hamming distance of the symmetry charges, does not exceed a global threshold. Although whether the condition is satisfied in generic disordered systems is unclear, the rigorous proof for DTC properties applies beyond the models considered here and extends to other existing DTCs realized by kicking disordered MBL systems, where the condition is automatically satisfied and conventional MBL-DTCs can be regarded as a special case of the subspace-thermal DTC with the subspace dimension being one, thus offering a systematic way to construct discrete time crystal models. We also introduce the notion of DTC charges that allow us to probe observables that spontaneously break the time-translation symmetry in both regular discrete time crystals and subspace-thermal discrete time crystals. Moreover, our discrete time crystal models can be generalized to systems with higher spin magnitudes or qudits, as well as to higher spatial 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.