Discrete Time Crystal Order in Spin-Chains Enabled by Floquet Flat-Bands

Abstract

We propose a novel protocol to realize discrete time-crystal (DTC) order in clean, periodically driven spin-$1/2$ chains. In each drive cycle, a global spin flip is followed by a two-tone flat-band segment. This flat-band segment engineers a fully degenerate Floquet quasienergy spectrum, suppresses thermalization, and stabilizes a robust period-doubled subharmonic response. Using exact time evolution, we identify a pronounced subharmonic peak at half the drive frequency in the Fourier spectrum of the order parameter, thereby providing clear evidence for the emergence of stable DTC. The resulting phase is insensitive to system size, interaction strength, and interaction range; however, it remains sensitive to spin-rotation errors ($\varepsilon_r$), which can destabilize the subharmonic response. Compared with disorder-induced many-body localized (MBL) and disorder-free dynamically many-body localized (DMBL) DTCs, we find that the exact flat-band protocol offers a broader tunability of drive parameters, whereas MBL and DMBL based DTCs are more resistant to $\varepsilon_r$. In particular, the $\varepsilon_r$ sensitivity can be suppressed by incorporating additional spin-spin interactions that have modest deviations from the ideal flat-band protocol. This manifests itself in a robust DTC response over a finite window of spin-coupling strengths and drive frequencies. Our results establish flat-band driving as a versatile and experimentally relevant route to DTC order in disorder-free spin systems and motivate further exploration of non-equilibrium phases.

Figures (12)

• Figure 1: Three time dependent drives $\lambda_{1,2,s}(t)$ protocol for the proposed flat-band DTC model. The first time interval, $T_1$, consists of a spin-flip drive, while the second time interval, $T_2$, consists of a flat-band protocol.
• Figure 2: Quasi-energy spectrum of the proposed flat-band protocol within the DTC model. Quasi-energy levels plotted against the quasi-energy eigenstates (n). The quasi-energy levels exhibit all-degeneracy at zero energy, indicating the emergence of the flat-band. The corresponding $\hat{H}_{\mathrm{FB}}(T_2)$ drive protocol is illustrated in the inset.
• Figure 3: Temporal evolution of the magnetization $\expval{\hat{\sigma}^z(t)}$ for the proposed DTC model in spin chains of sizes $N=8$ and $N=10$, with the spin rotational error fixed at $\varepsilon_r = 0$. The magnetization oscillates within the range [-1, 1] at each discrete time period, demonstrating a clear 2T periodicity, as depicted in panels (b) and (c). The left panel (a) illustrates the FFT of the magnetization, highlighting a distinct subharmonic peak at $\omega/2$ for both system sizes, thereby confirming the realization of the DTC phase.
• Figure 4: The top panel (a) illustrates the temporal variation of the magnetization $\expval{\hat{\sigma}^z(t)}$ for the proposed DTC model in a spin chain of size $N=8$, subjected to varying spin-spin interaction strengths. The bottom panel (b) presents the FFT of the corresponding magnetization, revealing the presence of a subharmonic peak at $\omega/2$ for all values of $\lambda_0$.
• Figure 5: Temporal evolution of the magnetization order parameter, $Z(nT) = (-1)^n \expval{ \hat{\sigma}^z(nT)}$, at stroboscopic times for system sizes $N = 8, 10$ and various spin-spin interaction ranges ($\beta = 0, 1.5, 2.5, \infty(\text{inf})$).