Integrable Floquet Time Crystals in One Dimension

TL;DR

The paper addresses the challenge of realizing robust discrete-time crystals in strictly one-dimensional, disorder-free systems. It introduces a Floquet-engineered, integrable 1D lattice with a tunable NNN parameter $oldsymbol{ ext{lambda}}$ that preserves integrability while enabling Floquet-gap engineering to pin a subharmonic mode at momentum $k_0$, under an equilibrium resonance condition $g_1=2oldsymbol{ extomega}$. The authors map a rich phase portrait in the $(g_0,oldsymbol{ extlambda})$ plane, identifying robust DTC regions as well as FTC, OSL, and PM phases, and they diagnose the dynamics via near-unit long-time fidelity $ar F_{k_0}$ and stroboscopic correlators $ar C_z$. Finite-size scaling reveals an algebraic melting-time scaling $oldsymbol{t_m}(N) obreak \\sim N$ in the deep DTC regime, with a two-Lorentzian FFT fit and RANSAC regression confirming distinct scaling across phases, all framed within a disorder-free, integrable mechanism for subharmonic stabilization. These results suggest a practical, disorder-free route to long-lived DTCs in 1D and point toward experimental realizations in near-term quantum simulators.

Abstract

We demonstrate the realization of a discrete-time crystal (DTC) phase in a family of periodically driven, one-dimensional quadratic lattice Hamiltonians that can be obtained using spin chains. These interactions preserve integrability while opening controllable gaps at resonant quasienergies and pinning the emergent quasienergy modes that are responsible for subharmonics. We demonstrate that the DTC phase is rigid in the parameter space of transverse field and an additional interaction like NNN coupling strength, with the drive frequency optimized to produce the strongest subharmonic response. We also provide a detailed phase portrait of the model, exhibiting a variety of new dynamical phases, such as a fragile time crystal and both spin-liquid and paramagnetic phases, as well as sharp quantum phase transitions between them. Finite-size scaling of the Floquet quasienergy splitting between the emergent subharmonic mode and its conjugate shows that the DTC lifetime diverges exponentially with system size. Our work thus establishes a novel mechanism for realizing robust, long-lived DTCs in one dimension, and paves the way for their experimental realization in near-term quantum simulators. Motivation for this work stems from the limitations of disorder-based stabilization schemes that rely on many-body localization and exhibit only prethermal or finite-lived plateaus, eventually restoring ergodicity. Disorder-free routes are therefore highly desirable. Integrable (or Floquet-integrable) systems provide an attractive alternative because their extensive set of conserved quantities and constrained scattering strongly restrict thermalization channels. Our construction exploits these integrable restrictions together with short-range NNN engineering to produce a clean, robust DTC that avoids the prethermal fragility of disordered realizations.

Figures (6)

• Figure 1: Panels depicting the Bogolon energies $\pm E_{k}(g_0, \lambda)$, obtained from the text are presented as a function of momentum $k\in\left[-\pi,\pi\right]$ across various parameters $g_{0}, \lambda$. Each row in the plots corresponds to a fixed value of $g_{0}$, while different values of $\lambda$ are systematically displayed across columns. The values assigned to each row for $g_{0}$ and the values for $\lambda$ are clearly indicated above the topmost panels and after the rightmost panels, respectively. Additionally, the values of $\pm E_{k}$ at the high-symmetry point $k=0$ (non high-symmetry points at $k=\arccos{(-1/2\lambda)}$) are indicated by blue (magenta) colored dots.
• Figure 2: Panels depicting the Floquet quasienergies $\pm\theta_{k}$ (in units of time period $T$), derived from Eqn. \ref{['eq:Ak']}, corresponding to the Floquet Hamiltonian as expressed in Eqn. \ref{['eq:floq:hamilt:spinor']}, are presented in the same manner and layout as in Fig. \ref{['fig:bogolon:energies']}. Additionally, the parameters $g_{1}$ and $T$ have been specifically set to $1$ and $2.3$, respectively. The horizontal lines at $\theta_k/T = \pm {\omega}/{4}$ are indicated by dashed lines. A subharmonic solution exists if the quasienergy curve intersects these lines.
• Figure 3: Density plot of the long-time average (strobed at even multiples of $T$) of the fidelity $\overline{F}_{k_{0}}$ at the optimal momentum $k_{0}$ (left panel), and the correlations $\overline{C_z}$ . In the right panel, the FBZ was discretized into $N=1000$ equally spaced points between $-\pi$ and $\pi$, and $k_{0}$ explicitly included in the chosen sum. The time average is performed over $10^4$ sets of $2T$-intervals, where $T=2\pi/\omega$. The parameter space is chosen to be $g_{0},\lambda$. For each point, the value of $\omega$ is optimized using the trust-region method to minimize the cost function in Eqn. \ref{['eq:costfunction']}. The parameters $g_{1}$ is set to $2\omega$, the equilibrium resonance condition described in the text. The solid lines indicate the gapless points in the Floquet quasienergy spectrum (Eqs. \ref{['eq:gapless:hs']} and \ref{['eq:gapless:nohs']}), where the lines that correspond to gaplessness in the HS (non-HS) points are colored blue (magenta). The gapless points divide the parameter space into Time Crystal and spin-liquid phases, as labeled in the figure.
• Figure 4: Density plot of the long-time average of the fidelity $\overline{F}_{k_{0}}$ at the optimal momentum $k_{0}$, averaged over $10^4$ sets of $2T$-intervals, where $T=2\pi/\omega$. All other quantities are identical to Fig. \ref{['fig:fidelity_long_time_avg']}, except that $g_{1}$ is varied across the panels as indicated.
• Figure 5: Center: Floquet phase diagram in the $(g_{0},\lambda)$ plane at equilibrium resonance $g_{1}=2\omega$, indicating regions of discrete time crystal (DTC, yellow), fragile time crystal (FTC, light yellow), oscillatory spin-liquid (OSL, green) and paramegnet (PM, pale green). Solid blue and magenta lines mark analytically obtained Floquet gapless loci at high-symmetry and non-HS momenta [Eqs. (\ref{['eq:gapless:hs']}, \ref{['eq:gapless:nohs']})]. Black dots (a-h, x, y) identify parameter points used in the finite-size analysis. Surrounding panels: normalized power spectra |FFT| of the stroboscopic subharmonic at the optimal momentum (with incommensurate $k_{0}$ replaced by its nearest lattice momentum $k_{0}^{R}(N)$) for several system sizes (solid traces: increasing N; dashed verticals: fitted peak centers). Each panel reports the fitted scaling $\delta\Omega\sim N^{\alpha}$ obtained from the two-Lorentzian peak splitting $\delta\Omega=|\Omega_{1}-\Omega_2|$; insets display the log-log data and the RANSAC fit.