Hierarchical time crystals

TL;DR

The work investigates how nested temporal order can emerge in a non-equilibrium quantum system by coupling a continuous-time crystal (CTC) and a discrete-time crystal (DTC) in a time-independent setup. It develops a Lindblad-based framework and derives mean-field equations in the thermodynamic limit for various coupling schemes (coherent, dissipative, and spin-exchange), then corroborates results with finite-size analyses. A two-step hierarchical symmetry breaking is revealed: the CTC acquires an emergent period $T_{\text{CTC}}$, which the DTC discretely locks to, yielding $T_{\text{DTC}}=nT_{\text{CTC}}$, i.e., an HTC with integer and, in some cases, fractional locking. The HTC phase is robust across parameter ranges and coupling types, suggesting observable hierarchical dissipative phase transitions in platforms like cavity QED and Bose–Einstein condensates.

Abstract

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 two-fold temporal symmetry breaking, resulting in a hierarchical time crystal phase. Interestingly, one of the subsystems breaks an emergent discrete temporal symmetry that does not exist in the dynamical generator but rather emerges dynamically, leading 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.

Figures (6)

• Figure 1: Sketch of the model and hierarchical symmetry breaking. (a) The time-independent coupled time crystal model features a CTC subsystem, realized as a collective spin ensemble, which interacts with a DTC subsystem of long-range coupled spins at interaction strength $\eta$. Each spin-$1/2$ system consists of a ground state $|{0}\rangle$ and an excited state $|{1}\rangle$. Both CTC and DTC are subjected to constant drives with strength $\Omega$ and $h$, respectively, with the CTC additionally exhibiting collective emission at a rate $\kappa$. (b) Illustration of spontaneous symmetry breaking (SSB) possibilities: no SSB, where the observables $\langle \hat{O} \rangle_\alpha$ are time-independent for both CTC and DTC; one-fold SSB, where only the CTC breaks a temporal symmetry, leading to synchronized periodic oscillations of both DTC and CTC with period $T_{\text{CTC}}$; and two-fold SSB, where the DTC discretely breaks the emergent periodicity of the CTC phase, leading to subharmonic oscillations with a period $T_{\text{DTC}} = nT_{\text{CTC}}$. The latter phase is termed a hierarchical time crystal (HTC).
• Figure 2: Hierarchical symmetry breaking via coherent coupling. In panel (a), stroboscopic Fourier amplitudes of the DTC magnetization z component $\mathcal{F}_s[m_{\text{D}}^z]$ are plotted against the rescaled inter-TC coupling strength $\eta/\kappa$. The relative frequency $\omega_r=\omega_{\text{DTC}}/\omega_{\text{CTC}}$ exhibits distinct locking plateaus at integer values for the DTC order $n$, where $\omega_{\text{DTC}}$($\omega_{\text{CTC}}$) represents the dominant frequency of DTC (CTC). System parameters are fixed to $\Omega/\kappa=2$, $J/\kappa=0.1$, and $h/\kappa=0.25$, while the initial state is $(m^x_\text{C,D},m^y_\text{C,D},m^z_\text{C,D})=(0,0,1)$. The main panel (inset) probes $m_2^z$ over 1000 (3000) CTC periods. The HTC phase for the specific case of a $4$-DTC (diamond in (a)) is detailed in panels (b-e). Panel (b) presents the normalized Fourier spectrum $\mathcal{F}[m_{i}^z]$ of both CTC (blue) and DTC (red). The respective dominant frequencies are labeled. In panel (c), the time evolution of $m_{\text{C}}^z$ (blue) and $m_{\text{D}}^z$ (red) is displayed for finite system sizes $N\in \{ 20,40,\dots,100\}$ with ascending opacity, which approaches the mean-field solutions indicated with dashed lines. Panels (d) and (e) show the dependence of the real and imaginary parts of the dominant Liouvillian eigenvalues on the system size. The dashed lines indicate linear fits to the CTC and DTC eigenvalues, confirming the spectral gap closure with purely imaginary eigenvalues, which match the frequencies obtained from the mean-field analysis in panel (b).
• Figure 3: Hierarchical symmetry breaking via dissipative coupling. (a) Stroboscopic Fourier amplitudes of the DTC magnetization z component $\mathcal{F}_s[m_{\text{D}}^z]$ are plotted against the rescaled inter-TC coupling strength $\eta/\kappa$. The dominant component of the relative frequency $\omega_r$ exhibits distinct locking plateaus at both integer and fractional values for the DTC order $n$. System parameters are fixed to $\Omega/\kappa=2$, $J/\kappa=0.08$, and $h/\kappa=0.2522$, while the initial state is $(m^x_\text{C,D},m^y_\text{C,D},m^z_\text{C,D})=(0,0,1)$. The main panel and inset probe $m_2^z$ over 1000 and 3000 CTC periods, respectively. The system parameters are chosen to target a (b,c) $3$-DTC (diamond in (a)) and a (d,e) fractional $10/3$-DTC (star in (a)), respectively. (b),(d) Normalized Fourier spectra of both CTC (blue) and DTC (red) with labeled dominant frequencies. (c),(e) Time evolution of $m_{\text{C}}^z$ (blue) and $m_{\text{D}}^z$ (red) for finite system sizes $N\in \{ 20,40,\dots,100\}$ displayed in ascending opacity, approaching the dashed lines indicating the corresponding mean field solution.
• Figure S1: HTC parameter regimes for coherent coupling. (a,c) Stroboscopic Fourier amplitudes of the corresponding $n$-DTC, which exceed 0.5, are marked in their respective color. The CTC drive strength is fixed at $\Omega/\kappa=2$, while the initial state is $(m^x_\text{C,D},m^y_\text{C,D},m^z_\text{C,D})=(0,0,1)$. (a) DTC all-to-all interaction strength $J/\kappa$ vs. inter-TC coupling strength $\eta/\kappa$ with $h/\kappa=0.25$. (c) DTC drive strength $h/\kappa$ vs. inter-TC coupling strength $\eta/\kappa$ with $J/\kappa=0.1$. (b,d) Stroboscopic Fourier amplitudes of the DTC magnetization z component $\mathcal{F}_s[m_{\text{D}}^z]$ are plotted against the inter-TC coupling strength $\eta/\kappa$ for a fixed parameter set. The CTC is fixed in its time crystal regime with $\Omega/\kappa=2$ and $m_{\text{D}}^z$ is probed over 1000 CTC periods. (b) Fixed DTC all-to-all interaction strength at $J/\kappa=0.125$ corresponding to the dashed line in (a). Locking intervals of $\eta/\kappa$ coincide with the predicted intervals in (a), as exemplarily visualized with grey dotted lines for the $6$-DTC. (d) Fixed DTC drive strength at $h/\kappa=0.45$ corresponding to the dashed line in (c). The in (c) predicted locking intervals are clearly distinguishable (e.g. grey dotted lines for the $3$-DTC). Additionally, a new, fractional locking plateau emerges at $\eta/\kappa \approx 0.92$, which has not been tracked in panel (c).
• Figure S2: HTC parameter regimes for dissipative coupling. Relative Fourier amplitudes of the corresponding $n$-DTC, which exceed 0.5, are marked in their respective color. The CTC drive strength is fixed at $\Omega/\kappa=2$, while the initial state is $(m^x_\text{C,D},m^y_\text{C,D},m^z_\text{C,D})=(0,0,1)$. (a) DTC all-to-all interaction strength $J/\kappa$ vs. inter-TC coupling strength $\eta/\kappa$. (b) DTC drive strength $h/\kappa$ vs. inter-TC coupling strength $\eta/\kappa$.