Floquet time crystals in clock models

We construct a class of period-$n$-tupling discrete time crystals based on ${\mathbb{Z}}_n$ clock variables, for all the integers $n$. We consider two classes of systems where this phenomenology occurs: disordered models with short-range interactions and fully connected models. In the case of short-range models, we provide a complete classification of time-crystal phases for generic $n$. For the specific cases of $n=3$ and $n=4$, we study in detail the dynamics by means of exact diagonalization. In both cases, through an extensive analysis of the Floquet spectrum, we are able to fully map the phase diagram. In the case of infinite-range models, the mapping onto an effective bosonic Hamiltonian allows us to investigate the scaling to the thermodynamic limit. After a general discussion of the problem, we focus on $n=3$ and $n=4$, representative examples of the generic behavior. Remarkably, for $n=4$ we find clear evidence of a crystal-to-crystal transition between period $n$-tupling and period $n/2$-tupling.

Figure 1

Pictorial representation of a clock model (with $n=6$) in a one-dimensional chain. Each site, labeled by the index $i$, has an $n$-dimensional local Hilbert space, and the blue points on the circle indicate the possible states of the clocks. The red arrows represent the action of the $\tau$ operators, and hence the action of the kicks, as discussed in the main text. In the case shown here, the clocks interact through a nearest-neighbour coupling of amplitude $J_i$. The coupling between the sites is indicated by the double black arrow.

Figure 2

The dots on the circle indicate the possible values of $e^{i\phi }{s}_i^{*}{s}_{i+1}$. In the nonchiral case (for example, $\phi =0$ and $\phi =\pi /3$), different values of $e^{i\phi }{s}_i^{*}{s}_{i+1}$ have the same real part: The spectrum is degenerate. In the chiral case (for example, $\phi =\pi /9$ and $\phi =\pi /6$), there is no degeneracy.

Figure 3

Time evolution of the order parameter ${\mathcal{Z}}_3^{\left[\widehat{\sigma }\right]}\left(t\right)$ for $n=3$ with distinct transverse fields $h_x$ and distinct chiralities $\phi$. We notice the absence of time crystal in the nonchiral case $\phi =0$. Numerical parameters are $\epsilon =0, T=1, J_z=1, h_z=0.9, {\phi }_z=0$, and ${\phi }_x=0$ and the results are averaged over 100 disorder realizations. Error bars (not shown in the plots) are of the order ${10}^{-2}$.

Figure 4

Dependence of $t^{*}$ as defined in the text on the size of the system for different values of $h_x$ and $\phi$. In the chiral case, $t^{*}$ grows exponentially for sufficiently small $h_x$ and becomes independent of the size for large $h_x$. As we get closer to the nonchiral case, the dependence gets flatter until at $\phi =0$ we do not see the exponential growth for any value of $h_x$.

Figure 5

Scaling of the quantity $\overline{{log}_{10}({\mathrm{\Delta }}^{\alpha }/{\mathrm{\Delta }}_0^{\alpha }})$ with the system size, for different values of the chirality parameter $\phi$.

Figure 6

(a) Averaged value of the logarithms of the spectral gaps $\overline{{log}_{10}{\mathrm{\Delta }}_0^{\alpha }}, \overline{{log}_{10}{\mathrm{\Delta }}^{\alpha }}$ as a function of ${log}_{10}{h}_x$. (b) Quasienergy level statistic ratio $r$ as a function of ${log}_{10}{h}_x$. Dashed lines represent the value for Poisson statistics (0.386) and for Wigner-Dyson (0.527). Data were obtained for the chiral case $\phi =\pi /18$.

Figure 7

Averaged value of the logarithms of the spectral gaps $\overline{{log}_{10}{\mathrm{\Delta }}_0^{\alpha }}, \overline{{log}_{10}{\mathrm{\Delta }}^{\alpha }}$ as a function of ${log}_{10}{h}_x$. Data were obtained for the nonchiral case $\phi =0$.

Figure 8

Averaged values of $\overline{{log}_{10}{\mathrm{\Delta }}^{\alpha }}-\overline{{log}_{10}{\mathrm{\Delta }}_0^{\alpha }}$ as a function of $L({log}_{10}{h}_x-{log}_{10}{h}_c)$, for different system sizes and $\phi =\pi /18$. In the inset, the same quantity is plotted vs ${log}_{10}{h}_x$: Dashed lines are the result of the fitting procedure in the region of small $h_x$. The vertical gray line corresponds to the critical value $h_c$.

Figure 9

The curve represents the critical value $h_c$ as a function of the chirality parameter $\phi$. It corresponds to the transition from the time crystal phase to a normal phase.

Figure 10

Dependence of $t^{*}$ (time before period-tripling oscillations decay) on the system size for distinct kicking perturbations $\epsilon$. We consider here the chiral case $\phi =\pi /6$.

Figure 11

[(a), (b)] Time evolution of the order parameters ${\mathcal{Z}}_4^{\left[\sigma \right]}\left(t\right)$ (period-4 time crystal) and ${\mathcal{Z}}_2^{\left[{\sigma }^2\right]}\left(t\right)$ (period-doubling time crystal), for varying $\eta$ parameters. (c) The upper (lower) plot shows the time of the decay of period-4 (period-2) oscillations. Results are obtained with the following choice of parameters: $J_i$ from the uniform distribution $[1/2,3/2], h_{z,i}$ from [0,1], $h_{x,i}$ from [0,1], $\phi =\pi /3, \delta =0.1$

Figure 12

Time-crystal behavior for a system with $n=3$. (Numerical parameters: $J=1, h=0.5$, and $N_T=2^{15}$). (a) Fourier transform of the order parameter ${〈\widehat{\sigma }〉}_t$. The initial state here is given by a symmetry-breaking ground state of the static Hamiltonian ${\widehat{H}}_{n=3}^{\left(LR\right)}$, the period $T=1$, and there is no perturbation in the kicking operator ($\epsilon =0$). [(b), (c)] Finite-size scaling for the position of the dominant frequencies (${\omega }_p$) and the height of the corresponding peak ($f_{{\omega }_p}^{\left[\sigma \right]}$); different initial states and different perturbations to the dynamics are considered.

Figure 13

Time evolution for the order parameter Eq. (45) for a system with $n=3, J=1, h=0.5$, and $T=1$. (a) Time evolution for the absolute value $|{〈\widehat{\sigma }〉}_t|$ for different system sizes. (b) Time evolution for the phase $\varphi \left(t\right)$ across the Rabi oscillation period $t_b$. (c) Scaling of the Rabi oscillation period $t_b$ with the system size.

Figure 14

Distinct time-crystal phases in the model Hamiltonian ${\widehat{H}}_{n=4,\eta }^{\left(LR\right)}$. [(a), (b)] Fourier transforms for the order parameters $\widehat{\sigma }$ and ${\widehat{\sigma }}_{\left[2\right]}$, considering a fixed system size $L=8$. [(c)–(d)] Scaling with the system size for (c) the height of the dominant peak in the Fourier transforms and (d) its corresponding frequencies, for $\eta =0.1$ and 0.9. (Numerical parameters: $n=4,J=J^{\prime }=1,h=h^{\prime }=0.5$, and $N_T=2^{15}$).

Figure 15

The value of $m_{1\mathrm{eq}}=m_{2\mathrm{eq}}$ at the minimum point of the Hamiltonian Eq. (52) vs $h$. For $h<0.77$ it is nonvanishing, marking a ${\mathbb{Z}}_3$ symmetry-breaking phase (numerical parameters $J=1$).

Figure 16

(a) Evolution of $m_1\left(t\right)$ with the Hamiltonian (52) and the kicking (42), with $n=3$. (b) Corresponding Fourier transform [see Eq. (6)]: We see a marked peak at the period-tripling frequency $\omega \left(3\right)=2\pi /\left(3T\right)$ (numerical parameters: $h=0.36, J=1.0, T=0.1, N_T=2048$).

Figure 17

Evolution of $m_1\left(t\right)$ with the Hamiltonian (52) without any kicking, for different initial conditions. Notice the oscillations around an average value different from 0, proving the existence of an interval of energies above the minimum where the corresponding trajectories are ${\mathbb{Z}}_3$ symmetry breaking (numerical parameters: $h=0.36, J=1.0$).

Figure 18

(Upper panels) Evolution of $m\left(t\right)$ with the Hamiltonian Eq. (52) and the $\epsilon \ne 0$-periodic kicking Eq. (42) (numerical parameters: $h=0.36, J=1.0, T=0.1$, and $m_1\left(0\right)=m_2\left(0\right)=m_{\mathrm{eq}}$ in the left panel and $m_1\left(0\right)=m_2\left(0\right)=-0.6$ in the right one). (Central panels) Corresponding Fourier transforms for $N_T=2048$ periods: For small $\epsilon$, we see a marked peak at the period-tripling frequency $\omega \left(3\right)=2\pi /\left(3T\right)$. (Lower panels) Position ${\omega }_p$ (left) and height $|f_{{\omega }_p}^{\left[m_1\right]}{|}^2$ (right) of the main peak of the Fourier power spectrum vs $\epsilon$ for different values of $m_1\left(0\right)=m_2\left(0\right)=m_{\mathrm{ini}}$. We see persistent period-tripling oscillations (main peak at ${\omega }_p=\omega \left(3\right)=2\pi /3$) for $\epsilon$ small enough ($\epsilon \le 0.08$).

Figure 19

The values of $m_{j\mathrm{eq}}$ and $|{\sigma }_{\mathrm{eq}}|$ at the minimum point of the Hamiltonian Eq. (54) vs $h$. For $h<1$ they are nonvanishing, marking a ${\mathbb{Z}}_4$ symmetry-breaking phase (numerical parameters: $J=1$).

Figure 20

Evolution of $m_1\left(t\right)$ with the Hamiltonian Eq. (54) and no kicking. Initial conditions ${\theta }_1\left(0\right)={\theta }_2\left(0\right)={\theta }_3\left(0\right)=0, m_2\left(0\right)=m_{2\mathrm{eq}}, m_1\left(0\right)=m_3\left(0\right)=m_{1\mathrm{eq}}+\delta m$. $J=1, h=0.36$.

Figure 21

Dynamics of $m_1\left(t\right)$ with the Hamiltonian [Eq. (54)] and perturbed kicking [Eq. (43)]. Time domain (left panels) and frequency domain (right panels). For $\epsilon =0.01$, we see the period-4-tupling oscillations (appearing as a peak at $\omega \left(4\right)=\pi /\left(2T\right)$ which disappear for larger $\epsilon$). Initial conditions: ${\theta }_1\left(0\right)={\theta }_2\left(0\right)={\theta }_3\left(0\right)=0, m_2\left(0\right)=m_{2\mathrm{eq}}, m_1\left(0\right)=m_3\left(0\right)=m_{1\mathrm{eq}}+\delta m$; we consider two different initial conditions, $\delta m=0$ in the upper panels and $\delta m=0.08$ in the lower ones. Numerical parameters: $J=1, h=0.36, T=0.1$.

Figure 22

Order parameters vs $\eta$ in the minimum-energy state of the Hamiltonian Eq. (57) (numerical parameters $h=h^{\prime }=0.5, J=J^{\prime }=1.0$).

Figure 23

Evolution of $m_1\left(t\right)$ with the Hamiltonian Eq. (57) and the kicking Eq. (43). Both period 4-tupling for $\eta =0.36$ (upper panel) and period doubling are present for $\eta =0.82$ (central panel; the factor ${(-1)}^k$ makes the period-doubling oscillations appear as an almost constant object). No time crystal whatsoever for $\eta =0.96$ and $\epsilon =0.01$ (lower panel). (Numerical parameters $h=h^{\prime }=0.5, J=J^{\prime }=1.0$.)

Figure 24

(Upper left panel) Peak in the Fourier transform at the period-4-tupling frequency ($|f_{{\omega }_{p}n=4}^{\left[m_1\right]}|$) vs $\eta$; we notice that it disappears at a value ${\eta }_c\sim 0.8$ where the period-4-tupling time-crystal phase ends up. (Upper right panel) Peak in the Fourier transform at the period doubling frequency ($|f_{{\omega }_{p}n=2}^{\left[m_1\right]}|$) vs $\eta$; we notice that it disappears at a value ${\eta }_{c1}>{\eta }_c$ when $\epsilon \ne 0$. For ${\eta }_c<\eta <{\eta }_{c1}$, there is a period-doubling time-crystal phase; for $\eta >{\eta }_{c1}$, there is no time crystal. (Bottom left panel) In the period-4-tupling phase, the peak at the period-4-tupling frequency sticks at $\omega \left(4\right)=\pi /\left(2T\right)$. (Bottom right panel) Peak at the period-doubling frequency $\omega \left(2\right)=\pi /T$ (numerical parameters: $h=h^{\prime }=0.5, J=J^{\prime }=1.0$).

Figure 25

Time evolution of the order parameters $Z\left(t\right)$ (period-4 time crystal) and $Z_{\left[2\right]}\left(t\right)$ (period-doubling time crystal), for varying $\lambda$ parameters. Results are obtained with the following choice of parameters: $J_i$ from the uniform distribution $[1/2,3/2], h_{z,i}$ from [0,1], $g_i$ from [0,1], ${\alpha }_1=\frac{e^{i\pi /3}}{2}, \epsilon =0.1$.

Figure 26

[(a) and (b)] Scaling of the $n$-fold gap $E_{n-1}-E_0$ with the system size, for $\eta =0$ and $n=3$ and 4, respectively. (c) Under a small perturbation $\widehat{W}$ explicitly breaking the symmetry of the Hamiltonian, the (nondegenerated) ground state acquires a macroscopic value for the order parameter ${\langle {\widehat{\sigma }}_i\rangle }_{\mathrm{GS}}\sim 1$ in the region where the $n$-fold gap is roughly smaller than the perturbation. System sizes: $L=50$ and 30 for $n=3$ and 4, respectively; $J=1$ in all the plots.

Figure 27

Finite-size effects and scaling with the size of the system for the order parameter $\langle {\sigma }_i\rangle$ in the ground state of the Hamiltonian ${\widehat{H}}_n^{\left(\mathrm{LR}\right)}+\widehat{W}$, with $\widehat{W}=-\delta {\sum }_{i=1}^L({\widehat{\sigma }}_i+{\widehat{\sigma }}_i^{†})$, a small perturbation breaking explicitly the symmetry of the model. We show here results for the case with $n=3$. In the left panel, we set $h/J=0.5$ and see the exponential corrections to the order parameter when $\delta \lesssim (E_n-E_0)$, while in the opposed case it scales to a finite value. In the right panel, we set $h/J=0.01$, where we clearly see the polynomial scaling of the order parameter to a finite value in the thermodynamic limit.

Figure 28

(a) Low-energy spectrum (shifted by the ground-state energy) of the Hamiltonian ${\widehat{H}}_{n,\eta =0}^{\left(LR\right)}$ for $n=3$ and $n=4$: The spectrum is organized in $n$-tuplets. In panels (b) and (c), we show the scaling of the $n$-tuplets' energy splittings [${\mathrm{\Delta }}_{q=n,\alpha }$ as defined in Eq. (E1)] with the system size. In panel (b), we show the case $n=3$ with $h=0.5$, and in panel (c) $n=4$ with $h=0.1$. In the panels (d) and (e), we show the scaling of the order parameter averaged over such $n$-tuplets, i.e., the averaged order parameter over the $n\alpha$ lowest eigenstates [${\sigma }_{\mathrm{av}}\left(n\alpha \right)$ as defined in Eq. (E2)]. In panel (d), we show the case $n=3$ with $h=0.5$, and in panel (e) $n=4$ with $h=0.1$. $J=1$ in all the plots.

Figure 29

$|{〈{\widehat{\sigma }}_i〉}_{\mu }|$ vs $E_{\mu }/L$ for (a) $n=3$ with $h=0.5$ and (b) $n=4$ with $h=0.1$ in a symmetry-breaking phase. $J=1$ in both cases. We see an extensive number of symmetry-breaking states below the broken-symmetry edge.

Figure 30

Different symmetry breakings in ${\widehat{H}}_{4,\eta }^{\left(LR\right)}$. (a) Scaling of the fourfold gap $E_3-E_0$ and the twofold gap $E_1-E_0$ with the size of the system, for parameters $\eta =0.1$ and 0.9. (b) Under a small perturbation ${\widehat{W}}_{\left[2\right]}$ breaking explicitly the symmetry of the Hamiltonian, the (nondegenerated) ground state acquires a macroscopic value for the order parameters ${\langle {\widehat{\sigma }}_i\rangle }_{\mathrm{GS}}\sim 1$ or ${\langle {\widehat{\sigma }}_i^2\rangle }_{\mathrm{GS}}\sim 1$, in the region where its corresponding $q$-fold gap is roughly smaller than the perturbation strength $\delta \sim {10}^{-8}$. Here the system size is $L=30$ and the twofold gap is negligible ($<{10}^{-10}$) and omitted from the figure. (c) Scaling of ${\mathrm{\Delta }}_{4,\alpha }$ and ${\mathrm{\Delta }}_{2,\alpha }$ [Eq. (E1)] with the system size. (d) Scaling of ${\sigma }_{\mathrm{av}}\left(\alpha \right)$ and ${\left({\sigma }^2\right)}_{\mathrm{av}}\left(\alpha \right)$ [Eq. (E2)] with the system size. ${\sigma }_{\mathrm{av}}\left(\alpha \right)$ is null in the case of $\eta =0.9$, and we thus omit it from the figure.