Floquet time crystal in the Lipkin-Meshkov-Glick model

In this work we discuss the existence of time-translation symmetry breaking in a kicked infinite-range-interacting clean spin system described by the Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under perturbations of the kicking protocol, its existence being intimately linked to the underlying ${\mathbb{Z}}_2$ symmetry breaking of the time-independent model. We show that the model being infinite range and having an extensive amount of symmetry-breaking eigenstates is essential for having the time-crystal behavior. In particular, we discuss the properties of the Floquet spectrum, and show the existence of doublets of Floquet states which are, respectively, even and odd superposition of symmetry-broken states and have quasienergies differing of half the driving frequencies, a key essence of Floquet time crystals. Remarkably, the stability of the time-crystal phase can be directly analyzed in the limit of infinite size, discussing the properties of the corresponding classical phase space. Through a detailed analysis of the robustness of the time crystal to various perturbations we are able to map the corresponding phase diagram. We finally discuss the possibility of an experimental implementation by means of trapped ions.

Figure 1

Some broken symmetry eigenstates of the Hamiltonian Eq. (1) with $J=1$, $h<1$, and energies below the broken symmetry edge $E^{*}=-Nh$. We plot their (real) amplitudes in the ${\widehat{S}}^z$ representation. Numerical parameters: $N=200,h=0.5$.

Figure 2

Stroboscopic magnetization ($t=n\tau$) for $N=100$ and kicking given by Eq. (2) with $h_{\mathrm{i}}=0.32$ and $\varphi =\pi$. We see persistent oscillations with period ${\tau }_B=2\tau$ occurring for many different initial conditions $\left|{\psi }_{\mathrm{GS}}^{-}\left(h_{\mathrm{i}}\right)\right.〉$ (see the main text).

Figure 3

Beatings of $|m_n^z|$ for different values of $N$. Numerical parameters $h_{\mathrm{i}}=h=0.32,\varphi =\pi ,\tau =0.6$.

Figure 4

(In all the plots $\tau =0.6,\varphi =\pi$.) Upper panel: Power spectrum of the Fourier transform of $m_n^z$ for different values of $N$; notice the doublet of marked peaks around ${\omega }_B=\pi /\tau$ (here $h_{\mathrm{i}}=h=0.32$) which shrink into a single one as $N$ increases. Lower panel, main figure: Power spectrum of the Fourier transform of $m_n^z$ for different $N$, $h_{\mathrm{i}}=0.7,h=0.32$. Notice the same peaks here. Lower panel, inset: Plot of the height of the maximum peaks around $\omega =\pi /\tau$ vs $N$. It converges to a constant for $N\to \infty$: this highlights the existence of the time-crystal behavior in the thermodynamic limit. (The Fourier transforms have been performed over 32 768 driving periods in the main figures and over $K=65536$ periods in the insets.)

Figure 5

Main figures (upper panel): $m_n^z$ for $N=100$ and different values of $\varphi$; if $\varphi$ deviates too much from $\pi$ we see that the oscillations of period $2\tau$ die away. (Lower panel) Corresponding power spectrum of the Fourier transform of the $z$ magnetization: when there is period doubling, two peaks around ${\omega }_B=\pi /\tau$ appear. Insets (upper panel): Dependence on $N$ of the main peaks frequencies ${\omega }_{\pm }(\varphi ,N)$ in the Fourier power spectrum: whenever there is the time crystal, these frequencies tend to $\pi /\tau$. (Lower panel) Dependence on $1/N$ of the main peaks height $|m_{{\omega }_{\pm }(\varphi ,N)}^z{|}^2$ in the Fourier power spectrum: whenever there is the time crystal, this height tends to a constant for $N\to \infty$, otherwise it tends to 0 as a power law (notice the logarithmic scale on both axes). (Numerical parameters $h=0.32,\tau =0.6$. The Fourier transforms in the insets are performed over $K=65536$ driving periods, in the main figures over $K=32768$ periods.)

Figure 6

(Upper panel) Dependence on $\varphi$ of the main peaks height $|m_{{\omega }_{\pm }(\varphi ,N)}^z{|}^2$ for different $N$ and initial conditions: we see a full region around $\varphi =\pi$ where it is different from 0. (Lower panel) Dependence on $\varphi$ of the main peaks frequencies: when $|m_{{\omega }_{\pm }(\varphi ,N)}^z{|}^2$ is different from 0 the frequencies equal ${\omega }_B$ up to terms exponentially small in $N$ and the system is a time crystal. (Numerical parameters $h=0.32,\tau =0.6$, the Fourier transforms have been performed over $K=65536$ driving periods.)

Figure 7

Main figures (upper panel): Evolution of $m_n^z$ for $N=100$ and different values of $\lambda$. (Lower panel) Power spectra $|m_{\omega }^z{|}^2$ of the corresponding Fourier transforms. Insets (upper panel): Dependence on $N$ of the frequencies ${\omega }_{\pm }(\lambda ,N)$ of the main peaks in the power spectrum. (Lower panel) Height $|m_{{\omega }_{\pm (\lambda ,N)}}^z{|}^2$ of the main peaks versus $1/N$ (double logarithmic scale). (Numerical parameters: $h_{\mathrm{i}}=h=0.32,\tau =0.6$. The Fourier transforms are performed over $K=65536$ driving periods in the insets and $K=32768$ periods in the main figures.)

Figure 8

(Upper panel) Three copies of the Floquet spectrum vs $\alpha$, horizontally shifted with respect to each other by $1+N/2$ and divided by $\pi /\tau$. The three curves constantly differ from each other by 1 along the vertical axis: each quasienergy has its own partner shifted by $\pm \pi /\tau$. (Lower panel) The corresponding Floquet states are organized in pairs: in each pair the quasienergies differ by $\pi /\tau$ and the two states are even and odd superpositions of symmetry broken states [we plot the (real) amplitudes of the members of one of such pairs in the basis of the eigenstates of ${\widehat{S}}^z$]. Numerical parameters: $N=100,\tau =0.006,h_0=0.32,\varphi =\pi ,\lambda =0$.

Figure 9

(a) (Upper plot) Plot of $〈{log}_{10}{\mathrm{\Delta }}_0〉$ and $〈{log}_{10}{\mathrm{\Delta }}_{\pi }〉$ [see Eq. (8)] vs $\varphi$ fixing $\lambda =0$; (lower plot) plot of the corresponding fitting exponents $\alpha \left({\mathrm{\Delta }}_0\right)$ and $\alpha \left({\mathrm{\Delta }}_{\pi }\right)$ vs $\varphi$ [see Eq. (10)]. (b) (Upper plot) Plot of $〈{log}_{10}{\mathrm{\Delta }}_0〉$ and $〈{log}_{10}{\mathrm{\Delta }}_{\pi }〉$ [see Eq. (8)] vs $\lambda$ fixing $\varphi =\pi$; (lower plot) plot of the corresponding fitting exponents $\alpha \left({\mathrm{\Delta }}_0\right)$ and $\alpha \left({\mathrm{\Delta }}_{\pi }\right)$ vs $\lambda$ [see Eq. (10)]. As expected, there is $\pi$-spectral pairing—${10}^{〈{log}_{10}{\mathrm{\Delta }}_0〉}\ll {10}^{〈{log}_{10}{\mathrm{\Delta }}_{\pi }〉}$ and $\alpha \left({\mathrm{\Delta }}_{\pi }\right)>\alpha \left({\mathrm{\Delta }}_0\right)$—when there is time-crystal behavior. (Numerical parameters: $\tau =0.6,h=0.32,N=1600$ in the upper plots.)

Figure 10

(Upper and central panels) Classical phase space for different values of the phase $\varphi$ and $\lambda =0$: the blue point corresponds to the initial symmetry-breaking ground state with $h_{\mathrm{i}}=h$. (Lower panel) Dynamics of the classical $m_n^z$ (left panel) and its Fourier transform (right panel): we initialize with the point in the classical phase space corresponding to the ground state for $h_{\mathrm{i}}=h$. (Numerical parameters: $h=0.32,\tau =0.6,\lambda =0$. The Fourier transform is performed over $K=65536$ driving periods.)

Figure 11

(Upper left panel) Dependence on $\varphi$ of the height of the main peak $|m_{{\omega }_p}{|}^2$ in the power spectrum of the Fourier transform of the classical $z$ magnetization and the quantum one with $N=400$ ($\lambda =0$). (Lower left panel) Corresponding value of the frequency ${\omega }_p$ vs $\varphi$: we can see an interval where it is locked to ${\omega }_B=\pi /\tau$. (Upper right panel) $|m_{{\omega }_p}{|}^2$ versus $\lambda$ in the classical and in the quantum case with $N=700$ ($\varphi =\pi$). (Lower right panel) The same plot for the corresponding ${\omega }_p$. (Numerical parameters: $h_i=h=0.32,\tau =0.6$; the Fourier transform has been performed over $K=65536$ driving periods.)

Figure 12

(Upper panels) Classical phase space for different values of $\lambda$. (Lower left panel) Stroboscopic evolution in time of the classical $z$ magnetization. (Lower right panel) The power spectrum of the corresponding Fourier transform. (Numerical parameters $h_{\mathrm{i}}=h=0.32,\varphi =\pi ,\tau =0.6$. The Fourier transforms have been performed over $K=65536$ driving periods.)