Stark time crystals: Symmetry breaking in space and time

The compelling original idea of a time crystal has referred to a structure that repeats in time as well as in space, an idea that has attracted significant interest recently. While obstructions to realize such structures became apparent early on, focus has shifted to seeing a symmetry breaking in time in periodically driven systems, a property of systems referred to as discrete time crystals. In this work, we introduce Stark time crystals based on a type of localization that is created in the absence of any spatial disorder. We argue that Stark time crystals constitute a phase of matter coming very close to the original idea and exhibit a symmetry breaking in space and time. Complementing a comprehensive discussion of the physics of the problem, we move on to elaborating on possible practical applications, and we argue that the physical demands of witnessing genuine signatures of many-body localization in large systems may be lessened in such physical systems.

Figure 1

Phase diagram of the discrete time-crystalline phases for the initial state being either ferro- (main panel) or antiferromagnetic (lower right inset). The false color plot shows $\overline{M}$, a quantity measuring the strength of the symmetry breaking in time [see Eq. (3) for the definition], in dependence of the potential difference $h$ and the imperfection of drive $\epsilon$. A dotted red line serves as a guide to the eye separating regions of rigid symmetry breaking from those where no rigid time-crystalline behavior is found (quantified by $\overline{M}=0.5$). At $h/J$ being $2\pi$ or $3\pi$ (white vertical dashed line), the time-crystalline phase is weakened substantially. This is due to the coherent destruction of the localization by the external drive. The other parameters are $U/J=1$, $L=100$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 2

Spatiotemporal structure of the symmetry breaking for a domain-wall initial state. We consider the modified correlation function ${\tilde{C}}_{zz}\left(t\right)$, which in time remains close to 1 for a perfect time-crystal (yellow in the false color plot). Deviations from yellow show that the time-crystal structure is broken. At $h/J=0$ no time crystal forms due to the lack of Stark many-body localization. Ballistic jets are found propagating from the domain-wall kink highlighted by a white dashed line. Those jets are barely visible on a linear color scale, but very pronounced on a logarithmic one; see also the Appendix. As the energy-gradient $h$, localizing the system, is increased, filaments of time-crystalline structure (yellow structures extended in time) are found with a filament density linearly increasing with $h$. When the density of filaments fills the entire system in space, a time crystal is established (compare also the Appendix). At this point, the transport originating from the domain wall is logarithmic (highlighted by a dashed black line for $h/J=4$). For values of $h$ close to integer multiples of $\mathrm{\Omega }=2\pi /T=\pi J$ (such as $h/J=6$), the time crystal is weakened again, which relates to coherent self-destruction. The other parameters are $U/J=1$, $L=100$, $\epsilon /J=0.2$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 3

Coherent self-destruction of a Stark time crystal. Top panel: the time-averaged strength of the time crystal $\overline{M}$ in dependence of the localization strength $h$ in units of the drive frequency $\mathrm{\Omega }=2\pi /T$ for two different drive imperfections $\epsilon$. Bottom panels: zoom into the region where $h$ is an even (left) or odd (right) multiple of $\mathrm{\Omega }$. As $h$ approaches integer multiples of $\mathrm{\Omega }=2\pi /T=\pi J$, the time-crystal structure is weakened in a resonant fashion. The weakening is stronger for $h$ being an even integer than for $h$ being an odd integer of $\mathrm{\Omega }$. The other parameters are $U/J=1$, $L=100$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 4

Comparison of the time-crystalline behavior for a ferromagnetic, domain-wall, or antiferromagnetic initial state (left to right columns) for two values of $h/J=0$ and 8. The first row shows the different initial states considered. The second row gives the correlation function $C_{zz}\left(t\right)$, which displays the spontaneous period doubling only when the system is Stark many-body localized ($h/J=8$). The third row demonstrates the modified correlation function ${\tilde{C}}_{zz}\left(t\right)$ measuring the stability of the period doubling (time crystal). The fourth row displays the fast Fourier transform of $C_{zz}\left(t\right)$. Period doubling (signal at frequencies $\omega =\mathrm{\Omega }/2$) is robust for all initial states only in the presence of Stark many-body localization ($h/J=8$). The other parameters are $U/J=1$, $L=100$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 5

Typical time evolution showing all times calculated (instead of only the stroboscopic ones, which we concentrate on in the main text) of the correlation function $C_{zz}\left(t\right)$ for two values of $h/J$ being in the time-crystalline or trivial region. The other parameters are $U/J=1$, $L=100$, $\epsilon /J=0.2$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 6

Stability of the results with respect to variations in the linearity of the gradient $\alpha$. The findings are insensitive. The other parameters are $U/J=1$, $L=100$, $\epsilon /J=0.2$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 7

Demonstration that the time-crystalline behavior is interaction-induced. The other parameters are $h/J=10$, $L=100$, $\epsilon /J=0.2$, and $JT/2=J{T}_1=J{T}_2=1$.

Figure 8

Same as Fig. 2 but with $\epsilon /J=0.4$.

Figure 9

Comparison between the filament density ${\rho }_{\mathrm{filament}}$, defined by the number of filaments found in Fig. 2 divided by half the system size, and the quantity $\overline{M}$ quantifying the stability of the time crystal. As the density of filaments ${\rho }_{\mathrm{filament}}$ approaches unity, the time crystal condenses and $\overline{M}\approx 1$. The parameters are the same as in Fig. 2.

Figure 10

The same as in Fig. 2, but using from a ferromagnetic initial state instead of a domain wall and comparing two system sizes $L=100$ (top row) and $L=200$ (bottom row) for two values of $h/J=0.05$ (left column) and $h/J=0.25$ (right column).

Figure 11

The same as in the topmost left panel of Fig. 2 of the main text, but instead of showing ${\tilde{C}}_{zz}^i\left(t\right)$ we show the log of the ab- solute value of 1 minus this quantity. In this way of depiction of the data, the ballistic jets mentioned in the main text are clearly visible.

Figure 12

Main panel: Same as Fig. 3 but for different system sizes $L$. Inset: full width at half-maximum (FWHM) of the peak centered at $h/\mathrm{\Omega }=2$. System sizes of $L\approx 50$ are needed to correctly capture the transition at small $h/\mathrm{\Omega }$. The central peak structures relating to coherent self-destruction scale as $1/L$. At small system sizes $L\approx 10$ the central features are washed out to a large degree.

Figure 13

Same as Fig. 2 but with $\epsilon /J=0.25$ and quenched disorder drawn uniformly from $[0,h)$.