Discrete Time Crystals Enforced by Floquet-Bloch Scars

We analytically identify a new class of quantum scars protected by spatiotemporal translation symmetries, dubbed Floquet-Bloch scars. They are distinguished from previous (quasi-)static scars by a rigid spectral pairing only possible in Floquet systems, where strong interaction and drivings equalize the quasienergy corrections to all scars and maintain their spectral spacings against generic bilinear perturbations. Scars then enforce the spatial localization and rigid discrete time crystal (DTC) oscillations as verified numerically in a trimerized kagome lattice model relevant to recent cold atom experiments. Our analytical solutions offer a potential scheme to understand the mechanisms for more generic translation-invariant DTCs.

Figure 1

(a) Trimerized kagome lattice and the schematic illustration of DTC dynamics. Triangles with strong (weak) bonds are denoted by black (gray) colors, with $+i$ hopping directions indicated by arrows. (b) Particle dynamics $n_{\mu }(NT)={\sum }_{\mathit{r}}⟨{\psi }_{\mathrm{ini}}|({\widehat{U}}_F^{†}{)}^N{n}_{\mathit{r}\mu }{\widehat{U}}_F^N|{\psi }_{\mathrm{ini}}⟩$, with the initial state $|{\psi }_{\mathrm{ini}}⟩$ at $t/T=1$ when all particles locate at a single site $\mathit{r}=\mathbf{0}$, $\mu =0$. To facilitate reading, data are grouped into three sets at $t$ mod $3T=0$, 1, 2 respectively. For comparison, the noninteracting ${\varphi }_2=0$, $L=3$ case is plotted at all $t$ in the upper panel as translucent dots. (c) Temporal correlation functions indicating infinite-time response frequencies ($L=3$). Unless denoted otherwise, $N_b=5$, ${\varphi }_1=2\pi /3\sqrt{3}$, $\lambda =0.1$, ${\varphi }_2=1.1$, and ${\theta }_{1,2,3}=(0.1,0.2,-0.3)$.

Figure 2

(a) $⟨r⟩$ shows generic ergodicity. (b) Subsystem for computing EE. (c) and (d) Eigenstate EE in (c) proximate-integrable and (d) DTC regimes, where low-entropy scars in (d) are highlighted by larger dots. (e) Lowest EE approaching size-insensitive values at small $\lambda$ and volume law $S_{\mathrm{ent}}\sim (N_{\mathrm{sub}}/3L^2)\ln (D_L)$ at large $\lambda$. Here $D_L$ is the total Hilbert space dimension and $N_{\mathrm{sub}}$ the subsystem site number enclosed in (b). Inset: $\mathrm{\Delta }{S}_{\mathrm{ent}}=S_{\mathrm{ent}}^{(L=3)}-S_{\mathrm{ent}}^{(L=2)}$ near the crossing ${\lambda }_0\approx 0.135$. Unless specified otherwise, in all plots parameters are the same as in Fig. 1. Blue (or red) colors denote $L=2$ (or $L=3$) respectively.

Figure 3

Structure of Floquet eigenstates in $\mathit{k}=\mathbf{0}$ sector. Other $\mathit{k}$ sectors show essentially the same results. (a) Most eigenstates involve an extensive number of basis $|\mathit{k},\{n_{\mathit{r}\mu }\}⟩$ leading to vanishing IPR, except for the three FBSs. (b) Expand for instance one FBS in the basis $|\mathit{k},\{n_{\mathit{r}\mu }\}⟩$; we see it is dominated by three components depicted in (c), exactly as given by Eq. (4). (d) Scaling of the maximal IPR, where $\mathrm{\Delta }{\mathrm{IPR}}_{\max }={\mathrm{IPR}}_{\max }^{(L=3)}-{\mathrm{IPR}}_{\max }^{(L=2)}$ in the inset. Parameters are the same as in Fig. 1, and $L=3$ for (a) and (b).