Discrete time crystals detected by time-translation twist

TL;DR

The paper introduces a time-translation twist boundary condition to diagnose dynamical phases in Floquet systems, formalized through the spectral form factor $K(t)=| ext{Tr}\,U_F^t|^2$. By drawing an analogy to the Little-Parks effect, the authors show that DTC order exhibits a periodic response to the twist parameter $a$, and they validate this using the disordered kicked Ising model. They further develop two twists—fractional twists and twists applied after a preparation time—to probe DTC formation and rigidity, with short-time vanishing SFF signaling DTC and long-time twists reducing the SFF plateau in DTC regions. Overall, the boundary-twist approach provides a robust, boundary-condition–based diagnostic of dynamical order in non-equilibrium quantum matter and suggests routes toward experimental implementation via tensor-network representations and related platforms.

Abstract

We introduce a boundary condition twisted by time translation as a novel probe to characterize dynamical phases in periodically driven (Floquet) quantum systems. Inspired by twisted boundary conditions in equilibrium systems, this approach modifies the temporal evolution of the system upon completing a spatial loop, enabling the identification of distinct Floquet phases, including discrete time crystals (DTCs). By studying the spectral form factor (SFF) and its response to the twist, we uncover signatures of time-crystalline order, which exhibits periodic dependence on the twist parameter analogous to the Little-Parks effect in superconductors. We apply this framework to the kicked Ising model, demonstrating that our twist can distinguish time-crystalline phases.

Figures (4)

• Figure 1: Tensor-network representation of the time-translation twist. (a-b) The product of Floquet operators by the ordinary temporal transfer matrix and that of twisted Floquet operators with $a=1$, respectively. (c-d) The product of fractionalized Floquet operators and of twisted Floquet operators with $a=1/3$, respectively. A consecutive set of rows highlighted by orange rectangles is the unit of the temporal transfer matrix. Black, gray, and white squares in (c) and (d) represent different roles of operators in a single period. (e) The boundary condition is switched from the periodic one to the one with time-translation twist with $a=2$ at $t_0$. In this figure, the trace that connects the top and bottom legs is included.
• Figure 2: (a-b) The ratio of half-twisted and untwisted SFFs with $L=30$ is shown for $t=2$ and $t=4$, respectively. The number of disorder configurations is 10000 for (a) and 1000 for (b). (c-d) The SFF as a function of twist for some points indicated in (a) and (b). The frequency of the largest eigenvalue state of the bulk column-to-column transfer matrix for (e) $t=2$ and (f) $t=4$.
• Figure 3: (a) The phase diagram of the kicked Ising model. Notice that when $h_j=0$, the phase diagram is invariant under $b\to b+\pi$, $\bar{J}\to \bar{J}+\pi$, $b\to -b$ and $\bar{J}\to -\bar{J}$ due to the corresponding symmetry of the Floquet operator (\ref{['eq:kickedising_floquetoperator']}). (b) The phase diagram showing the ratio of the averaged SFFs at $t=2^8$ for $a=1$ and $2$, where the preparation time is $t_0=2^7$. (c) The averaged SFF at $t=2^8$ and $t=2^8+1$ as a function of $b$ at $\bar{J}=\pi/4$ [red line in (b)]. (d) The finite-size scaling of the SFF ratio from $L=4$ to $8$ at $\bar{J}=\pi/8$ [green line in (b)]. (e)-(h), the averaged SFF of $L=8$ when the boundary condition is switched at $t_0=2^7$ is shown for $a=0,1$, and $2$. Each figure corresponds to the parameter given in (b). We averaged over 1000 samples for (b) and $a=3$ in (c), while 10000 samples otherwise.
• Figure 4: (a) The phase diagram of the kicked Ising model. (b)-(g) The SFF averaged over 10000 samples with length $L=8$, longitudinal field mean $\bar{h}=0.6$, variance $\pi/10$, and Ising coupling variance $\pi/10$. Each figure corresponds to (b) the self-dual point $(b,\bar{J})=(\pi/4,\pi/4)$, (c) the 0SG phase $(b,\bar{J})=(\pi/16,\pi/4)$, (d)-(e) the $\pi$SG (DTC) phase ($b,\bar{J})=(7\pi/16,\pi/4)$ and $(b,\bar{J})=(\pi/2,\pi/4)$, (f) the PM phase $(b,\bar{J})=(\pi/4,\pi/8)$, and (g) the 0$\pi$PM phase $(b,\bar{J})=(\pi/4,3\pi/8)$, respectively. In each figure, the SFF without twist (black), with a half twist $a=1/2$ (purple) and with a single twist $a=1$ (orange) are shown. In (b), the SFF of the random matrix theory (COE and CUE) is shown by dashed lines. In (d) and (e), the SFF at even $t$ and odd $t$ are shown independently by solid lines while lightly colored regions represent the oscillation.