Time-Crystalline Topological Superconductors

Time crystals form when arbitrary physical states of a periodically driven system spontaneously break discrete time-translation symmetry. We introduce one-dimensional time-crystalline topological superconductors, for which time-translation symmetry breaking and topological physics intertwine—yielding anomalous Floquet Majorana modes that are not possible in free-fermion systems. Such a phase exhibits a bulk magnetization that returns to its original form after two drive periods, together with Majorana end modes that recover their initial form only after four drive periods. We propose experimental implementations and detection schemes for this new state.

Figure 1

Proximitized quantum-dot array coupled to Ising spins. The Ising spins polarize the dot electrons—effectively producing a system of spinless fermions $c_j$. In any Ising configuration, the fermions can realize topological superconductivity with unpaired Majorana zero modes ${\gamma }_{1,2}$ that intertwine with the adjacent spins.

Figure 2

Phase diagram for Eq. (4) assuming (a) fully polarized and (b) random Ising spins. In (a) a nonzero chemical potential ${\mu }^{\prime }=|a|$ generates the trivial phase, and the system is gapless along the thick black lines. Data in (b) were generated from transfer-matrix simulations at ${\mu }^{\prime }=|b|=|a|/4$ with ${10}^6$ sites. Data points indicate sharp peaks in the localization length, as expected at a topological phase transition. The red diagonal line ${\varphi }_a={\varphi }_b$ is relevant for the physical quantum-dot setup from Fig. 1. As the dashed arrow illustrates, the topological phase along this line can be deformed to the zero-correlation-length limit with ${\varphi }_a=\pi /4$, ${\varphi }_b=-\pi /4$ (and also $|a|=|b|$, ${\mu }^{\prime }=0$) without crossing a phase boundary. Increasing the magnitude of ${\mu }^{\prime }$ tends to thicken the trivial regions, while altering the relative magnitudes of $|a|$ and $|b|$ shifts the boundaries separating the topological and trivial phases.

Figure 3

Time evolution for the time-crystalline topological superconductor generated by Eq. (10) at $\epsilon =0$. Each period $T$ globally flips all Ising spins, yielding doubled-periodicity bulk response, whereas the Floquet Majorana modes ${\gamma }_{1,2}$ exhibit quadrupled-periodicity response that can be probed in the junction with the static topological superconductor on the right. The inner Majorana modes ${\gamma }_{2,3}$ hybridize with coupling strength $\lambda$. Since ${\gamma }_3$ is static while ${\gamma }_2$ evolves nontrivially after each period $T$, the junction’s energy inherits the latter’s quadrupled periodicity.

Figure 4

Fourier transform of the quantities shown in the legend following time evolution via Eq. (10) with $\epsilon =0.2$ and parameters specified in the main text. Data are normalized by setting the maximum of each Fourier spectrum to 1, and frequency $\omega$ on the horizontal axis is normalized by $\mathrm{\Omega }=2\pi /T$, with $T$ the drive period. Here $m_{10}^z$ represents an Ising spin at the center of the chain, $c_0$ is an auxiliary zero-energy static fermion that enables probing the Floquet Majorana mode periodicity, and $c_1$ is the fermion at the left end of the quantum-dot chain. For initialization we use random Ising configurations and random fermionic states that entangle $c_0$ with the rest of the system. Runs were repeated 150 times for disorder averaging with maximum bond dimension $\chi =50$; similar results were obtained with $\chi =25$. For $\overline{a}T=2$ sharp peaks persist at $\mathrm{\Omega }/2$ and $3\mathrm{\Omega }/4$—despite “imperfect” driving generated by $\epsilon \ne 0$—indicating “rigid” doubled-periodicity Ising spins and quadrupled-periodicity Floquet Majorana modes characteristic of time-crystalline topological superconductivity. For $\overline{a}T=0.2$, the imperfect drive pushes the peak frequencies away from these quantized values, indicating a loss of rigid time crystallinity.