Discrete Time Crystal in quantum Sherrington-Kirkpatrick model

Abstract: Discrete time crystals (DTC) have emerged as a significant phase of matter for out-of-equilibrium many-body systems. We study how long-range interactions and disorder contribute to the stability of the DTC phase. Generally, a stable DTC phase is believed to be realized in disordered systems with short-range interactions. In this work, we study periodically driven quantum Sherrington-Kirkpatrick (SK) model of Ising spin-glass in which all spins are randomly coupled. We investigate the possibilities of DTC phase in the SK model within three different driving protocols and found that the quantum SK model exhibits a robust DTC phase despite the long-range nature of interactions. The DTC phase in quantum SK model persists for a larger range of parameters if the $XY$ coupling or a transverse field is also random. This suggests that disorder in the $XY$ coupling or transverse field is also crucial for stabilising a broad DTC phase, despite the SK model's random couplings. Our analysis shows that the stability of the DTC phase is determined by the non-ergodic nature of the eigenstates of the quantum SK model, with the DTC order-parameter closely following the Shannon entropy of eigenstates. We compare the periodically driven SK model to alternative models of long-range interactions with uniform coefficients and found that the DTC phase is absent in these models for most of the driving protocols.