The robustness condition for general disordered discrete time crystals, and subspace-thermal DTCs from phase transitions between different n-tuple DTCs

Abstract

We propose a new Floquet time crystal model that responds in arbitrary multiples of the driving period. Such an $n$-tuple discrete time crystal is theoretically constructed by permuting spins in a disordered chain and is well suited for experimental implementations. Transitions between these time crystals with different periods give rise to a novel phase of matter that we call subspace-thermal discrete time crystals, where states within subspaces of definite charges are fully thermalized at an early time. However, the whole system still robustly responds to the periodic driving subharmonically, with a period being the greatest common divisor of the original two periods. Existing theoretical analysis from many-body localization cannot be used to understand the rigidity of such subspace-thermal time crystal phases. To resolve this, we develop a new theoretical framework for the robustness of DTCs from the perspective of the robust $2\pi/n$ quasi-energy gap. Its robustness is rigorously proved if the system satisfies a certain condition where the mixing length, defined by the Hamming distance of the symmetry charges, does not exceed a global threshold. The proof applies beyond the models considered here to other existing DTCs realized by kicking disordered systems, where conventional MBL-DTCs can be regarded as a special case of the subspace-thermal DTC with the subspace dimension being one, thus offering a systematic way to construct new discrete time crystal models. We also introduce the notion of DTC-charges that allow us to probe the observables that spontaneously break the time-translation symmetry in both the regular discrete time crystals and subspace-thermal discrete time crystals. Moreover, our discrete time crystal models can be generalized to systems with higher spin magnitudes or qudits, as well as to higher spatial dimensions.