Higher-order and fractional discrete time crystals in clean long-range interacting systems

Andrea Pizzi,Johannes Knolle,Andreas Nunnenkamp

doi:10.1038/s41467-021-22583-5

Abstract

Discrete time crystals are periodically driven systems characterized by a response with periodicity nT, with T the period of the drive and n > 1. Typically, n is an integer and bounded from above by the dimension of the local (or single particle) Hilbert space, the most prominent example being spin-1/2 systems with n restricted to 2. Here, we show that a clean spin-1/2 system in the presence of long-range interactions and transverse field can sustain a huge variety of different ‘higher-order’ discrete time crystals with integer and, surprisingly, even fractional n > 2. We characterize these (arguably prethermal) non-equilibrium phases of matter thoroughly using a combination of exact diagonalization, semiclassical methods, and spin-wave approximations, which enable us to establish their stability in the presence of competing long- and short-range interactions. Remarkably, these phases emerge in a model with continous driving and time-independent interactions, convenient for experimental implementations with ultracold atoms or trapped ions.

Highlights

Discrete time crystals are periodically driven systems characterized by a response with periodicity nT, with T the period of the drive and n > 1
We overcome this limitation by considering a system amenable to a set of complimentary methods, which enable us to discover an unusually rich dynamical phase diagram hosting a zoo of novel, exotic nonequilibrium phases of matter
Higher-order discrete time crystals (DTCs) in clean long-range interacting systems are qualitatively distinct from DTCs of MBL Floquet systems[32]

Summary

Results

We consider a one-dimensional chain of N spins in the thermodynamic limit (N → ∞), driven according to the following. The plateau at ν = 0 for h ≈ 0 signals the tendency of the spins to remain aligned along z in a dynamical ferromagnetic phase (F) This corresponds to macroscopic quantum self-trapping of weakly driven bosons in a double well[34], which can be exactly mapped to the LMG limit (details in Supplementary Note 1). We observe that the higher-order DTCs are stable (at least in a prethermal fashion) for sufficiently long-range interactions (i.e., sufficiently small λ and α), whereas thermalization quickly sets in for shorter-range interactions (Fig. 4a) The transition between these two dynamical phases is sharp and can be located comparing the spin-wave density time average 〈ε〉t with a threshold 0.1 (Fig. 4b). We find small peaks at incommensurate frequencies that do (do not) vary when slightly perturbing the drive (initial conditions)

Discussion

Methods

Code availability