Abstract
We investigate the transverse field Ising model subject to a two-step periodic driving protocol and quasiperiodic modulation of the Ising couplings. Analytical results on the phase boundaries associated with Majorana edge modes and numerical results on the localization of single-particle excitations are presented. The implication of a region with fully localized domain-wall-like excitations in the parameter space is eigenstate order and exact spectral pairing of Floquet eigenstates, based on which we conclude the existence of time crystals. We also examine various correlation functions of the time crystal phase numerically, in support of its existence.
Highlights
Our understanding of the out of equilibrium phase structures of periodically driven (Floquet) quantum many-body systems has made impressive progress over the past decade, see for example the review [1] and the references therein
We study the out of equilibrium phase structure of a driven transverse field Ising model (TFIM) with quasiperiodic (QP) modulation of Ising coupling
We have investigated the integrable TFIM subject to periodic driving and quasiperiodic modulation of the Ising coupling
Summary
Our understanding of the out of equilibrium phase structures of periodically driven (Floquet) quantum many-body systems has made impressive progress over the past decade, see for example the review [1] and the references therein. We study the out of equilibrium phase structure of a driven transverse field Ising model (TFIM) with quasiperiodic (QP) modulation of Ising coupling. III we present our results regarding the Majorana edge modes, localization of single-particle excitations, long-range order of excited states in order Based on these we conclude the existence of time crystal phase and present numerical results supporting our statement. Quasiperiodicity implies that the wavelength 1/Q is incommensurate to the lattice constant a, namely Q is an irrational number (we set a = 1 hereinafter) To analyze such a system theoretically, one can start by approximating the modulation by a sequence of periodic functions, which retrieve translational invaria√nce. We have confirmed in our numerics that the phase φJ is irrelevant to magnetic order and localization properties of single-particle excitations away from the phase boundaries, similar to what was observed in the undriven case [53, 54]
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