where the correlation function changes sign many times, corresponding to the presence of many pinned domain walls. For weak modulation, all Ising couplings have the same sign and the phase transition is in the conventional Ising universality class, which is verified by the linear asymptotics of the spectrum around the transition points, see Fig. The action of the two unitaries on a single Majorana fermion can be easily written down [see Eq. We will see that this brings further simplifications, allowing us to obtain analytic expressions for some of the phase boundaries. [■] (a) shows that the spectrum in the weakly modulated region has gaps both around and , and only one of the two gaps can disappear at the phase boundaries or in the strongly modulated region. We have investigated the integrable TFIM subject to periodic driving and quasiperiodic modulation of the Ising coupling. a lot of interest. [■] pairing of Floquet eigenstates, based on which we conclude the existence of For integrable systems (like models solvable by the JW transformation), the fluctuation in the equal-time correlation function is solely determined by and is related to the property of the single-particle spectrum A crucial feature of the Floquet operator is that it consists of two unitaries acting on disconnected Majorana pairs. [■] Furthermore, in the incommensurate limit the sum in the right hand side of Eq. . [■] [■] This fact implies ) have a non-analytical behavior at the edges of the weak modulation region. [■] , analogous to that of phases transitions in equilibrium systems. )] is indicated by the hatched region. ( We define the time at which the autocorrelation function decays to its minimum for the first time as the lifetime of the time crystal in a finite system and investigate its scaling with the system size (see insets in Fig. To describe the time evolution in terms of noninteracting fermions, we introduce a pair of Majorana fermions for each spin-, The Majorana operators satisfy the anticommutation relation . We apply a worm algorithm to simulate the quantum transverse-field Ising model in a path-integral representation of which the expansion basis is taken as the spin component along the external-field direction. For a given Floquet eigenstate expressed in the Majorana representation, the IPR is defined by. Then any excited Floquet eigenstate is given by hosting many excitations with the aforementioned vacuum states. [■] ( The Ising symmetry, denoted by the symmetry operator , is inherited from the Hamiltonian ( As the and [defined in Eq. Many-body eigenstates may be classified by their broken symmetries Furthermore, due to the symmetry (see the discussion in Sec. (a) we have and the weak modulation regime [see Eq. . , for example the review  [■] (a). ), the and phases eigenstates are composed of domain walls. ( Here we prove that the vector is an eigenvector of the transfer matrix , with its eigenvalue satisfying Eq. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). )] and, with in the bulk, yields, Finally, based on Eq. A remarkable consequence of the localization of excitations in the FM phases is the preservation of long-range order in a generic “excited states”, which is expressed as the Slater determinant of single-particle excitations. [■] For our purpose of seeking time crystals, we will mainly focus on the case in the rest of the paper. Analytical results on the phase boundaries associated with Majorana edge modes and numerical results on the localization of single-particle excitations are presented. We show here how to get the transfer matrix from the eigenequations for the edge modes of a semi-infinite chain. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. In fact, expanding the initial state in the basis of Floquet eigenstates where the index accounts for possible degeneracy of eigenstates, we can separate the stationary value from the oscillation as This explains what we see in Fig. [■] This is best illustrated in an open chain. (b)). However, the correlation function never decay to zero at long distance. [■] , no edge states imply trivial band topology and absence of long-range order, namely a paramagnetic (PM) phase; While one edge state, no matter if it is at or quasienergy, indicates nontrivial band topology and long-range ferromagnetic (FM) order. The author is a research scientist at Dartmouth College. Nonvanishing long-range order in excited states follows from the fact that in the FM phase excitations are domain-wall-like, and a single domain wall simply revert the sign of the correlator of the vacuum state. Generic non-integrable Floquet systems will be heated up to the infinite temperature ensemble due to persistent pumping of energy by the driving crystal phase numerically, in support of its existence. [■] ) and satisfies . Another simple observation is that we can assume , as a change of sign is equivalent to a phase shift . [■] Astrophysical Observatory. We see that in both cases the localization-delocalization transitions depend on energy, signaling the presence of a mobility edge. [■] [■] Note also that there is a boundary term if periodic boundary conditions are imposed on the spin chain. ). where denotes the generalized “diagonal ensemble” when the systematic degeneracy of Floquet eigenstates is taken into consideration, and the function The analytical estimation in Ref. [■] The transfer matrix of the 2-D Ising model . (b) as overwhelmingly modulated. Define the Hamiltonian as . and the references therein. (b) we show a representative phase diagram with , when the Ising coupling are always strongly modulated and the phase boundaries are more complex. In Fig. [■] )] consist of odd-even and even-odd Majorana pairs respectively, we can write down explicitly their action on a single Majorana fermion, With these, one can easily derive the eigenequations for the . [■] [■] In Fig. that the long time steady states of both cases always have trivial correlations, thus excludes any non-trivial phase structures. [■] Damian Sowinski . ) can be approximated by an integral, giving, where in the integrand and we set , to simplify the notation (we follow this choice from now on). The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. For the clean model, the correlator decays exponentially to zero, indicating vanishing long-range order in the stationary state; while for the quasiperiodic model, it quickly relaxes to a nonvanishing value with some persistent fluctuation.