The equal-time correlation detects the presence or absence of FM order in the long run. The phase boundaries determined by Eqs. Note also that there is a boundary term if periodic boundary conditions are imposed on the spin chain.
boundary term is marginal in detecting bulk properties. For a given Floquet eigenstate expressed in the Majorana representation, the IPR is defined by. . Our work will be a good starting point for future works on robustness of the time-crystalline order when integrability-breaking perturbations are included. (
As expected, the states close to the lines of vanishing gap are forced to be extended, as a result of the Ising universality class, whereas in the vicinity of all states remain localized (marked by black lines in Fig. In the fully localized phase close to (see Fig. We will see that this brings further simplifications, allowing us to obtain analytic expressions for some of the phase boundaries.
(
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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. Finally, we consider the reflections and . Theoretical Quantum Technologies (TQT). For the reflection, preserves its form after applying local rotations to all the sites, such that .
non-trivial long range order. Agreement NNX16AC86A, Is ADS down? Both average magnetization per spin along the z-direction <σz> and along the x-direction <σx> are calculated in this picture. [■]
The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. [■]
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We analyzed two consequences of fully-localized single-particle excitations, namely long-range FM order of eigenstates and spectral pairing, based on which a time crystal phase was anticipated. Astrophysical Observatory. On the other hand, Fig. [■]
Damian Sowinski . [■]
The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. This can be remedied by examining the autocorrelation function . 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. However, this makes no trouble in our case as we will consider a semi-infinite chain in the study of edge modes, while for sufficient long chains the
R. F. acknowledges partial financial support from the Google Quantum Research Award. we write explicitly the relevant eigenproblems in terms of the coefficients of Eq. [■]
When , Eq. , where it was found that in the strong modulation regime new gapless phases with localized or multifractal excitations emerge. We investigate the transverse field Ising model subject to a two-step periodic driving protocol and quasiperiodic modulation of the Ising couplings. Furthermore, the Floquet operator simply changes sign under the shift or , which can be compensated by shifting the quasienergy while leaving the Floquet eigenstate unchanged. [■]
) and (
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Instead, it is simply a reference state from which one can obtain the other Floquet eigenstates by applying the quasiparticle creation operators, and all the physical properties of the model are independent of the choice for . [■]
However, we emphasize that the choice of has no special role in the present driven model. and soon
This fact implies
Generic non-integrable Floquet systems will be heated up to the infinite temperature ensemble due to persistent pumping of energy by the driving
Our interest will be in the long-time behavior of the driven system, so it suffices to consider the Floquet operator of (
Among the many interesting discoveries made in this context, the prediction and the consequent observation of
The Majorana representation is superior to the fermionic one in that it allows us to treat the problem of edge modes and localization of excitations on equal footing. (
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For simplicity, although the notion of ground state is meaningless for Floquet systems, we will sometimes refer to the eigenstates generated by as single-particle ‘excitations’. [■]
), the and phases eigenstates are composed of domain walls.