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arXiv:2312.17070 · Physical review. B./Physical review. B

Swapping Floquet time crystal

Roberto Gargiulo, Gianluca Passarelli, Procolo Lucignano, Angelo Russomanno

Published May 22, 2024• 6 citations
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Abstract

We propose a Floquet period-doubling time-crystal model based on a disordered interacting long-range spin chain where the periodic swapping of nearby spin couples is applied. This protocol can be applied to systems with any local spin magnitude $s$ and in principle also to systems with nonspin (fermionic or bosonic) local Hilbert space. We explicitly consider the cases $s=1/2$ and $s=1$, using analytical and numerical methods to show that the time-crystal behavior appears in a range of parameters. In particular, we study the persistence of period-doubling oscillations in time and the time-crystal properties of the Floquet spectrum (quasienergy $\ensuremath{\pi}$-spectral pairing and long-range correlations of the Floquet states), and introduce a quantity (the local imbalance) to assess what initial states give rise to a period-doubling dynamics. We also consider the quasienergy average level-spacing ratio, and find that the interval of parameters where the system dose not thermalize and persistent period doubling is possible corresponds to the one where the Floquet spectrum shows time-crystal properties.

DOI
10.1103/physrevb.109.174310

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We propose a Floquet period-doubling time-crystal model based on a disordered interacting long-range spin chain where the periodic swapping of nearby spin couples is applied. This protocol can be applied to systems with any local spin magnitude sss and in principle also to systems with nonspin (fermionic or bosonic) local Hilbert space. We explicitly consider the cases s=1/2s=1/2s=1/2 and s=1s=1s=1, using analytical and numerical methods to show that the time-crystal behavior appears in a range of parameters. In particular, we study the persistence of period-doubling oscillations in time and the time-crystal properties of the Floquet spectrum (quasienergy \ensuremathπ\ensuremath{\pi}\ensuremathπ-spectral pairing and long-range correlations of the Floquet states), and introduce a quantity (the local imbalance) to assess what initial states give rise to a period-doubling dynamics. We also consider the quasienergy average level-spacing ratio, and find that the interval of parameters where the system dose not thermalize and persistent period doubling is possible corresponds to the one where the Floquet spectrum shows time-crystal properties.
We show that time-reflection symmetry in periodically driven (Floquet) quantum systems enables an inherently nonequilibrium phenomenon structurally similar to quantum-mechanical supersymmetry. In particular, we find Floquet analogs of the Witten index that place lower bounds on the degeneracies of states with quasienergies 0 and π. Moreover, we show that in some cases time-reflection symmetry can also interchange fermions and bosons, leading to fermion-boson pairs with opposite quasienergy. We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies 0 and π, which are robust as long as the time-reflection symmetry is preserved. Floquet supersymmetry manifests itself in the evolution of certain local observables as a period-doubling effect with dramatic finite-size scaling, providing a clear signature for experiments.
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Time crystals break the discrete time translational invariance of an external periodic drive by oscillating at an integer multiple of the driving period. In addition to this fundamental property, other aspects are often considered to be essential characteristics of a time crystal, such as the presence of disorder or interactions, robustness against small variations of system parameters, and the free choice of the initial quantum state. We study a finite-length polarized XX spin chain engineered to display a spectrum of equidistant energy levels without drive and show that it keeps a spectrum of equidistant Floquet quasienergies when subjected to a large variety of periodic driving schemes. Arbitrary multiples of the driving period can then be reached by adjusting parameters of the drive, for arbitrary initial states. This behavior is understood by mapping the XX spin chain with N+1N+1N+1 sites to a single large spin with S=N/2S=N/2S=N/2 invoking the closure of the group SU(2). Our simple model is neither intrinsically disordered nor is it an interacting many-body system (after suitable mapping), and it does not have a thermodynamic limit in the conventional sense. It does, however, show controllable discrete time translational symmetry breaking for arbitrary initial states and a degree of robustness against perturbations, thereby carrying some characteristic traits of a discrete time crystal.
Floquet time crystals are characterized by the subharmonic behavior of temporal correlation functions. Studying the paradigmatic time crystal based on the disordered Floquet quantum Ising model, we show that its temporal spin correlations are directly related to spectral characteristics and that this relation provides analytical expressions for the correlation function of finite chains, which compare favorably with numerical simulations. Specifically, we show that the disorder-averaged temporal spin correlations are proportional to the Fourier transform of the splitting distribution of the pairs of eigenvalues of the Floquet operator, which differ by π to exponential accuracy in the chain length. We find that the splittings are well described by a log-normal distribution, implying that the temporal spin correlations are characterized by two parameters. We discuss possible implications for the phase diagram of Floquet time crystals.
The critical properties characterizing the formation of the Floquet time crystal are investigated analytically in the periodically driven O(N)O(N)O(N) model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for N→∞N\to\inftyN→∞, we determine the exponent ν\nuν that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent β\betaβ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent θ\thetaθ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents ν,β,θ\nu, \beta, \thetaν,β,θ are found to be identical to those in the absence of a drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector ω/(2v)\omega/(2 v)ω/(2v), vvv being the velocity of the quasiparticles, and ω\omegaω being the drive frequency.
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