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arXiv:2405.14216 · Physical review. B./Physical review. B

Emergence of spatial patterns and synchronization in superconducting time crystals

Bo Fan, Zi Cai, Antonio M. Garcı́a-Garcı́a

Published Oct 11, 2024• 3 citations
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Abstract

We identify a time crystal phase characterized by a frequency half of the driving frequency in disordered superconductors by employing the time-dependent Bogoliubov--de Gennes formalism at zero temperature with a periodically driven coupling constant. After a period of exponential increase of spatial inhomogeneities and exponential suppression of the order parameter amplitude, the time crystal develops islands of different sizes. Each of these islands is a time crystal with the same frequency albeit with a phase shift $\ensuremath{\pi}$ with respect to the homogeneous time crystal. After its emergence, the island gradually becomes smaller, though the phase shift persists, until it is abruptly synchronized at a time that it depends on its initial size. We find a critical disorder strength, still deep in the metallic phase, at which the time crystal phase terminates. For even stronger disorder, the order parameter oscillates with the driving frequency in regions where localization effects are not important.

DOI
10.1103/physrevb.110.134307

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We identify a time crystal phase characterized by a frequency half of the driving frequency in disordered superconductors by employing the time-dependent Bogoliubov--de Gennes formalism at zero temperature with a periodically driven coupling constant. After a period of exponential increase of spatial inhomogeneities and exponential suppression of the order parameter amplitude, the time crystal develops islands of different sizes. Each of these islands is a time crystal with the same frequency albeit with a phase shift \ensuremathπ\ensuremath{\pi}\ensuremathπ with respect to the homogeneous time crystal. After its emergence, the island gradually becomes smaller, though the phase shift persists, until it is abruptly synchronized at a time that it depends on its initial size. We find a critical disorder strength, still deep in the metallic phase, at which the time crystal phase terminates. For even stronger disorder, the order parameter oscillates with the driving frequency in regions where localization effects are not important.
We study the quench dynamics of a two dimensional superconductor in a square lattice of size up to 200× 200 200 × 200 employing the self-consistent time dependent Bogoliubov-de Gennes (BdG) formalism. In the clean limit, the dynamics of the order parameter for short times, characterized by a fast exponential growth and an oscillatory pattern, agrees with the Bardeen-Cooper-Schrieffer (BCS) prediction. However, unlike BCS, we observe for longer times a universal exponential decay of these time oscillations. We show explicitly that the origin of this exponential decay is the full emergence of spatial inhomogeneities of the order parameter characterized by the exponential growth of its variance. The addition of a weak disorder does not alter these results qualitatively. In this region, the spatial inhomogeneities rapidly develop into an intricate spatial structure consisting of ordered fragmented stripes in perpendicular directions where the order parameter is heavily suppressed. As the disorder strength increases, the fragmented stripes gradually turn into a square lattice of approximately circular spatial regions where the condensate is heavily suppressed. A further increase of disorder leads to the deformation and ultimate destruction of this lattice. We show these emergent spatial patterns are sensitive to the underlying lattice structure. We explore suitable settings for the experimental confirmation of these findings.
We show, using a Bogoliubov-de Gennes (BdG) mean field theory, that the local pairing amplitude Δ(r) becomes highly inhomogeneous with increasing disorder in an s-wave superconductor. The probability distribution P(Δ) is peaked about the BCS value at low disorder, but with increasing disorder, progressively develops into a broad distribution with significant build up of weight near Δ≈0. At high disorder, the system is found to form superconducting "islands" separated by a non-superconducting sea. Surprisingly, a finite energy gap persists into the highly disordered state in spite of many sites having negligible pairing amplitude and is understood in detail within the BdG framework. Once the pairing amplitude becomes inhomogeneous, the role of quantum phase fluctuations becomes crucial in driving a superconductor-insulator transition at a critical disorder. The insulator is unusual as it has a finite gap for all disorder strengths in marked contrast to the Anderson insulator in non-interacting systems. We treat the phase fluctuations within a self consistent harmonic approximation and obtain the superfluid stiffness as a function of disorder, which agrees well with our earlier quantum Monte Carlo studies.
We study a simple model of a two-dimensional s-wave superconductor in the presence of a random potential as a function of disorder strength. We first use the Bogoliubov--de Gennes (BdG) approach to show that, with increasing disorder, the pairing amplitude becomes spatially inhomogeneous, and the system cannot be described within conventional approaches for studying disordered superconductors that assume a uniform order parameter. In the high-disorder regime, we find that the system breaks up into superconducting islands, with large pairing amplitude, separated by an insulating sea. We show that this inhomogeneity has important implications for the physical properties of this system, such as superfluid density and the density of states. We find that a finite spectral gap persists in the density of states, even in the weak-coupling regime, for all values of disorder, and we provide a detailed understanding of this remarkable result. We next generalize Anderson's idea of the pairing of exact eigenstates to include an inhomogeneous pairing amplitude, and show that it is able to qualitatively capture many of the nontrivial features of the full BdG analysis. Finally, we study the transition to a gapped insulating state driven by quantum phase fluctuations about the inhomogeneous superconducting state.
The effect of nonmagnetic impurities on 2D sss-wave superconductors is studied beyond the weak disorder regime. Within the Bogoliubov--de Gennes (BdG) framework, the local pairing amplitude develops a broad distribution with significant weight near zero with increasing disorder. Surprisingly, the density of states continues to show a finite spectral gap. The persistence of the spectral gap at large disorder is shown to arise from the breakup of the system into superconducting ``islands.'' Superfluid density and off-diagonal correlations show a substantial reduction at high disorder. A simple analysis of phase fluctuations about the highly inhomogeneous BdG state is shown to lead to a transition to a nonsuperconducting state.
We study the effect of disorder on the superconductor-insulator transition in an inhomogeneous d-wave superconductor using the kernel polynomial method. As the Bogoliubov-de Gennes equations of the two-dimensional square lattice are solved self-consistently for the cases with more than 100000 unit cells, it is possible to observe the spatial fluctuations of the superconducting order parameters at the nanoscale. We find that strong spatial fluctuation of the superconducting order parameters can be introduced by disorder, and some superconducting specific order parameters are even enhanced. Moreover, we find that some isolated superconducting “islands” can survive the strong disorder, giving a boson insulator with some localized Cooper pairs. Our numerical calculations predict the existence of two sequential transitions with the increasing disorder strength: a d-wave to s-wave superconductor transition, and then an s-wave superconductor to insulator transition. The possibility of the appearance of a metallic phase between the superconducting and insulating phases is excluded by performing the lattice-size scaling of the generalized inverse participation ratio. In addition, we also discuss the effect of disorder on the optical conductivity of the d-wave superconductors.
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.