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 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 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.
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.
We consider excitation of Higgs modes via the modulation of the BCS coupling within the Migdal-Eliashberg-Keldysh theory of time-dependent superconductivity. Despite the presence of phonons, which break integrability, we observe Higgs amplitude oscillations reminiscent of the integrable case. The dynamics of quasiparticles follows from the effective Bogolyubov-de Gennes equations, which represent a Floquet problem for the Bogoliubov quasiparticles. We find that when the Floquet-Bogoliubov bands overlap, the homogeneous solution formally leads to a negative superfluid density, which is no longer proportional to the amplitude of the order parameter. This result indicates an instability, which we explore using spatially-resolved BdG equations. Spontaneous appearance of spatial inhomogeneities in the order parameter is observed and they first occur when the superfluid density becomes unphysical. We conclude that the homogeneous solution to time-dependent superconductivity is generally unstable and breaks up into a complicated spatial landscape via an avalanche of topological excitations.
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 random on-site disorder on s-wave (BCS) superconductors described by a two dimensional negative-U Hubbard model is studied using Bogoliubov-de Gennes (BdG) method. The mean field pairing amplitudes become spatially inhomogeneous at large values of disorder where the system breaks up into superconducting islands with large pairing amplitudes, separated by insulating strips. The amplitude fluctuations are correctly accounted for via BdG calculations, however it misses phase fluctuations which are inherent to low dimensions. The phase fluctuations affect superconducting properties strongly, and the effect is more pronounced in the limit of large disorder. We provide a close estimate of the actual transition temperature, Tcby incorporating phase fluctuations about the inhomogeneous BdG state. This is obtained by relating the jump in renormalized Ds(obtained from a self consistent Harmonic approximation on a phase-only Hamiltonian) at the Kosterlitz-Thouless (KT) transition and the KT transition temperature, TKTobtained from Ds(T) = (2/π)TKTas temperature tends to TKTfrom below. This yields opening of a large region sandwiched between [Formula: see text] (obtained by the vanishing of [Formula: see text]) and TKTwhere there is no phase coherence between the pairs, however amplitude correlations continue to exist, reminiscent of a pseudogap phase in high-Tcsuperconductors which is marked by short ranged preformed pairs without any definite phase relation between them. Further, the appearance of the superconducting islands from a homogeneous phase indicates the evolution of a system consisting of large and overlapping pairs to one that contains short and tightly bound pairs - a scenario termed as BCS-BEC crossover. We have investigated in details the crossover phenomenon as a function of disorder and confirmed it's existence at small values of electron concentration, however, absent at larger densities.
We study the interplay of superconductivity and disorder by solving numerically the Bogoliubov--de Gennes equations in a two-dimensional lattice of size $80\ifmmode\times\else\texttimes\fi{}80$, which makes investigating the weak-coupling limit possible. In contrast with results in the strong-coupling region, we observe enhancement of superconductivity and intriguing multifractal-like features such as a broad log-normal spatial distribution of the order parameter, a parabolic singularity spectrum, and level statistics consistent with those of a disordered metal at the Anderson transition. The calculation of the superfluid density, including small phase fluctuations, reveals that, despite this intricate spatial structure, phase coherence still holds for sufficiently weak disorder. It breaks down only for stronger disorder but before the insulating transition takes place.