Inhomogeneous phase stiffness in two-dimensional $s$-wave disordered superconductors

TL;DR

This work addresses how white-noise disorder affects local phase stiffness and thermodynamics in a 2D $s$-wave superconductor by deriving a phase-only action from a local attractive Hamiltonian and mapping it to a random-coupling XY model via BdG solutions. The local couplings $J_{ij}=J_{ij}^{(1)}+J_{ij}^{(2)}$ exhibit a transition from a single-peaked to a bimodal distribution with negative values as disorder grows, indicating frustration and potential glassiness. Monte Carlo analysis reveals that strong disorder smears the BKT transition, induces anomalous low-temperature behavior where $J_s$ can rise with temperature, and is accompanied by a nonzero Edwards-Anderson order parameter, signaling a phase-glass state near a disorder-driven SIT. These findings illuminate how spatial inhomogeneity in phase stiffness modifies superconducting coherence and suggest glassy phases in disordered 2D superconductors, with caveats due to neglect of quantum fluctuations and avenues for further study.

Abstract

We investigate the effect of white-noise disorder on the local phase stiffness and thermodynamic properties of a two-dimensional $s$-wave superconductor. Starting from a local attractive model and using path-integral formalism, we derive an effective action by decoupling the superconducting order parameter into amplitude and phase components in a gauge-invariant manner. Perturbative techniques are applied to the phase fluctuation sector to derive an effective phase-only XY model for disordered superconducting systems. Solving the saddle-point Green's function using Bogoliubov-de Gennes theory, we calculate the distributions of nearest-neighbor couplings for various disorder strengths. A single-peak distribution is observed for low disorder strength, which becomes bimodal with one peak at negative couplings as the disorder strength increases. The local phase stiffness remains randomly distributed throughout the lattice and shows no correlation with pairing amplitudes. The temperature dependence of the superfluid stiffness ($J_s$) is studied using Monte Carlo simulations. At strong disorder and low temperatures, $J_s$ increases with increasing temperature, exhibiting anomalous behavior that may indicate the onset of a glassy transition. Additionally, calculations of the Edwards-Anderson order parameter in this disorder regime suggest the emergence of a $phase$-$glass$ state at very low temperatures.

Figures (6)

• Figure 1: (color online) Diagrammatic representations of possible contributions in the expressions of $J_{ij}^{(1)}$ (a) and $J_{ij}^{(2)}$ (b)--(f) due to single and double hopping respectively between the $i^{th}$ and $j^{th}$ sites (red dots). Lattice points are represented by dots and a line between the $i^{th}$ and $j^{th}$ sites represent ${\cal G}_{ij}$ with an arrow headed towards $i^{th}$ site. (a) Direct coupling between the $i^{th}$ and $j^{th}$ sites, (b) onsite contributions without any involvement of intersite hopping, (c) contribution for double-hopping between the $i^{th}$ and $j^{th}$ sites, (d) and (e) contributions for hoppings involving the $i^{th}$ and $j^{th}$ sites and one of these sites with three other neighboring sites (green, blue, and yellow dots), and (f) the combination of one hopping from the $i^{th}$ and the other from the $j^{th}$ site such that the end sites become nearest neighbor (no hopping is involved between the $i^{th}$ and $j^{th}$ sites).
• Figure 2: (color online) Probability distribution functions of nearest neighbor coupling $J_{ij}$ for different disorder strengths $V$ in a $48 \times 48$ square lattice. Four panels correspond to different ranges of $V$.
• Figure 3: (color online) (a) Color-coded maps of the lattice variable $\Delta_i$ and the nearest-neighbor bond-variable $J_{ij}$ on a $48\times48$ square lattice for a disorder realization with strength $V=1.5$. Lesser (greater) values of $\Delta_i$ are marked with black (yellow) color. (b)--(e) Probability distributions functions $P(\Delta_i)$ and $P(J_{ij})$ for the selected zones A--D in (a) respectively. Insets: Magnified view of the selected zones.
• Figure 4: (color online) In a $48\times 48$ square lattice for a realization of disorder with $V=1.5$, onsite disorder potential $V_i$ and bond-variable (between nearest neighbors) $J_{ij}$ are shown with color-coding. $\vert V_i\vert <1\, (>1)$ is represented by yellow (indigo) points. Inset: Magnified view of a selected square-shaped zone with red-colored boundary.
• Figure 5: (color online) Temperature dependence of the superfluid stiffness (a), its diamagnetic contribution (b), and its paramagnetic contribution (c). The dashed lines in (a) correspond to $J_s = (2/\pi)T$ as per the BKT criterion for BKT transition from superfluid state to the normal state. All figures share the same color legend. Inset: Variation of the critical temperature $T_c$ with disorder strength $V$.