Superfluid Density and Phase Relaxation in Superconductors with Strong Disorder

We consider the attractive Hubbard model with on-site disorder as a prototype of a disordered superconductor. We solve the Bogoliubov–de Gennes equations on two-dimensional finite clusters at zero temperature and evaluate the electromagnetic response to a vector potential. We find that the standard decoupling between transverse and longitudinal response does not apply in the presence of disorder. Moreover, the superfluid density is strongly reduced by the relaxation of the phase of the order parameter already at mean-field level when disorder is large. We also find that the anharmonicity of the phase fluctuations is strongly enhanced by disorder. Beyond mean field, this provides an enhancement of quantum fluctuations inducing a zero-temperature transition to a nonsuperconducting phase of disordered preformed pairs. Finally, the connection of our findings with the glassy physics for extreme dirty superconductors is discussed.

Figure 1

Distribution of the local current (arrows) and lines of constant $\tilde{\theta }$ superimposed to the map of the order parameter, neglecting phase relaxation (a) and allowing for it (b). We applied a transverse field $A=0.003$ for a disorder amplitude $V_0/t=2$. Notice that in (b), while the percolative pattern of the current goes along the pattern of largest ${\Delta }_i$ values, the gauge-invariant phase gradient is larger along the locations of smaller ${\Delta }_i$ values.

Figure 2

Main panel: Comparison between the superfluid stiffness computed using Eq. (5) and its BCS counterpart as a function of disorder. We averaged over 10–20 disorder configurations for each disorder strength. Upper right inset: Same data plotted on a logarithmic scale to demonstrate increasing deviations for large $V_0/t$. Upper left inset: Ratio between anharmonic ($Q$) and quadratic coefficient ($D_s$) of the expansion Eq. (6) (with $Q/D_s=1.01$ at $V=0$). Note that for the given parameter set ${\xi }_0\approx 1a$ so that $J^{\prime }/J\approx Q/D_s$.