Perturbative studies of the conductivity in the vortex-liquid regime

Abstract

We calculate the Aslamazov-Larkin term of the conductivity in the presence of a magnetic field applied along the $c$ axis from the time-dependent Ginzburg-Landau equation perturbatively using two approaches. In the first a uniform electric field is explicitly applied; in the second the Kubo formula is used to extract the linear response. The former yields a version of the flux-flow formula for the uniform $\mathrm{ab}$-plane conductivity, ${\ensuremath{\sigma}}_{\mathrm{xx}}(\mathbf{k}=0)$, that holds to all orders of perturbation theory. Obtaining the same result from the Kubo formula requires considerable cancellation of terms. We also use the Kubo calculation to examine the nonlocal $\mathrm{ab}$-plane conductivity, ${\ensuremath{\sigma}}_{\mathrm{xx}}(\mathbf{k}\ensuremath{\ne}0)$ (where the cancellations no longer occur), as well as the nonlocal $c$-axis conductivity ${\ensuremath{\sigma}}_{\mathrm{zz}}(\mathbf{k}\ensuremath{\ne}0)$, and look for the perturbative precursors of the growing viscous length scales. In addition, we consider the effects of weak disorder---both uncorrelated (point defects) and correlated (columnar and planar defects).