Persistent equilibrium currents in one-dimensional disordered normal-metal rings

Using rigorous results for the Green’s function of one-dimensional free electrons in a Gaussian white-noise random potential, we calculate the disorder-averaged energy F of electrons in a mesoscopic ring threaded by a magnetic flux φ. The chemical potential is considered to be independent of the randomness. The energy can be expanded as F=${\mathit{tsum}}_{\mathit{n}=0\mathrm{}}^{\mathrm{\infty }}$cos(2πnφ/${\mathrm{\phi }}_0$)${\mathit{F}}_{\mathit{n}}$, where ${\mathrm{\phi }}_0$=hc/e is the flux quantum, and all Fourier coefficients ${\mathit{F}}_{\mathit{n}}$ are explicitly calculated. From the first derivative of F with respect to the flux, we obtain the disorder-averaged persistent equilibrium current I. From the second derivative of F with respect to the flux, we obtain ${\lim }_{\mathrm{\omega }\to 0}$[ω Imσ(ω)], where σ(ω) is the frequency-dependent conductivity of the system. Comparison of our results with previous numerical and approximate analytical calculations confirms their validity.