ac conductivity of mesoscopic rings: The discrete-spectrum limit

We present an extensive study of the current response of isolated mesoscopic rings (noninteracting electrons in the diffusive regime) to a small ac flux superimposed on a dc Aharonov-Bohm flux. This response can be very different from the conductance of the same ring connected to a voltage source. We emphasize the importance of the inelastic rate γ compared to the level spacing Δ and the driving frequency ω. We essentially focus on the discrete-spectrum limit γ≪Δ. The conductivity has an imaginary component which, at low frequency, is related to the flux derivative of the persistent current through the ring; as previously shown this quantity presents an ensemble average which is zero in the grand canonical case but which is finite in the canonical statistical ensemble. At frequencies larger than γ we show evidence of an extra contribution to the imaginary conductivity which is finite and temperature independent in the grand canonical ensemble, and increases with temperature in the canonical case. The real part of the average conductivity exhibits ${\mathrm{\Phi }}_0$/2-periodic flux oscillations which are also very sensitive to the statistical ensemble: for the canonical ensemble, these oscillations can assume either sign compared to the Altshuler, Aharonov, and Spivak oscillations observed in long cylinders and connected arrays of rings. We indeed identify different contributions giving rise to flux oscillations of opposite sign. An off-diagonal contribution related to interlevel transitions which is connected with the flux dependence of the level statistics and gives rise to a negative low-field magnetoconductance, and a diagonal contribution related to the flux dependence of the occupation of the different levels. This past quantity is proportional to the average square of the single-level persistent current and to the inelastic scattering time, and gives rise to a positive low-field magnetoconductance. It can be much larger than the Drude conductance. Our results rely on numerical simulations; most of them are also justified analytically.