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Quantum Corrections to the Polarizability and Dephasing in Isolated Disordered Metals

We study the quantum corrections to the polarizability of isolated metallic mesoscopic systems using the loop-expansion in diffusive propagators. We show that the difference between connected (grand-canonical ensemble) and isolated (canonical ensemble) systems appears only in subleading terms of the expansion, and can be neglected if the frequency of the external field, $ω$, is of the order of (or even slightly smaller than) the mean level spacing, $Δ$. If $ω\ll Δ$, the two-loop correction becomes important. We calculate it by systematically evaluating the ballistic parts (the Hikami boxes) of the corresponding diagrams and exploiting electroneutrality. Our theory allows one to take into account a finite dephasing rate, $γ$, generated by electron interactions, and it is complementary to the non-perturbative results obtained from a combination of Random Matrix Theory (RMT) and the $σ$-model, valid at $γ\to 0$. Remarkably, we find that the two-loop result for isolated systems with moderately weak dephasing, $γ\sim Δ$, is similar to the result of the RMT+$σ$-model even in the limit $ω\to 0$. For smaller $γ$, we discuss the possibility to interpolate between the perturbative and the non-perturbative results. We compare our results for the temperature dependence of the polarizability of isolated rings to the experimental data of Deblock \emph{et al} [\prl \ {\bf 84}, 5379 (2000); \prb \ {\bf 65}, 075301 (2002)], and we argue that the elusive 0D regime of dephasing might have manifested itself in the observed magneto-oscillations. Besides, we thoroughly discuss possible future measurements of the polarizability, which could aim to reveal the existence of 0D dephasing and the role of the Pauli blocking at small temperatures.