Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices

TL;DR

This paper investigates the joint statistical distribution of capacitance and charge relaxation resistance in chaotic quantum cavities using random matrix theory, revealing their average behaviors, fluctuations, correlations, and phase transition phenomena.

Contribution

It provides a detailed analysis of the joint distribution of mesoscopic capacitance and charge relaxation resistance in chaotic cavities, including large deviation properties and phase transitions.

Findings

Average charge relaxation resistance R_q equals the dc resistance h/(Ne^2).

Fluctuations of C_q and R_q are of order 1/N and strongly correlated.

A second order phase transition in the Coulomb gas describes large deviation behavior.

Abstract

We consider the AC transport in a quantum RC circuit made of a coherent chaotic cavity with a top gate. Within a random matrix approach, we study the joint distribution for the mesoscopic capacitance $C_\mu=(1/C+1/C_q)^{-1}$ and the charge relaxation resistance $R_q$, where $C$ is the geometric capacitance and $C_q$ the quantum capacitance. We study the limit of a large number of conducting channels $N$ with a Coulomb gas method. We obtain $\langle R_q\rangle\simeq h/(Ne^2)=R_\mathrm{dc}$ and show that the relative fluctuations are of order $1/N$ both for $C_q$ and $R_q$, with strong correlations $\langle \delta C_q\delta R_q\rangle/\sqrt{\langle \delta C_q^2\rangle\,\langle \delta R_q^2\rangle}\simeq+0.707$. The detailed analysis of large deviations involves a second order phase transition in the Coulomb gas. The two dimensional phase diagram is obtained.