Generalized circular ensemble of scattering matrices for a chaotic cavity with nonideal leads

We consider the problem of the statistics of the scattering matrix S of a chaotic cavity (quantum dot), which is coupled to the outside world by nonideal leads containing N scattering channels. The Hamiltonian H of the quantum dot is assumed to be an M×M Hermitian matrix with probability distribution P(H)∝det[${\lambda }^2$+(H-ɛ${)}^2$ ${]}^{\mathrm{-}(\mathrm{\beta }\mathit{M}+2\mathrm{}\mathrm{-}\mathrm{\beta })/2\mathrm{}}$, where λ and ɛ are arbitrary coefficients and β=1,2,4 depending on the presence or absence of time-reversal and spin-rotation symmetry. We show that this ‘‘Lorentzian ensemble’’ agrees with microscopic theory for an ensemble of disordered metal particles in the limit M→∞, and that for any M≥N it implies P(S)∝‖det(1-S¯ ${}_{}{}^{\mathrm{°}}{}_{}{}^{}$S)${\mathrm{\Vert }}^{\mathrm{-}(\mathrm{\beta }\mathit{M}+2\mathrm{}\mathrm{-}\mathrm{\beta })}$, where S¯ is the ensemble average of S. This ‘‘Poisson kernel’’ generalizes Dyson’s circular ensemble to the case S¯≠0 and was previously obtained from a maximum entropy approach. The present work gives a microscopic justification for the case that chaotic motion in the quantum dot is due to impurity scattering.