Microscopic theory of the Andreev gap

We present a microscopic theory of the Andreev gap, i.e. the phenomenon that the density of states (DoS) of normal chaotic cavities attached to superconductors displays a hard gap centered around the Fermi energy. Our approach is based on a solution of the quantum Eilenberger equation in the regime $t_D\ll t_E$, where $t_D$ and $t_E$ are the classical dwell time and Ehrenfest-time, respectively. We show how quantum fluctuations eradicate the DoS at low energies and compute the profile of the gap to leading order in the parameter $t_D/t_E$ .