Level statistics inside the vortex of a superconductor and symplectic random-matrix theory in an external source

In the core of the vortex of a superconductor, energy levels appear inside the gap. We discuss here through a random matrix approach how these levels are broadened by impurities. It is first shown that the level statistics is governed by an ensemble consisting of a symplectic random. potential added to a nonrandom matrix. A generalization of previous work on the unitary ensemble in the presence of an external source (which relied on the Itzykson-Zuber integral) is discussed for this symplectic case through, the formalism introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit formulas for the density of states and for the correlation functions, which describe the crossover from the clean to the dirty limits. [S0163-1829(99)00429-7].