Abstract:
We study the average density of states in a small metallic grain
coupled to two superconductors with the phase difference $\pi$,
in a magnetic field. The spectrum of the low-energy excitations
in the grain is described by the random matrix theory whose symmetry
depends on the magnetic field strength and coupling to the
superconductors.
In the limiting cases,
a pure superconducting symmetry class is realized.
For intermediate magnetic fields or couplings
to the superconductors, the system experiences a crossover
between different symmetry classes. With the help of the supersymmetric
$\sigma$-model we derive the exact expressions for the average density
of states in the crossovers between the symmetry classes A–C and CI–C.
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