Fractional charging and reentrance in granular superconductors

Two cases of quantum-mechanical states of corresponding to non-integer numbers of superconducting pairs are discussed within the context of granular superconductors. The first correspond to states that appear to be half-pairs. The physical consistency of such states is established on the basis of the Anderson pseudo-spin picture of superconductivity. In the second case, it appears that the number of pairs in a given system may range over a continuum. In this case, the continuum arises in connection with a modification of the usual Hubbard-Stratonovich transformation used to discuss superconductivity. Physically, there is a situation in which coupling to external degrees of freedom break the symmetry associated with an integer counting of pairs. A formal device, called a frustrated H-S transformation, allows discussion of this symmetry breaking as a charge continuum. A pair-sensitive device making a measurement on such a continuum, would always measure an integer pair number, the average over an ensemble of identical systems yielding a new quantum number, which we call the crystal charge, corresponding to the continuum. These continuous states give rise to a reentrant phase transition in a granular superconductor