Supersymmetry and the superconductor-insulator transition

We present a theory of supersymmetric superconductivity and discuss its physical properties. We define the supercharges Q and Q(+) satisfying Q psi(BCS) = Q+psi(BCS) = 0 for the Bardeen-Cooper- Schrieffer state psi(BCS). They possess the property expressed by Q(2) = (Q(+))(2) = 0, and psi(BCS) is the ground state of the supersymmetric Hamiltonian H = E(QQ(+) + Q(+)Q) for E > 0. The superpartners psi(g) and psi(BCS) are shown to be degenerate. Here psi(g) denotes a fermionic state within the superconducting gap that exhibits a zero-energy peak in the density of states. A supers,,' mmetric model of superconductivity with two bands is presented. On the basis of this modei we argue that the system of interest goes into a superconducting state from an insulator if an attractive interaction acts between states in the two bands. There are many unusual properties of this model due to an unconventional gap equation stemming from the two-band effect. The model exhibits an unconventional insulator-superconductor first-order phase transition. In the ground state, a first-order transition occurs at the supersymmetric point. Ve show that certain universal relations in the BCS theory, such as that involving the ratio Delta(0)/k(B)T(c) do not hold in the present model.