Spin Permutation at Work in the BCS Hamiltonian

Abstract

Owing to conservation of the total projected spin in the elementary two-body scattering process, two electron spin transpositions are shown to secure an exact block-diagonal splitting of the whole BCS Hamiltonian matrix H. Because the transpositions do not commute with the Hamiltonian due to the Pauli principle, the Hilbert space does not supply any representation of the group of transpositions. Likewise every other eigenstate of H is found to be an (odd) eigenstate of a spin transposition operator while the remaining eigenstates of H are not eigenstates of the spin transpositions. Each block-diagonal submatrix is characterized by its own signature indicative of whether all of its eigenstates are eigen-state of each spin transposition operator or not. The submatrix sizes range from a lowest value (= 1 at half-filling) up to the largest value d _M . The groundstate is included in the largest submatrix or in a smaller one according to whether the number of electron pairs is odd or even. The ratio d ₀ / d _M where d ₀ designates the dimension of the BCS Hilbert space grows exponentially with the electron concentration. This result holds for any dimension of space, sign of the two-body interaction and electron filling. The whole spectrum of eigenvalues has been obtained for all possible values of the total momentum of the eigenstates of H up to d ₀ ~ 10⁵. Possible extensions towards much higher do values are outlined.