Berry phases in superconducting transitions

I generalize the concept of Berry's geometrical phase for quasicyclic Hamiltonians to the case in which the ground state evolves adiabatically to an excited state after one cycle, but returns to the ground state after an integer number of cycles. This allows to extend the charge Berry phase $\gamma _{\rm c}$ related to the macroscopic polarization, to many-body systems with fractional number of particles per site. Under certain conditions, $\gamma _{\rm c}$ and the spin Berry phase $\gamma _{\rm s}$ jump in $\pi $ at the boundary of superconducting phases. In the extended Hubbard chain with on-site attraction U and nearest-neighbor interaction V at quarter filling, the transitions detected agree very well with exact results in two limits solved by the Bethe ansatz, and with previous numerical studies. In chains with spin SU(2) symmetry, $\gamma _{\rm s}$ jumps when a spin gap opens