Flux quantization in rings for Hubbard (attractive and repulsive) and t-J-like Hamiltonians

The ground-state energy of rings with 10 or 16 sites and 2 or 4 fermions (holes or electrons) is computed as a function of the flux ? created by a vector potential constant along the ring (Aharonov-Bohm setup) for three different model Hamiltonians: the attractive (U<0) and repulsive (U>0) Hubbard and a t-J-like Hamiltonian. In all cases the flux is found to be quantized in units hc/2e, showing that the charge carriers are pairs. The possible role of phonon fluctuations in discriminating among the different models is also investigated.