  The superfluid weight, free carrier density, and specific heat of the
three-dimensional tJ model are calculated by renormalization-group theory. We
find that optimal hole doping for superfluidity occurs in the electron density
range of n_i approximately between 0.63 - 0.68, where the superfluid weight
n_s/m* reaches a local maximum. This density range is within the novel tau
phase, where the electron hopping strength renormalizes to infinity, the system
remains partially filled at all length scales, and the electron-hopping
expectation value remains distinctively non-zero at all length scales. The
calculated superfluid weight drops off sharply in the overdoped region. Under
hole doping, the calculated density of free carriers increases until optimal
doping and remains approximately constant in the overdoped region, as seen
experimentally in high-T_c materials. Furthermore, from calculation of the
specific heat coefficient gamma, we see clear evidence of a gap in the
excitation spectrum for the tau phase.
