The theoretical solution of quantum mechanical electronic conduction systems is essential for the understanding of high-$T_c$ superconductivity. We have made progress along this line, by solving the Hubbard and t-$J$ models in $d=3$ with renormalization-group theory and obtaining their phase diagrams and thermodynamic properties for all temperatures, densities, and coupling strengths. In the Hubbard model, at low temperatures and around half filling, an antiferromagnetic phase, of purely quantum mechanical origin, is obtained. Furthermore: (1) At strong coupling and 30-35% electron or hole doping from half filling; (2) At weak and intermediate coupling and 10-18% electron or hole doping; two distinct novel phases, which we call $tau$ phases, were found. The distinguishing character of these phases is that the electron or hole hopping expectation value is non-zero at all length scales. The weak-intermediate coupling $tau$ phase exhibits, as in BCS superconductivity, an excitation spectrum gap and, in the specific heat, a low-temperature exponential decay and a cusp phase transition singularity. The strong coupling $tau$ phase exhibits, as in BEC superconductivity, in the specific heat, an $alphasim -1$ phase transition singularity and a pair-formation peak above the phase transition temperature. In the limit of large Coulomb repulsion, the t-$J$ model is obtained. In our calculations for this model, we find that the superfluid weight increases with doping, passes through a maximum within the $tau$ phase at 32-37% doping, and decreases, and that the free carrier density also increases to a maximum value at 32-37% doping but remains at this value for further doping. These characteristics, seen experimentally in high-$T_c$ superconductors, have not been obtained in previous theoretical studies.

This research has been performed with my doctorate student Michael Hinczewski.

I.T.U., M.I.T., and Gürsey Research Center