Two superconducting phases in the d=3 Hubbard model

Abstract

The phase diagram of the d=3 Hubbard model is calculated as a function of temperature and electron density 〈ni〉, in the full range of densities between 0 and 2 electrons per site, using renormalization-group theory. An antiferromagnetic phase occurs at lower temperatures, at and near the half-filling density of 〈ni〉= 1. The antiferromagnetic phase is unstable to hole or electron doping of at most 15%, yielding to two distinct“τ" phases: for large coupling U/t, one such phase occurs between 30–35% hole or electron doping, and for small to intermediate coupling U/t another such phase occurs between 10–18% doping. Both τ phases are distinguished by non-zero hole or electron hopping expectation values at all length scales. Under further doping, the τ phases yield to hole- or electron-rich disordered phases. We have calculated the specific heat over the entire phase diagram. The low-temperature specific heat of the weak-coupling τ phase shows an exponential decay, indicating a gap in the excitation spectrum, and a cusp singularity at the phase boundary. The strong-coupling τ phase, on the other hand, has a critical exponent α≈-1, and an additional peak in the specific heat above the transition temperature possibly indicating pair formation. In the limit of large Coulomb repulsion, the phase diagram of the tJ model is recovered.