Van Hove Exciton-Cageons and High-T$_c$ Superconductivity, XA: Role of Spin-Orbit Coupling in Generating a Diabolical Point

Spin-orbit coupling plays a large role in stabilizing the low-temperature orthorhombic phase of La$_{2-x}$Sr$_x$CuO$_4$. It splits the degeneracy of the van Hove singularities (thereby stabilizing the distorted phase) and completely changes the shape of the Fermi surfaces, potentially introducing diabolical points into the band structure. The present paper gives a detailed account of the resulting electronic structure. A slave boson calculation shows how these results are modified in the presence of strong correlation effects. A scaling regime, found very close to the metal-insulator transition, allows an analytical determination of the crossover, in the limit of zero oxygen-oxygen hopping, $t_{OO}\rightarrow 0$. Extreme care must exercised in chosing the parameters of the three-band model. In particular, $t_{OO}$ is renormalized from its LDA value. Furthermore, it is suggested that the slave boson model be spin-corrected, in which case the system is close to a metal-insulator transition at half filling.