Vortex Core Structure and Hall Effect in d-wave Superconductors

The sign of the Hall conductivity σ_xy in metals corresponds to the sign of the charge carriers. In type II superconductors σ_xy was predicted to have the same sign as in the normal state; that is, the same as the charge carriers. However, Hall measurements showed that σ_xy changes sign as a function of temperature in cuprate high temperature superconductors such as YBCO shortly after their discovery, contradicting the existing models. Recently, another anomaly in the form of a transition from σ_xy ∝ 1+p to σ_xy ∝ p, where p is the hole doping was measured in YBCO. We show that both of these effects are exhibited by a simple model for a d-wave superconductor. We solve this model using self-consistent mean field, and calculate its Hall conductivity via the relation between σ_xy and Chern numbers. We show that the sign of σ_xy corresponds to the sign of excess charge in the vortex core, while the transition from σ_xy ∝ 1+p to σ_xy ∝ p is triggered by the appearance of antiferromagnetism in the core. We then proceed to study the t-J model, widely considered to be a minimal model for the cuprates, using variational Monte Carlo. Once again we find both effects, suggesting that the peculiarities of the Hall conductivity in the cuprates may be entirely explained by their vortex cores' structure.