Anomalous Hall effect in the Abrikosov lattice of type-II superconductors

We study the temperature and magnetic field dependence of the vortex-core charge in the Abrikosov lattice of an $s$-wave superconductor based on the augmented quasiclassical equations with the Lorentz force and the pair-potential gradient (PPG) force. It is shown that the charging is dominated by the spatial derivative terms of the pair potential in the PPG force, where the PPG force terms are divided into the two parts, namely, (i) the spatial derivative of the pair potential and (ii) the product of the vector and pair potential. This originates from the fact that the Lorentz force and vector potential terms almost cancel each other out. We also clarify that the vortex-core charging due to the PPG force is caused by the anomalous Hall effect on circulating supercurrents, since the charge is large even in an isolated vortex and in the limit of high-$\kappa$, which is also the zero field limit, and originates primarily from the phase of the pair potential, i.e. supercurrents. Moreover, we see that the vortex-core charge peaks near zero temperature and about half the upper critical field, which makes it the most suitable region for a measurement of the charge.