  Analogous to Peierls' arguments for the `anomalous' Hall in metals I
demonstrate that the Hall anomaly in the mixed state of superconductors, the
sign change of the Hall resistivity, is a property of a vortex many-body
correlation, and show that the anomaly is due to the competition between vortex
vacancies and interstitials. Within this vortex many-body effect picture, many
features of the complicated Hall effect can be understood in quantitative
terms. For example, the expression for the vacancy formation energy is
obtained, the scaling relation between the Hall and longitudinal resistivities
with the power varying between 1 and 2 is found to depend on sample details,
and near the superconducting transition temperature and for small magnetic
fields the Hall conductivity is proportional to the inverse of the magnetic
field and to the quadratic of the difference between the measured and the
transition temperatures.
