Poisson's equation in the vortex state of type II superconductors

We derive Poisson's equation in the single-vortex state of type II superconductors in a conserved approximation in both dynamical and static cases. The charge and current vertex functions in the chain approximation are expressed in terms of the solutions of the Bogoliubov-de Gennes equation. We rigorously prove that these vertex functions satisfy the Ward-Takahashi relation. The vertex-correction in the chain approximation is, thus, consistent with the single-electron states satisfying the Bogoliubov-de Gennes equation in the single-vortex state. It is also shown that the vertex correction for the charge-charge correlation function vanishes in the static limit. The static vortex charge, thus, can be calculated by solving Poisson's equation in the lowest-order approximation.