H}_{rp}$ takes the form $${\cal H}=\sum_p \, \mu_0 \, |\phi_p|^2 + \sum_{\bf k} \biggl({b
\over {2 \xi_0^2 d}} \, v_{\bf k} \, |{\tilde \rho}_{\bf k}|^2 +
(u_{-{\bf k}} + f_{-{\bf k}} \, {\bf k}^2 \xi_0^2 ) \, v^{1/2}_{\bf k}
{\tilde \rho}_{\bf k} \biggr), \eqno(2.1)$$ where $h=2 \pi \xi_0^2 B/\phi_0$, $\mu_0=-1+h+T/T_{c0}$, $B$ the magnitude of the applied field, $v_{\bf k}=\exp(-{\bf k}^2/(2 h))$, and $${\tilde \rho}_{\bf k}={{\xi_0} \over L} \sum_p \exp({\rm i} p k_1/h)
\,
\phi^*_{p-k_2/2} \, \phi_{p + k_2/2} \eqno(2.2)$$ with linear system size $L$. Since the characteristic microscopic length of a vortex state is the magnetic length $r_B \equiv \sqrt{\phi_0/(2 \pi
B)}$, the factor $k^2$ of the random-flux term implies that the random potential $f$ is accompanied by the factor $h$ and hence that the pinning disorder will be enhanced with increasing field. This trend is also valid in the phase-only model with no $u$-potential term and will be valid in general at least in type II limit \[20,21\]. The Gaussian ensemble for the $u$-potential will be assumed: ${\overline {u({\bf r}) \, u({\bf r}')}}=\Delta
\delta^{(2)} ({\bf r} - {\bf r}')$. For just simplicity of our presentation, the $f$-potential term will be omitted for a while.

First, let us consider, as 