transition, $V_{\bf k}$ will take a form of RPA type such as $$V^{\rm liq}_{\bf k} \simeq {{v_{\bf k}} \over {1 + 2 \, x \, v_{\bf
k}}}. \eqno(2.7)$$ On the other hand, one will notice by comparing eq.(2.6) with its mean field expression \[30\] that $V_{\bf k}$ in the limit of a perfect solid takes the form $$V^{\rm sol}_{\bf k}=V^{\rm liq}_{\bf k}(x \gg 1) + \delta V_{\bf k}
=
{1 \over {2 x}} ( \, 1 - N_v \sum_{{\bf G} \neq 0} \delta_{{\bf k}, {\bf
G}} \, ), \eqno(2.8)$$ where ${\bf G}$’s are the reciprocal lattice vectors of the vortex solid. Note that the first term corresponds to the $x \to \infty$ (i.e., low $T$) limit of $V^{\rm liq}_{\bf k}$, while the second term $\delta
V_k$ corresponds to the structure factor of a vortex state. In a solid-like vortex state with a long but finite positional correlation length, $\delta
V_{\bf k}$ near ${\bf G}$ will take, for instance, a Gaussian form like $$\delta V_{\bf k} \sim - {1 \over {2 \, x \, \epsilon}} \exp{\biggl(-
{{({\bf k} - {\bf G})^2}
\over {2 \epsilon h}} \biggr)}, \eqno(2.9)$$ where $\epsilon^{-1}$ ($\gg 1$) corresponds to the positional correlation area $N_{\rm cor}$. One can 