  We introduce an Allen-Cahn type functional, $\text{BE}_{\epsilon}$, that
defines an energy on separating hypersurfaces, $Y$, of closed Riemannian
Manifolds. We establish $\Gamma$-convergence of $\text{BE}_{\epsilon}$ to the
area functional, and compute first and second variations of this functional
under hypersurface pertrubations. We then compute an explicit expansion for the
variational formula as $\epsilon \to 0$. A key component of this proof is the
invertibility of the linearized Allen-Cahn equation about a solution, on the
space of functions vanishing on $Y$. We also relate the index and nullity of
$\text{BE}_{\epsilon}$ to the Allen-Cahn index and nullity of a corresponding
solution vanishing on $Y$. We apply the second variation formula and index
theorems to show that the family of $2p$-dihedrally symmetric solutions to
Allen-Cahn on $S^1$ have index $2p - 1$ and nullity $1$.
