  We compute the second inner variation of the Abelian Yang--Mills--Higgs and
Ginzburg--Landau energies. Given a sequence of critical points with energy
measures converging to a codimension $2$ minimal submanifold, we use the second
inner variation formula to bound the morse index of the submanifold by the
index of the critical points. The key tools are the convergence of the energy
measures and the stress-energy tensors of solutions to Abelian
Yang--Mills--Higgs and Ginzburg--Landau equations.
