Inverse-density Abelian Higgs model ($V=\frac{1}{2}\int [B^2/h+|D\Phi {|}^2+h(1-h{)}^2/4],h=|\Phi {|}^2$)

This planar Abelian Higgs model replaces the usual magnetic energy by $B^2/h$. Use $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$. The magnetic term is singular at a zero of the Higgs field, so the admissible class must have finite energy; formal field equations are first derived where $h>0$. Its positive-flux Bogomolny equations are $(D_1+i{D}_2)\Phi =0$ and $B=h(1-h)/2$. On those solutions $B/h=(1-h)/2$ extends regularly across zeros.

Write $\Phi =e^{u/2+i\chi }$ away from its zeros and use the oriented planar Hodge star with $\ast d{x}^1=d{x}^2$. The first Bogomolny equation gives $A=d\chi -\frac{1}{2}\ast du$. A zero of multiplicity $n_j$ contributes $u=2n_j\log |x-p_j|+\text{smooth}$ and phase winding $2\pi n_j$. Combining the smooth magnetic field with these delta sources gives the displayed equation and the topological boundary condition $u\to 0$. Prescribed zero positions and multiplicities reduce multi-vortex construction to this scalar elliptic problem; a regular solution reconstructs both smooth fields.

Let $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identityand completing the square give, when the current boundary integral vanishes,For flux $2\pi N>0$, the lower bound is $\pi N$, saturated by the two first-order Bogomolny equations. Division by $h$ is interpreted in the finite-energy sense at its zeros.