Dual free energies in Poisson-Boltzmann theory

Poisson-Boltzmann theory allows one to study soft matter and biophysical systems involving point-like charges of low valencies. The inclusion of fluctuation corrections beyond the mean-field approach typically requires the application of loop expansions around a mean-field solution for the electrostatic potential $\phi({\bf r})$, or sophisticated variational approaches. Recently, Poisson-Boltzmann theory has been recast, via a Legendre transform, as a mean-field theory involving the dielectric displacement field ${\bf D}({\bf r})$. In this paper we consider the path integral formulation of the dual theory. Exploiting the transformation between $\phi$ and ${\bf D}$, we formulate a dual Sine-Gordon field theory in terms of the displacement field and provide a strategy for precise numerical computations of free energies beyond the leading order.