Charged Plate Beyond Mean-field: One-loop Corrections by Salt-density Fluctuation

We present an exact field-theoretic formulation for a fluctuating, generally asymmetric, salt density in the presence of a charged plate. The non-linear Poisson-Boltzmann equation is obtained as the saddle-point of our field theory action. Focusing on the case of symmetric salt, we systematically compute first-order corrections arising from electrolytes fluctuation to the free energy density, which can be explicitly obtained in closed form. We find that for systems with low to moderate salt density, fluctuation corrections to the free-energy depends sensitively on the salt concentration as well as their charge valency. Further, we find that electrolyte fluctuation leads to a reduced electrostatic repulsion between two point-charges when they are close to the charged plate.