Gauge-invariant de Gennes model

Abstract

A gauge-invariant formulation of the de Gennes model for the nematic—to—smectic-$A$ transition is presented. In this formulation the energy associated with the gauge field $\overrightarrow{A}$ reduces to the Frank elastic energy with the application of the constraint ${\overrightarrow{n}}_0·\overrightarrow{A}=0\mathrm{}$ where ${\overrightarrow{n}}_0$ is the uniform equilibrium director and $\overrightarrow{A}$ is to be identified with deviations $\delta \overrightarrow{n}$ of the director from equilibrium. It is shown that thermodynamic quantities and renormalization-group recursion relations are gauge invariant. All gauge dependence appears in the exponent $\eta$ describing order-parameter correlations. The gauge invariance of a negative dielectric anisotropy smectic-$A$ in an external electric field is also studied.