# Exact solution for the order parameter profiles and the Casimir force in $^4$He superfluid films in an effective field theory

## TL;DR

This paper provides an exact analytical solution for the order parameter profiles and Casimir force in $^4$He superfluid films near the transition, aligning well with experimental data and predicting force characteristics.

## Contribution

It introduces an exact solution within an effective field theory framework for superfluid helium films, connecting theory with experimental observations.

## Key findings

- The Casimir force is attractive in the model.
- The extremum of the force occurs at x=π, matching experiments.
- The theory's predictions are independent of the adjustable parameter M.

## Abstract

We present an analytical solution of an effective field theory which, in one of its formulations, is equivalent to the Ginzburg's $\Psi$-theory for the behavior of the Casimir force in a film of $^4$He in equilibrium with its vapor near the superfluid transition point. We consider three versions of the theory, depending on the way one determines its parameters from the experimental measurements. We present exact results for the behavior of the order parameter profiles and of the Casimir force within this theory, which is characterized by $d=3$, $\nu=2/3$ and $\beta=1/3$, where $d$ is the bulk spatial dimension and $\nu$ and $\beta$ are the usual critical exponents. In addition, we revisit relevant experiments \cite{GC99} and \cite{GSGC2006} in terms of our findings. We find reasonably good agreement between our theoretical predictions and the experimental data. We demonstrate analytically that our calculated force is attractive. The position of the extremum is predicted to be at $x_{\rm min}=\pi$, with $x=(L/\xi_0)(T/T_\lambda-1)^{1/\nu}$, which value effectively coincides with the experimental finding $x_{\rm min}=3.2\pm 0.18$. Here $L$ is the thickness of the film, $T_\lambda$ is the bulk critical temperature and $\xi_0$ is the correlation length amplitude of the system for temperature $T>T_\lambda$. The theoretically predicted position of the minimum does not depend on the one adjustable parameter, $M$, entering the theory.