# Thermal fluctuations in crystalline membranes with long-range dipole interactions

## TL;DR

This paper investigates how long-range dipole interactions influence the thermal fluctuations of crystalline membranes, showing they mainly renormalize elastic properties without altering universal scaling laws.

## Contribution

It demonstrates that dipole-dipole interactions do not change the fundamental scaling behavior of membranes, unlike scalar interactions, but significantly affect elastic coefficients.

## Key findings

- Dipole interactions renormalize elastic coefficients.
- Universal scaling laws remain valid with dipole interactions.
- Scalar interactions lead to different fluctuation scaling.

## Abstract

We study the effects of long-range electrostatic interactions on the thermal fluctuations of free-standing crystalline membranes exhibiting spontaneous electric polarization directed at each point along the local normal to the surface. We show that the leading effect of dipole-dipole interactions in the long-wavelength limit consists in renormalizations of the bending rigidity and the elastic coefficients. A completely different result was obtained in the case of scalar two-point interactions decaying as $R^{-3}$, where $R$ is the distance. In the latter case, which was addressed in previous theoretical research, the energy of long-wavelength bending fluctuations is controlled by power-law interactions and it scales with the wavevector $q$ as $q^{3}$, leading to a modified large-distance behaviour of correlation functions. By contrast, in the case of dipole interactions, the $q^{3}$ dependence of the bending energy vanishes. Non-local terms generated by the expansion of the electrostatic energy are suppressed in the limit of small wavevectors. This suggests that the universal scaling behaviour of elastic membranes holds even in presence of dipole interactions. At the same time, the shift of the Lam\'e coefficients and the bending rigidity induced by electrostatic interactions can be quantitatively important for two-dimensional materials with a permanent out-of-plane polarization.