Thermal fluctuations in crystalline membranes with long-range dipole interactions

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.