Elastic Anisotropies and Long-Ranged Interactions in Solid Membranes

Abstract

I study the effects of elastic anisotropy and long-ranged interactions on the fluctuations of solid membranes. Elastic anisotropies prove to be irrelevant in the absence of long-ranged interactions. Long-ranged interactions which fall off with distance $r$ like $r^{-\sigma }$ are relevant only for sufficiently small $\sigma$. In real (two-dimensional) membranes with dipolar interactions ($\sigma =3\mathrm{}$), the Lamé coefficients $\mu$ and $\lambda$ vanish like ${\left[\ln \right(\frac{1}{q}\left)\right]}^{-1\mathrm{}}$, as wave vector $q\to 0$. For larger but still relevant $\sigma \text{'}\mathrm{s}$, $\mu \propto q^{{\eta }_{\perp }}\propto \lambda$ with ${\eta }_{\perp }=2\mathrm{}\sigma -3D$, exactly, for $D$-dimensional membranes.