Thermal Excitations of Warped Membranes

Abstract

We explore thermal fluctuations of thin planar membranes with a frozen spatially varying background metric and a shear modulus. We focus on a special class of D-dimensional “warped membranes” embedded in a d-dimensional space with d≥D+1 and a preferred height profile characterized by quenched random Gaussian variables \(\{h_\alpha(q)\}\), \(\alpha=D+1,...,d\), in Fourier space with zero mean and a power-law variance \(\over{h\alpha(q_1)h_\beta(q_2)}\) \(\sim \delta_{\alpha,\beta} \delta_{q_1,−q_2} q_1^{-d_h}\). The case D=2, d=3, with \(d_h=4\) could be realized by flash-polymerizing lyotropic smectic liquid crystals. For \(D\lt max\{4,d_h\}\) the elastic constants are nontrivially renormalized and become scale dependent. Via a self-consistent screening approximation we find that the renormalized bending rigidity increases for small wave vectors q as \(\kappa_R \sim q^{−\eta_f}\), while the in-hyperplane elastic constants decrease according to \(\lambda_R, \mu_R \sim q^{+\eta_u}\). The quenched background metric is relevant (irrelevant) for warped membranes characterized by exponent \(d_h\gt 4−\eta^{(F)}_f (d_h\lt 4−\eta ^{(F)}_f)\), where \(\eta^{(F)}_f\) is the scaling exponent for tethered surfaces with a flat background metric, and the scaling exponents are related through \(\eta_u+\eta_f=d_h−D (\eta_u+2\eta_f=4−D)\).