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)).