Thermal excitations of warped membranes

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 {hα(q)}, α=D+1,...,d, in Fourier space with zero mean and a power-law variance hα(q1)hβ(q2) ̄∼δα,βδq1,-q2q1-dh. The case D=2, d=3, with dh=4 could be realized by flash-polymerizing lyotropic smectic liquid crystals. For D<max{4,dh} 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 κR∼q-ηf, while the in-hyperplane elastic constants decrease according to λR,μR∼q+ηu. The quenched background metric is relevant (irrelevant) for warped membranes characterized by exponent dh>4-ηf(F) (dh<4-ηf(F)), where ηf(F) is the scaling exponent for tethered surfaces with a flat background metric, and the scaling exponents are related through ηu+ηf=dh-D (ηu+2ηf=4-D).