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 {h(α)(q)}, α = D + 1,...,d, in Fourier space with zero mean and a power-law variance h(α)(q(1))h(β)(q(2)) ∼ δ(α,β)δ(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 < 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 κ(R) ∼ q(-η(f)), while the in-hyperplane elastic constants decrease according to λ(R),μ(R) ∼ q(+η(u)). The quenched background metric is relevant (irrelevant) for warped membranes characterized by exponent d(h) > 4-η(f)((F)) (d(h) < 4-η(f)((F))), where η(f)((F)) is the scaling exponent for tethered surfaces with a flat background metric, and the scaling exponents are related through η(u) + η(f) = d(h) -D (η(u) + 2η(f) = 4-D).