Thermal Excitations of Warped Membranes

Andrej Kosmrlj; David R. Nelson

doi:10.1103/PhysRevE.89.022126

paper

Thermal Excitations of Warped Membranes

arXiv:1312.4089 · doi:10.1103/PhysRevE.89.022126

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 $d \geq D + 1$ and a preferred height profile characterized by quenched random Gaussian variables ${h_{α} (q)}$ , $α = D + 1, \dots, d$ , in Fourier space with zero mean and a power law variance $h_{α} (q_{1}) h_{β} (q_{2}) \sim δ_{α, β} δ_{q_{1}, - q_{2}} q_{1}^{- d_{h}}$ . 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 non-trivially renormalized and become scale dependent. Via a self consistent screening approximation we find that the renormalized bending rigidity increases for small wavevectors $q$ as $κ_{R} \sim q^{- η_{f}}$ , while the in-hyperplane elastic constants decrease according to $λ_{R}, μ_{R} \sim q^{+ η_{u}}$ . The quenched background metric is relevant (irelevant) 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 $η_{u} + η_{f} = d_{h} - D$ ( $η_{u} + 2 η_{f} = 4 - D$ ).

14 pages, 3 figures

Thermal Excitations of Warped Membranes

Abstract

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