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 $d\ge D+1$ and a preferred height profile characterized by quenched random Gaussian variables $\{h_α({\bf q})\}$, $α=D+1,\ldots, d$, in Fourier space with zero mean and a power law variance $\overline{ h_α({\bf q}_1) h_β({\bf q}_2) } \sim δ_{α, β} \, δ_{{\bf q}_1, -{\bf q}_2} \, q_1^{-d_h}$. 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 ${\bf q}$ 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 $η_u + η_f = d_h - D$ ($η_u + 2 η_f=4-D$).

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