Scale without conformal invariance in membrane theory

We investigate the relation between dilatation and conformal symmetries in statistical mechanics of flexible crystalline membranes. analyze, particular, a well-known model which describes fluctuations continuum elastic medium embedded higher-dimensional space. In this theory, renormalization group flow connects non-interacting ultraviolet fixed point, where theory is controlled by linear elasticity, to an interacting infrared point. By studying structure correlation functions energy-momentum tensor, we show that, infrared, only scale-invariant: symmetry not enhanced full invariance. The shown present non-vanishing virial current which, despite being non-conserved, maintains scaling dimension exactly equal $D - 1$, even presence interactions. attribute absence anomalous dimensions under translations rotations embedding space, are realized as shifts phonon fields, protect several non-invariant operators. also note that closure algebra with both shift invariance would require, hypothesis phonons transform primary new expected hold on physical grounds. then consider alternative model, involving scalar effective phonon-mediated interactions local Gaussian curvatures. described two copies biharmonic conformal, but flows point argue be dilatation-invariant.