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