  We study the scaling behavior of two-dimensional (2D) crystalline membranes
in the flat phase by a renormalization group (RG) method and an
$\epsilon$-expansion. Generalization of the problem to non-integer dimensions,
necessary to control the $\epsilon$-expansion, is achieved by dimensional
continuation of a well-known effective theory describing out-of-plane
fluctuations coupled to phonon-mediated interactions via a scalar composite
field, equivalent for small deformations to the local Gaussian curvature. The
effective theory, which will be referred to as Gaussian curvature interaction
(GCI) model, is equivalent to theories of elastic $D$-dimensional manifolds
fluctuating in a $(D + d_{c})$-dimensional embedding space in the physical case
$D = 2$ for arbitrary $d_{c}$. For $D\neq 2$, instead, the GCI model is not
equivalent to a direct dimensional continuation of elastic membrane theory and
it defines an alternative generalization to generic internal dimension $D$. We
calculate explicitly RG functions at two-loop order and determine the exponent
$\eta$ characterizing the long-wavelength scaling of correlation functions to
order $\epsilon^{2}$ in an $\epsilon=(4-D)$-expansion. The self-consistent
screening approximation (SCSA) for the GCI model is shown to be exact to
O($\epsilon^{2}$). For $d_{c} = 1$, the O($\epsilon^{2}$) correction is
suppressed by a small numerical prefactor. As a result, despite the large value
of $\epsilon = 2$, extrapolation of the first and second order results to $D =
2$ leads to very close numbers, $\eta = 0.8$ and $\eta \simeq 0.795$. The
calculated exponent values are close to earlier reference results obtained by
non-perturbative RG, the SCSA and numerical simulations. These indications
suggest that a perturbative analysis of the GCI model could provide an useful
framework for accurate quantitative predictions of the scaling exponent even at
$D = 2$.
