  Motivated by recent developments in twisted bilayer graphene moir\'e
superlattices, we investigate the effects of electron-electron interactions in
a honeycomb lattice with an applied periodic potential using a
finite-temperature Wilson-Fisher momentum shell renormalization group (RG)
approach. We start with a low-energy effective theory for such a system, at
first giving a discussion of the most general case in which no point group
symmetry is preserved by the applied potential, and then focusing on the
special case in which the potential preserves a $D_3$ point group symmetry. As
in similar studies of bilayer graphene, we find that, while the coupling
constants describing the interactions diverge at or below a certain "critical
temperature" $T=T_c$, it turns out that ratios of these constants remain finite
and in fact provide information about what types of orders the system is
becoming unstable to. However, in contrast to these previous studies, we only
find isolated fixed rays, indicating that these orders are likely unstable to
perturbations to the coupling constants. Our RG analysis leads to the
qualitative conclusion that the emergent interaction-induced symmetry-breaking
phases in this model system, and perhaps therefore by extension in twisted
bilayer graphene, are generically unstable and fragile, and may thus manifest
strong sample dependence.
