  We study 2+1 dimensional conformal field theories (CFTs) with a globally
conserved U(1) charge, placed in a chemical potential which is periodically
modulated along the spatial direction $x$ with zero average: $\mu(x) = V
\cos(kx)$. The dynamics of such theories depends only on the dimensionless
ratio $V/k$, and we expect that they flow in the infrared to new CFTs whose
universality class changes as a function of $V/k$. We compute the
frequency-dependent conductivity of strongly-coupled CFTs using holography of
the Einstein-Maxwell theory in 4-dimensional anti-de Sitter space. We compare
the results with the corresponding computation of weakly-coupled CFTs,
perturbed away from the CFT of free, massless Dirac fermions (which describes
graphene at low energies). We find that the results of the two computations
have significant qualitative similarities. However, differences do appear in
the vicinities of an infinite discrete set of values of $V/k$: the universality
class of the infrared CFT changes at these values in the weakly-coupled theory,
by the emergence of new zero modes of Dirac fermions which are remnants of
local Fermi surfaces. The infrared theory changes continuously in holography,
and the classical gravitational theory does not capture the physics of the
discrete transition points between the infrared CFTs. We briefly note
implications for a non-zero average chemical potential.
