  We use the AdS/CFT correspondence to study models of entanglement and
correlations between two $d=4$ CFTs in thermofield double states at finite
chemical potential. Our bulk spacetimes are planar Reissner-Nordstr\"om AdS
black holes. We compute both thermo-mutual information and the two-point
correlators of large-dimension scalar operators, focussing on the
small-temperature behavior -- an infrared limit with behavior similar to that
seen at large times. The interesting feature of this model is of course that
the entropy density remains finite as $T \rightarrow 0$ while the bulk geometry
develops an infinite throat. This leads to a logarithmic divergence in the
scale required for non-zero mutual information between equal-sized strips in
the two CFTs, though the mutual information between one entire CFT and a
finite-sized strip in the other can remain non-zero even at $T=0$. Furthermore,
despite the infinite throat, there can be extremally charged operators for
which the two-point correlations remain finite as $T \rightarrow 0$. This
suggests an interestingly mixed picture in which some aspects of the
entanglement remain localized on scales set by the chemical potential, while
others shift to larger and larger scales. We also comment on implications for
the localized-quasiparticle picture of entanglement.
