  We use holographic duality to study the entanglement entropy (EE) of
Conformal Field Theories (CFTs) in various spacetime dimensions $d$, in the
presence of various deformations: a relevant Lorentz scalar operator with
constant source, a temperature $T$, a chemical potential $\mu$, a marginal
Lorentz scalar operator with source linear in a spatial coordinate, and a
circle-compactified spatial direction. We consider EE between a strip or sphere
sub-region and the rest of the system, and define the "entanglement density"
(ED) as the change in EE due to the deformation, divided by the sub-region's
volume. Using the deformed CFTs above, we show how the ED's dependence on the
strip width or sphere radius, $L$, is useful for characterizing states of
matter. For example, the ED's small-$L$ behavior is determined either by the
dimension of the perturbing operator or by the first law of EE. For
Lorentz-invariant renormalization group (RG) flows between CFTs, the "area
theorem" states that the coefficient of the EE's area law term must be larger
in the UV than in the IR. In these cases the ED must therefore approach zero
from below as $L \to \infty$. However, when Lorentz symmetry is broken and the
IR fixed point has different scaling from the UV, we find that the ED often
approaches the thermal entropy density from above, indicating area theorem
violation.
