  We study the behavior of the entanglement entropy in $(2+1)$--dimensional
strongly coupled theories via the AdS/CFT correspondence. We consider theories
at a finite charge density with a magnetic field, with their holographic dual
being Einstein-Maxwell-Dilaton theory in four dimensional anti--de Sitter
gravity. Restricting to black hole and electron star solutions at zero
temperature in the presence of a background magnetic field, we compute their
holographic entanglement entropy using the Ryu-Takayanagi prescription for both
strip and disk geometries. In the case of the electric or magnetic zero
temperature black holes, we are able to confirm that the entanglement entropy
is invariant under electric-magnetic duality. In the case of the electron star
with a finite magnetic field, for the strip geometry, we find a discontinuity
in the first derivative of the entanglement entropy as the strip width is
increased.
