  We study binary Bose-Einstein condensates subject to synthetic magnetic
fields in mutually parallel or antiparallel directions. Within the mean-field
theory, the two types of fields have been shown to give the same vortex-lattice
phase diagram. We develop an improved effective field theory to study
properties of collective modes and ground-state intercomponent entanglement.
Here, we point out the importance of introducing renormalized coupling
constants for coarse-grained densities. We show that the low-energy excitation
spectra for the two kindsof fields are related to each other by suitable
rescaling using the renormalized constants. By calculating the entanglement
entropy, we find that for an intercomponent repulsion (attraction), the two
components are more strongly entangled in the case of parallel (antiparallel)
fields, in qualitative agreement with recent studies for a quantum (spin) Hall
regime. We also find that the entanglement spectrum exhibits an anomalous
square-root dispersion relation, which leads to a subleading logarithmic term
in the entanglement entropy. All of these are confirmed by numerical
calculations based on the Bogoliubov theory with the lowest-Landau-level
approximation. Finally, we investigate the effects of quantum fluctuations on
the phase diagrams by calculating the correction to the ground-state energy due
to zero-point fluctuations in the Bogoliubov theory. We find that the
boundaries between rhombic-, square-, and rectangular-lattice phases shift
appreciably with a decrease in the filling factor.
