  Using the approach of low-energy effective field theory, the phase diagram is
studied for a mixture of two species of pseudospin-$\1/2$ Bose atoms with
interspecies spin-exchange. There are four mean-field regimes on the parameter
plane of $g_e$ and $g_z$, where $g_e$ is the interspecies spin-exchange
interaction strength, while $g_z$ is the difference between the interaction
strength of interspecies scattering without spin-exchange of equal spins and
that of unequal spins. Two regimes, with $|g_z| > |g_e|$, correspond to ground
states with the total spins of the two species parallel or antiparallel along
$z$ direction, and the low energy excitations are equivalent to those of
two-component spinless Bosons. The other two regimes, with $|g_e| > |g_z|$,
correspond to ground states with the total spins of the two species parallel or
antiparallel on $xy$ plane, and the low energy excitations are described by a
sine-Gordon model coupled with a free scalar field, where the effective fields
are combinations of the phases of the original four Boson fields. In
(1+1)-dimension, they are described by Kosterlitz-Thouless renormalization
group (RG) equations, and there are three sectors in the phase plane of a
scaling dimension and a dimensionless parameter proportional to the strength of
the cosine interaction, both depending on the densities. The gaps of these
elementary excitations are experimental probes of the underlying many-body
ground states.
