  We consider a two-dimensional nonlinear Schr{\"o}dinger equation proposed in
Physics to model rotational binary Bose-Einstein condensates. The nonlinearity
is a logarithmic modification of the usual cubic nonlinearity. The presence of
both the external confining potential and rotating frame makes it difficult to
apply standard techniques to directly construct ground states, as we explain in
an appendix. The goal of the present paper is to analyze the orbital stability
of the set of energy minimizers under mass constraint, according to the
relative strength of the confining potential compared to the angular frequency.
The main novelty concerns the critical case (lowest Landau Level) where these
two effects compensate exactly, and orbital stability is established by using
techniques related to magnetic Schr{\"o}dinger operators.
