Global phase diagram of two-component Bose gases in antiparallel   magnetic fields
  We study the ground-state phase diagram of two-dimensional two-component (or pseudospin-1/2) Bose gases in mutually antiparallel synthetic magnetic fields in the space of the total filling factor and the ratio of the intercomponent coupling $g_{\uparrow\downarrow}$ to the intracomponent one g>0. This time-reversal-invariant setting represents a bosonic analogue of spin Hall systems. Using exact diagonalization, we find that (fractional) quantum spin Hall states composed of a pair of nearly independent quantum Hall states are remarkably robust and persist for $g_{\uparrow\downarrow}$ up to as large as g. For $g_{\uparrow\downarrow}=-g$, we find the exact many-body ground state in which particles in different spin states form pairs. This gives the exact critical line beyond which the system collapses. 
Shunsuke Furukawa, Masahito Ueda
cond-mat.quant-gas cond-mat.str-el