  In Mott insulators with the t^4_{2g} electronic configuration such as of
Re^{3+}, Ru^{4+}, Os^{4+}, and Ir^{5+} ions, spin-orbit coupling dictates a Van
Vleck-type nonmagnetic ground state with angular momentum J=0, and the magnetic
response is governed by gapped singlet-triplet excitations. We derive the
exchange interactions between these excitons and study their collective
behavior on different lattices. In perovskites, a conventional Bose
condensation of excitons into a magnetic state is found, while an unexpected
one-dimensional behavior supporting spin-liquid states emerges in honeycomb
lattices, due to the bond-directional nature of exciton interactions in the
case of 90 degree d-p-d bonding geometry.
