We present a mixed-spin cluster expansion method to discuss the quantum phase transitions for the Haldane system in two and three dimensions. By mapping the $s=1\mathrm{}$ antiferromagnetic spin model on square and cubic lattices to the equivalent mixed-spin model, we study the competition among the Haldane, the dimer, and the magnetically ordered phases. The mixed-spin cluster expansion proposed here realizes the notion of the valence bond solid in a perturbation theory. This method allows us to directly deal with the Haldane phase, which may not be reached by standard series expansion methods. The zero-temperature phase diagram is thus determined rather precisely for the two- and three-dimensional Haldane systems.

• Received 21 October 1999

DOI:https://doi.org/10.1103/PhysRevB.61.6133