Quantum phase transitions for the Haldane system in higher dimensions: A mixed-spin cluster expansion approach

We present a mixed-spin cluster expansion method to discuss the quantum pha se transitions for the Haldane system in two and three dimensions. By mappi ng the s = 1 antiferromagnetic spin model on square and cubic lattices to t he equivalent mixed-spin model, we study the competition among the Haldane, the dimer, and the magnetically ordered phases. The mixed-spin cluster exp ansion proposed here realizes the notion of the valence bond solid in a per turbation theory. This method allows us to directly deal with the Haldane p hase, which may not be reached by standard series expansion methods. The ze ro-temperature phase diagram is thus determined rather precisely for the tw o- and three-dimensional Haldane systems.