Order and Disorder in AKLT Antiferromagnets in Three Dimensions

The models constructed by Affleck, Kennedy, Lieb, and Tasaki (PRL {\bf 59}, 799 (1987)) describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin $S$ is an integer multiple $M$ of half the lattice coordination number. The equal time quantum correlations in their ground states may be computed as finite temperature correlations of a classical $\textsf{O}(3)$ model on the same lattice, where the temperature is given by $T=1/M$. In dimensions $d=1$ and $d=2$ this mapping implies that all AKLT states are quantum disordered. We consider the $d=3$ case where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large $S$ some form of N{\'e}el order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered while for the unfrustrated diamond lattice the minimal $S=2$ state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal $S=3$ state are disordered. These are a significant addition to the catalog of magnetic Hamiltonians in $d=3$ with ground states known to lack order on account of strong quantum fluctuations.