Order, Disorder, and Transitions in AKLT-like States on the Decorated Bethe Lattice

We consider the AKLT state on the Bethe lattice (solved by Affleck, Kennedy, Lieb, and Tasaki in 1988 and Fannes, Nachtergaele, and Werner in 1991), and extend it, first to a variant with a chain of n spin-1 decorations on each edge. Having found the recurrence relations that define their long-range behavior in each case, we use two continuously-parametrized variants of those decorations to interpolate between those cases. In those systems we are able to find order-disorder transitions on Bethe lattices of any coordination number z>4. We analyze these critical points exactly, finding simple critical exponents.