Order, Disorder, and Transitions in Decorated AKLT States on Bethe Lattices

Returning to one of the original generalizations of the AKLT state, we extend prior analysis on the Bethe lattice (or Cayley tree) to a variant with a series of $n$ spin-1 decorations placed on each edge. The recurrence relations derived for this system demonstrate that such systems are critical for coordination numbers $z=3^{n+1}$, demonstrating order for greater and disorder for lesser coordination number. We then generalize further, effectively interpolating between systems with different values of $n$, using two realizations, one isotropic under local $SU(2)$ transformations and one anisotropic. Exact analysis of these recurrence relations allows us to deduce the location and behavior of order-disorder phase transitions for $z>4$.