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

TL;DR

This paper analyzes decorated AKLT states on Bethe lattices, revealing critical behavior and phase transitions between ordered and disordered states depending on lattice coordination and decoration parameters.

Contribution

It extends AKLT state analysis to decorated Bethe lattices, deriving recurrence relations and identifying phase transition points for various coordination numbers.

Findings

Systems are critical at coordination number z=3^{n+1}.

Order exists for z > 4, disorder for lower z.

Phase transitions depend on decoration level n and lattice anisotropy.

Abstract

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 $z>4$.