Symmetry group factorization and unitary equivalence among Temperley-Lieb integrable models

It is shown that there is a hidden connection between the two well-studied sequences of the Temperley-Lieb (TL) integrable models -- the $q$-state quantum Potts (QP) models at the self-dual points and the staggered ${\rm SU}(n)$ spin-$s$ chains with $n=2s+1$ ($s \ge 1$), in addition to the uniform ${\rm SU}(2)$ spin-$1/2$ Heisenberg model. For each sequence, symmetry group factorization arises, in the sense that if $q$ is factorized into $q_1$ and $q_2$, then the $q$-state QP model is unitarily equivalent to a combined QP model with the symmetry group ${\rm S}_{q_1} \times {\rm S}_{q_2}$ or if $n$ is factorized into $n_1$ and $n_2$, then the staggered ${\rm SU}(n)$ spin-$s$ chain with the symmetry group ${\rm SU}(n)$ is unitarily equivalent to a combined staggered ${\rm SU}(n_1) \times {\rm SU}(n_2)$ spin chain with the symmetry group ${\rm SU}(n_1) \times {\rm SU}(n_2)$, valid for both ferromagnetic (FM) and antiferromagnetic (AF) cases. Moreover, the FM (AF) staggered ${\rm SU}(n)$ spin-$s$ chain is unitarily equivalent to the AF (FM) $q$-state QP model with $q=n^2$, as long as the size of the AF (FM) staggered ${\rm SU}(n)$ spin-$s$ chain is doubled. A combination of the two distinct types of unitary equivalences yields a family of models such that they are essentially identical, but appear in different guises. Some physical implications for unitary equivalence among different TL integrable models are clarified.