also be understood based on a wavefunction proposed by Affleck, Lieb, Kennedy and Tasaki (the AKLT state) for $S=1$ spin chains [@AKLT] (see section IV) and have been observed experimentally in $S=1$ spin chain materials [@s1end]. In modern terminology, the end states of integer spin chains are a manifestation of symmetry-protected topological (SPT) order [@Gu09; @Chen12; @Pollmann12], which manifests itself as a boundary action that is protected by rotational (SO(3)) symmetry.[^1]

For half-odd-integer spin chains, the analysis is a bit more complicated. We start by rewriting Eq.  for $S_T^{(o)}$ as follows [@Ng94]: $$\begin{aligned}
      \label{WZo1}
         S_{T}^{(o)} & = &  {\hbar\over2}\left(4\pi{1\over2}Q+S\left(\Omega(\hat{n}(L))-\Omega(\hat{n}(0))\right)\right)  \\ \nonumber
      & = & {\hbar\over2}\left(4\pi{1\over2}Q+{1\over2}\left(\Omega(\hat{n}(L))-\Omega(\hat{n}(0))\right)\right.  \\ \nonumber
        & & +\left.(S-{1\over2})\left(\Omega(\hat{n}(L))-\Omega(\hat{n}(0))\right)\right)
    \end{aligned}$$ where we have replaced $S$ with $1/2$ in the usual topological (Pontryagin index) term and have divided the boundary Berry’s phase term into two parts; the first part, when combined with the Pontryagin index term, is the total Berry’s phase contribution for an open $S=1/2$ spin chain, and the second part is the additional contribution when $S>1/2$. Performing the strong coupling expansion as before, we find that the system behaves as an open spin-$1/2$ chain coupled to two end spins with a magnitude of ${1\over2}+{1\over2}(S-{1\over2})$. The problem of impurity 