the Pontryagin index for the topological term $S_T$. For an open chain of length $L$, $S_T$ is replaced by [@Haldane83PLA; @Affleck90; @Ng94] $$\begin{aligned}
      \label{WZo}
         S_{T}^{(o)} & = &{\hbar\over2}\int^L_0 d^dx{\partial S_B(\hat{n}(x))\over \partial x}  \\ \nonumber
         & = & 2\pi\hbar SQ+{\hbar S\over2}\left(\Omega(\hat{n}(L))-\Omega(\hat{n}(0))\right),
      \end{aligned}$$ where $2\pi SQ=\theta Q$ is the usual topological $\theta$ term that we obtain when $\Omega(\hat{n}(0))=\Omega(\hat{n}(L))$, i.e., when we consider periodic boundary conditions. An open chain differs from a closed chain in the existence of an additional boundary Berry’s phase term with an effective spin magnitude of $S/2$.

We now examine the effect of this additional Berry’s phase term. First, we consider integer spin chains. Following the previous discussion, we expect the spin chain to be described by the strong coupling limit of the effective Hamiltonian given in Eq. , except that the rotors at the two ends of the spin chain are subjected to monopole potentials of strength $S/2$, resulting in effective free spins of magnitude $S/2$ located at the ends of the spin chain. The two spins are coupled by a term $J_{eff}\sim JS^2e^{-L/\xi}$ when the coupling between rotors is considered, where $\xi\sim E^{-1}_g$ is the correlation length and $E_g$ is the spin gap. These end states can 