integer and half-odd-integer spin chains [@Haldane88-NLSM]. To proceed further, we first consider integer spin chains, where $e^{{i\over\hbar}S_T}\equiv 1$ and the system is described by the “pure" NL$\sigma$M $S_{\sigma}$.

### Integer spin chains

We start by asking the following question: what are the plausible ground states described by $S_{\sigma}$? For this purpose, it is more convenient to consider a lattice version of $S_{\sigma}$: $$\label{nlsm3}
    S_{\sigma}\rightarrow{1\over2}\int dt\sum_i \left({1\over J}\left({\partial\hat{n}_i\over\partial
 t}\right)^2+JS^2\hat{n}_i\cdot\hat{n}_{i+1}\right),$$ with the corresponding Hamiltonian $$\label{hrotor}
    H_{\sigma}={J\over2}\sum_i\left((\mathbf{L}_i)^2-S^2\hat{n}_i\cdot\hat{n}_{i+1}\right),$$ where $\mathbf{L}_i$ is the angular momentum operator for the $i^{th}$ rotor. The Hamiltonian contains two competing terms, and we expect that it may describe two plausible phases, a strong coupling phase, in which the kinetic energy (first) term dominates, and a weak coupling phase, in which the potential energy (second) term dominates. A natural control parameter for this analysis is the spin magnitude $S$, which dictates the magnitude of the potential energy. In the first case (small $S$), in which the potential energy term is small, we expect that the ground state can be viewed, to a first approximation, as a product of local spin singlets, i.e., $\mathbf{L}=0$ states, $$|G\rangle=|0\rangle_1|0\rangle_2\cdots|0\rangle_N,$$ where $|0\rangle_i$ represents the $\mathbf{L}=0$ state for the rotor on site $i$. The lowest-energy excitations are 