integer spin chains can always be viewed as a product state of local spin singlets, irrespective of the spin magnitude $S$. The lowest-energy excitations are gapped spin triplet ($\mathbf{L}=1$) excitations. This is the Haldane conjecture for integer spin chains.

### Half-odd-integer spin chains

The RG analysis cannot be straightforwardly applied to half-odd-integer spin chains because of the appearance of the topological term $S_T$. To understand why, let us again take the RG to the strong coupling limit and examine what happens in this case.

To zeroth order, the Hamiltonian of the system consists only of the kinetic energy term. However, the rotors are moving under the influence of effective monopole potentials originating from $S_T$. In particular, all half-odd-integer spin chains have the same $S_T$ with an effective magnetic monopole strength of $1/2$, corresponding to that of a spin-$1/2$ chain. In this case, the ground state of a single rotor has an angular momentum of $\mathbf{L}=1/2$ and is two-fold degenerate (see the discussion after Eq. ). The total degeneracy of the ground state is $2^N$, where $N$=number of lattice sites. This enormous degeneracy implies that the coupling between rotors cannot be neglected when we consider the rotor Hamiltonian given in Eq. 