Haldane conjecture ($H=J{\sum }_j{\mathbf{S}}_j\cdot {\mathbf{S}}_{j+1},J>0$)

For the uniform, nearest-neighbour, isotropic Heisenberg antiferromagnet chain, the Haldane conjecture distinguishes integer spin $S$, with a nonzero bulk excitation spectral gap, from half-odd-integer $S$, with gapless excitations in the thermodynamic limit. The continuum nonlinear sigma model contains a topological angle $\theta =2\pi S$: integer and half-odd-integer spins give $0$ and $\pi$ modulo $2\pi$. This is a different use of topology from the two-dimensional occupied-band Chern number. The $S=1$ chain belongs to the Haldane phase.^[ref]