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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]

The $S=1$ Heisenberg antiferromagnet chain realizes the Haldane phase: a gapped symmetry-protected topological phase with effective spin one-half degrees of freedom at open ends. Time-reversal symmetry is one possible protecting symmetry. Integer spin alone does not imply the same protected class: odd- and even-integer chains differ. The broader gap distinction is the Haldane conjecture.^[ref]