= Haldane conjecture
{c}
{title2=$H=J\sum_j\mathbf S_j\cdot\mathbf S_{j+1},\quad J>0$}
{wiki}
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>.<Nobel Lecture: Topological quantum matter by F. Duncan M. Haldane>[Haldane, individual Nobel lecture delivered 8 December 2016, published 2017, pp. 040502-8–9: integer versus half-odd-integer chains and topological angle]{ref}
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