O(3) Nonlinear $\sigma$ Model and the Topological Distinction between Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions

It is shown that the O(3) nonlinear $\sigma$ model derived from the quantum Heisenberg antiferromagnet in two spatial dimensions, under the assumption that the Néel field $\hat{\Omega }(\mathbf{r},t)$ is well defined at all points in (2+1)D space-time, has no Hopf term, in contradiction to recent speculation. However, the amplitudes for tunneling between configurations with different spatial topologies reveal a difference between integer and half-integer, and between even- and odd-integer, spins.