Nonlinear sigma models, antiperiodic boundary conditions, spin chains, and 't Hooft anomalies

We consider two sets of related models: initially, these are $SU(2)$ antiferromagnetic spin chains with $N$ sites of spin $S$, and the $O(3)$ nonlinear sigma model in two dimensions with topological coefficient $\Theta$ a multiple of $\pi$ (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with $\Theta=2\pi S$. We study these models with $N$ odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the $\mathbb{Z}_2$ inversion symmetry $\vec{n}\to-\vec{n}$, and amounts to a flux of a $\mathbb{Z}_2$ gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the $\mathbb{Z}_2$ inversion symmetry. We show directly that, if and only if $(-1)^{\Theta/\pi}=-1$, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space $\mathbb{R}\mathbb{P}^2\cong \mathbb{S}^2/\mathbb{Z}_2$, and hence this model does not exist for $\Theta=\pi$ (mod $2\pi$). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on $N$ modulo $4$ and $2S$ modulo $2$. For a large class of spin chains and associated sigma models we find similar results, but now $(-1)^{\Theta/\pi}$ is replaced by the value $\pm 1$ of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.