sensitive to the boundary conditions (see the discussion below), and we assume closed (periodic) boundary conditions in the following. The case of open boundary conditions is discussed afterward. For periodic boundary conditions, it is easy to see that $S_T$ is zero unless the integrand has a non-trivial topological structure.

To evaluate $\partial_x\Omega$, we recall that $\Omega(\hat{n})$ measures the area on the surface of the sphere bounded by the trajectory $\hat{n}(t)$. Thus, the variation $\delta \Omega(\hat{n})$ due to a small variation in the trajectory $\delta\hat{n}$ is simply $$\delta\Omega(\hat{n})=\int dt \delta\hat{n}\cdot(\hat{n}\times\partial_t\hat{n}),$$ and $$\label{WZ4}
         S_{T}={\hbar S\over2^d}\int d^dx\int dt\left[\left({\partial\over\partial_x^1}\cdots{\partial\over\partial_x^d}\right)\hat{n}\right]\cdot(\hat{n}\times\partial_t\hat{n}).$$ The total effective action describing the quantum antiferromagnet is $S=S_{\sigma}+S_T$.

The topological term is nonzero in one dimension and is usually written in the form

$$\label{topo1}
         {S_{T}\over\hbar}={\theta\over8\pi}\sum_{\mu,\nu=0,1}\int d^2x\varepsilon_{\mu\nu}\hat{n}\cdot(\partial_{\mu}\hat{n}\times\partial_{\nu}\hat{n}),$$

where $x_0=t$, $x_1=x$, $\theta=2\pi S$ and $\varepsilon_{\mu\nu}$ is the rank-2 Levi-Civita antisymmetric tensor [@Haldane85; @Affleck85]. The Pontryagin index $$\label{topo2}
         Q={1\over8\pi}\sum_{\mu,\nu=0,1}\int d^2x\varepsilon_{\mu\nu}\hat{n}\cdot(\partial_{\mu}\hat{n}\times\partial_{\nu}\hat{n})= integer$$

measures how many times the 2\[=1(space)+1(time)\]-dimensional spin configuration $\hat{n}$ has wrapped around the unit sphere. In two dimensions, $$S_T\rightarrow{\hbar\theta\over2}\int dy {\partial Q(y)\over\partial y}=0,$$ where $Q(y)$ is the Pontryagin index that arises from summing over all spin configurations in the $y^{th}$ column of the two-dimensional lattice. The sum is zero for smooth spin configurations because $Q$ 