is an integer and thus cannot “change smoothly" [@Haldane88-NLSM; @Wen88; @Fradkin88; @Dombre88]. For the same reason, $S_T$ vanishes for any number of dimensions greater than one. However, one should be cautioned that this conclusion is valid only when we restrict ourselves to smooth spin configurations $\hat{n}(\mathbf{x},t)$ when computing $S_T$. The Berry’s phase may have a nonzero contribution if we also allow singular spin configurations in the theory. This is the case in $2+1$D, where monopole-like spin configurations are allowed in 3D space [@Haldane88-NLSM; @ReadSachdev90].

Quantum spin chains and the Haldane conjecture
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We now study the predictions of the effective action for quantum spin chains. In one dimension, the quantum spin chains are described by the path integral $$\int D[\hat{n}(x,t)]e^{{i\over\hbar}(S_{\sigma}(\hat{n})+S_T(\hat{n}))}.$$

We first consider the topological term. We note that $S_T=2\hbar\pi SQ$ and $e^{{i\over\hbar}S_T}=(-1)^{2SQ}$ ($Q=$ integer). In particular, $e^{{i\over\hbar}S_T}\equiv 1$ for integer spin chains, and the Berry’s phase has no effect on the effective action. However, $e^{{i\over\hbar}S_T}=\pm1$ for half-odd-integer spin chains, depending on whether $Q$ is even or odd. There is no further distinction between spin chains with different spin values $S$ in $S_T$. This result leads to the first part of the Haldane conjecture, namely, that fundamental differences exist between 