end spins coupled to a spin-$1/2$ chain has been analyzed using the bosonization technique, through which it was found that after the screening induced by the spin-$1/2$ chain (essentially a Kondo effect), a free spin with a magnitude of ${1\over2}(S-{1\over2})$ is left at each end of the spin chain [@Eggert92]. Note that the existence of end states in half-odd-integer spin chains is rather non-trivial because the bulk spin excitations are [*gapless*]{}. As a result, the end spins at the two ends of a half-odd-integer spin chain are coupled by a term $J_{eff}\sim JS^2/(L\ln L)$, where $L$ is the length of the spin chain. The excitation energy of the end state is logarithmically lower than the energy of the bulk spin excitations, which have an energy of $\sim J/L$ [@Ng94]. These predictions for open chains and end states based on the NL$\sigma$M plus topological $\theta$ term analysis have been verified numerically by means of density matrix renormalization group (DMRG) calculations [@Qin95].

Higher dimensions and frustrated quantum antiferromagnets
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The NL$\sigma$M approach to quantum antiferromagnets has been extended to higher dimensions and to frustrated quantum antiferromagnets. For simple antiferromagnets, $S_T$ vanishes in dimensions of $d>1$, and we need only consider the NL$\sigma$M, 