Spin chains with a periodic array of impurities

We investigate a spin-chain model composed of a periodic array of two kinds of spins ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$, which allows us to study the spin chains with impurities, as well as the alternating spin chains, in a unified fashion. By using the Lieb-Shultz-Mattis theorem, we first study the model rigorously, and then by mapping it to the nonlinear σ model, we extensively investigate low-energy properties with particular emphasis on the competition between the massive and massless phases.