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An infinite number of spin chains are solved and it is derived that theground-state phase transitions belong to the universality classes with centralcharge c=m/2, where m is an integer. The models are diagonalized byautomatically obtained transformations, many of which are different from theJordan-Wigner transformation. The free energies, correlation functions, stringorder parameters, exponents, central charges, and the phase diagram areobtained. Most of the examples consist of the stabilizers of the cluster state.A unified structure of the one-dimensional XY and cluster-type spin chains isrevealed, and other series of solvable models can be obtained through thisformula.

Infinite number of solvable generalizations of XY-chain, with cluster state, and with central charge c=m/2