  An infinite number of spin chains are solved and it is derived that the
ground-state phase transitions belong to the universality classes with central
charge c=m/2, where m is an integer. The models are diagonalized by
automatically obtained transformations, many of which are different from the
Jordan-Wigner transformation. The free energies, correlation functions, string
order parameters, exponents, central charges, and the phase diagram are
obtained. 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 is
revealed, and other series of solvable models can be obtained through this
formula.
