Spin-Gap States of a Periodic Mixed Spin Chain
  We examine a chain of periodic arrays of 4 quantum spins with magnitudes of 1/2, 1, 3/2 and 1. There are four kinds of nearest-neighbour exchange parameters among them. We choose two independent parameters for concreteness: one represents the ratio of typical exchange parameters, and the other represents a distortion. We determine the phase diagram of the ground state in the parameter space. The phase boundaries appear as gapless lines which separate gapful disordered phases. They are determined by the gapless equation which was previously derived by mapping a general periodic spin chain to the nonlinear $\sigma$ model. 
Ken'ichi Takano