Ground State Properties of One Dimensional S=1/2 Heisenberg Model with   Dimerization and Quadrumerization
  The one dimensional S=1/2 Heisenberg model with dimerization and quadrumerization is studied by means of the numerical exact diagonalization of finite size systems. Using the phenomenological renormalization group and finite size scaling law, the ground state phase diagram is obtained in the isotropic case. It exhibits a variety of the ground states which contains the S=1 Haldane state, S=1 dimer state and S=1/2 dimer state as limiting cases. The gap exponent $\nu$ is also calculated which coincides with the value for the dimerization transition of the isotropic Heisenberg chain. In the XY limit, the phase diagram is obtained analytically and the comparison is made with the isotropic case. 
Wei Chen and Kazuo Hida