Phase diagram of theS=1/2 quantum spin chain with bond alternation

We study the ground-state properties of the bond-alternating S=1/2 quantum spin chain whose Hamiltonian is H= ${\mathit{tsum}}_{\mathit{j}}$ (${\mathit{S}}_{2\mathit{j}}^{\mathit{x}}$ ${\mathit{S}}_{2\mathit{j}+1\mathrm{}}^{\mathit{x}}$ +${\mathit{S}}_{2\mathit{j}}^{\mathit{y}}$ ${\mathit{S}}_{2\mathit{j}+1\mathrm{}}^{\mathit{y}}$ +λ${\mathit{S}}_{2\mathit{j}}^{\mathit{z}}$ ${\mathit{S}}_{2\mathit{j}+1\mathrm{}}^{\mathit{z}}$) +βt${sum}_{\mathit{j}}$ ${\mathbf{S}}_{2\mathit{j}\mathrm{-}1}$⋅${\mathbf{S}}_{2\mathit{j}}$. When β=0, the ground state is a collection of local singlets with a finite excitation gap. In the limit of strong ferromagnetic coupling β→-∞, this is equivalent to the S=1 XXZ Hamiltonian. It has several ground-state phases in the λ-β plane including the Haldane phase which has a finite excitation gap above the ground. They are characterized by a full breakdown, partial breakdowns, and a nonbreakdown of the hidden discrete ${\mathit{Z}}_2$×${\mathit{Z}}_2$ symmetry. The ground-state phase diagram is obtained by series expansions.