HiddenZ2×Z2symmetry breaking and the Haldane phase in theS=1/2 quantum spin chain with bond alternation

Abstract

We study the S=1/2 spin chain with the Hamiltonian H=${\mathcal{J}}_{\mathit{j}=1\mathrm{}}^{\mathit{L}\mathrm{-}1}$ {${\mathrm{\sigma }}_{2\mathit{j}}^{\mathit{x}}$ ${\mathrm{\sigma }}_{2\mathit{j}+1\mathrm{}}^{\mathit{x}}$+${\mathrm{\sigma }}_{2\mathit{j}}^{\mathit{y}}$ ${\mathrm{\sigma }}_{2\mathit{j}+1\mathrm{}}^{\mathit{y}}$ +λ${\mathrm{\sigma }}_{2\mathit{j}}^{\mathit{z}}$ ${\mathrm{\sigma }}_{2\mathit{j}+1\mathrm{}}$} +β${\mathcal{J}}_{\mathit{j}=1\mathrm{}}^{\mathit{L}}$ (1-${\mathrm{\sigma }}_{2\mathit{j}\mathrm{-}1}$⋅${\mathrm{\sigma }}_{2\mathit{j}}$), where ${\mathrm{\sigma }}_{\mathit{i}}$=(${\mathrm{\sigma }}_{\mathit{i}}^{\mathit{x}}$,${\mathrm{\sigma }}_{\mathit{i}}^{\mathit{y}}$,${\mathrm{\sigma }}_{\mathit{i}}^{\mathit{z}}$) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden ${\mathit{Z}}_2$×${\mathit{Z}}_2$ symmetry of the system. It has been argued that a similar hidden ${\mathit{Z}}_2$×${\mathit{Z}}_2$ symmetry of the S=1 chain is fully broken when and only when the system exhibits the Haldane gap. We prove that the present system exhibits both an excitation gap and a full breaking of the hidden ${\mathit{Z}}_2$×${\mathit{Z}}_2$ symmetry in a range of the parameter space including the line β=0, λ>-1. We argue that the range with such properties indeed extends to the limit β→∞ in which the present model reduces to the S=1 spin chain. This observation provides support of Hida’s conclusion that the Haldane gap in the S=1 chain is continuously connected to the gap in the decoupled S=1/2 system with β=0.