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Physical review. B, Condensed matter

Hidden Z 2 × Z 2 symmetry breaking and the Haldane phase in the S =1/2 quantum spin chain with bond alternation

Mahito Kohmoto, Hal Tasaki

Published Aug 1, 1992• 85 citations
Publisher

Abstract

We study the S=1/2 spin chain with the Hamiltonian H=${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}\mathrm{\ensuremath{-}}1}$ {${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{x}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{x}}$+${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{y}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{y}}$ +\ensuremath{\lambda}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{z}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}$} +\ensuremath{\beta}${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}$ (1-${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}$), where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$=(${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{x}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{y}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{z}}$) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry of the system. It has been argued that a similar hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ 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 ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry in a range of the parameter space including the line \ensuremath{\beta}=0, \ensuremath{\lambda}>-1. We argue that the range with such properties indeed extends to the limit \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty} 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 \ensuremath{\beta}=0.

DOI
10.1103/physrevb.46.3486

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We study the S=1/2 spin chain with the Hamiltonian H=Jj=1L\ensuremath−1{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}\mathrm{\ensuremath{-}}1}Jj=1L\ensuremath−1​ {{\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{x}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{x}}+{\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{y}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{y}} +\ensuremath{\lambda}{\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{z}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}} +\ensuremath{\beta}Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ (1-\ensuremathσ2j\ensuremath−1{\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}\ensuremathσ2j\ensuremath−1​\ensuremath{\cdot}\ensuremathσ2j{\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}\ensuremathσ2j​), where \ensuremathσi{\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}\ensuremathσi​=(\ensuremathσix{\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{x}}\ensuremathσix​,\ensuremathσiy{\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{y}}\ensuremathσiy​,\ensuremathσiz{\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{z}}\ensuremathσiz​) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry of the system. It has been argued that a similar hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ 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 Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry in a range of the parameter space including the line \ensuremath{\beta}=0, \ensuremath{\lambda}>-1. We argue that the range with such properties indeed extends to the limit \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty} 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 \ensuremath{\beta}=0.
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We show that the Haldane phase of the S=1 antiferromagnetic chain is closely related to the breaking of a hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry. When the chain is in the Haldane phase, this Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry is fully broken, but when the chain is in a massive phase other than the Haldane phase, e.g., the Ising phase or the dimerized phase, this symmetry is broken only partially or not at all. The hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry is revealed by introducing a nonlocal unitary transformation of the chain. This unitary transformation also leads to a simple variational calculation which qualitatively reproduces the phase diagram of the S=1 chain.
The ground-state properties of a spin S=1/2S=1/2S=1/2 tetrameric Heisenberg antiferromagnetic chain with alternating couplings AF1-AF2-AF1-F{\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}AF1​-AF2​-AF1​-F (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization mmm are found at m=0m=0m=0 and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.