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Physical review. B, Condensed matter

Hidden Z2 × Z2 symmetry breaking in Haldane-gap antiferromagnets

Tom Kennedy, Hal Tasaki

Published Jan 1, 1992• 476 citations
Publisher

Abstract

We show that the Haldane phase of the S=1 antiferromagnetic chain is closely related to the breaking of a hidden ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry. When the chain is in the Haldane phase, this ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ 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 ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ 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.

DOI
10.1103/physrevb.45.304

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Phase Diagram of S = 1 Bond-Alternating XXZ Chains Reflecting the Hidden Z2 × Z2 Symmetry

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Phase Diagram of S=1 Bond-Alternating XXZ Chains Reflecting the Hidden Z_2\times Z_2 Symmetry

Atsuhiro Kitazawa, K. Nomura

The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Néel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Néel and dimer-Néel). We determine the phase boundary and universality class using the recent developed techniques by the authors.

arXiv:cond-mat/9705179•May 19, 1997•0 citations
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.
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}$} +\ensuremath{\beta}Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ (1-${\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 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.
The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Neel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Neel and dimer-Neel). We determine the phase boundary and universality class using the recent developed techniques by the authors.
We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry is broken for 0.3<∣δ∣<0.50.3 < |δ| < 0.50.3<∣δ∣<0.5 while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.
We propose nonlocal unitary transformations on S =1 antiferromagnetic chains in a simple operator form. This is a generalization of the transformation recently introduced by Kennedy and Tasaki to clarify the hidden symmetry in the Haldane phase. The three operators ( T i α ≡ i S i α e i πS i β ) a p p e a r i n g i n t h e s e t r a n s f o r m a t i o n s a r e e x a m i n e d a n d s o m e s i m p l e a l g e b r a i c r e l a t i o n s a r e s h o w n t o h o l d a m o n g T_i^α a n d S_i^α. I t i s a l s o s h o w n t h a t t h r e e d i f f e r e n t f o r m s o f t r a n s f o r m e d H a m i l t o n i a n s a r e t r a n s f o r m e d i n t o e a c h o t h e r b y local u n i t a r y t r a n s f o r m a t i o n s .
We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1)\mathrm{SO}(2k+1)SO(2k+1) and SU(k)\mathrm{SU}(k)SU(k).
The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Néel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Néel and dimer-Néel). We determine the phase boundary and universality class using the recent developed techniques by the authors.