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Journal of the Physical Society of Japan

Nonlocal Unitary Transformations onS=1 Antiferromagnetic Spin Chains

Satoshi Takada, Kenn Kubo

Published Dec 15, 1991• 20 citations
Publisher

Abstract

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 .

DOI
10.1143/jpsj.60.4026

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arXiv:cond-mat/9802163•Physical review. B, Condensed matter•Aug 1, 1998•39 citations
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 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 author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.
We study the structure and properties of the ground state of an antiferromagnetic S=1 spin chain with single-ion anisotropy energy d and exchange anisotropy energy \ensuremath{\epsilon}. We apply the nonlocal unitary transformation of Kennedy and Tasaki to take into account the essential correlations and restrict our study to the subspace of spin-zero defects (solitons). We investigate the quantum phase transitions in the (\ensuremath{\epsilon},d) plane in a cluster approximation which takes into account correlations between spins at 3 neighboring sites in the unitarily transformed representation. We obtain quantitative results for the phase diagram, for the values of the hidden order parameter, and for amplitudes that describe the transition from the N\'eel to the Haldane phase in terms of the dissociation of soliton pairs.
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 S=12S=\frac{1}{2}S=21​ chain-boundary excitations occurring in the Haldane phase of S=1S=1S=1 antiferromagnetic spin chains are investigated. The bilinear-biquadratic Hamiltonian is used to study these excitations as a function of the strength of the biquadratic term, \ensuremath{\beta}, between $\ensuremath{-}1<~\ensuremath{\beta}<~1.$ At the Affleck-Kennedy-Lieb-Tasaki point, $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3},$ we show explicitly that these excitations are localized at the boundaries of the chain on a length scale equal to the correlation length $\ensuremath{\xi}=1/\mathrm{ln}3$, and that the on-site magnetization for the first site is 〈S1z〉=23.〈{S}_{1}^{z}〉=\frac{2}{3}.〈S1z​〉=32​. Applying the density matrix renormalization group we show that the chain-boundary excitations remain localized at the boundaries for $\ensuremath{-}1<\ensuremath{\beta}<1.$ As the two critical points \ensuremath{\beta}=\ifmmode\pm\else\textpm\fi{}1 are approached the size of the S=12S=\frac{1}{2}S=21​ objects diverges and their amplitude appears to vanish.