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Journal of the Physical Society of Japan

Nonlocal Unitary Transformation and Haldane State inS=1/2 Antiferromagnetic Ladder Model

Satoshi Takada, Hiroshi K. WATANABE

Published Jan 15, 1992• 33 citations
Publisher

Abstract

The Haldane state is investigated in the S =1/2 antiferromagnetic (AF) ladder model for open chains with the help of a nonlocal unitary transformation. It is shown that the transformation works well, as in the case of S =1 AF models, so that it transforms the model into a ferromagnetic model with Z 2 × Z 2 symmetry, and four degenerate Haldane states are obtained by a variational calculation in a molecular field approximation. The S =1/2 edge states are examined in these Haldane states.

DOI
10.1143/jpsj.61.39

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The Haldane state is investigated in the S =1/2 antiferromagnetic (AF) ladder model for open chains with the help of a nonlocal unitary transformation. It is shown that the transformation works well, as in the case of S =1 AF models, so that it transforms the model into a ferromagnetic model with Z 2 × Z 2 symmetry, and four degenerate Haldane states are obtained by a variational calculation in a molecular field approximation. The S =1/2 edge states are examined in these Haldane states.
We investigate the ground-state properties of an S =1/2 Heisenberg chain with bond alternation, which reduces to the S =1 antiferromagnetic (AF) chain in a strong coupling limit, using a nonlocal unitary transformation. It is shown that the transformation works well, as in the case of S =1 AF models, and hence it transforms the model into a ferromagnetic model with a Z 2 × Z 2 symmetry. The phase diagram is calculated in the transformed system with string order parameters by a molecular field approximation, and the result is shown to agree essentially with the numerical result obtained by Hida. We also argue the difference between Hida's string order parameter and one corresponding to the S =1 string order parameter of den Nijs and Rommelse.
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 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 .
The ground-state phase diagram of the S =1/2 X X Z ladder model is investigated by using the density-matrix renormalization-group method. This model is composed of two S =1/2 X X Z chains with ferromagnetic rung couplings (λ) and intra-chain anisotropy (Δ), and continuously changes from the two independent S =1/2 X X Z chains at λ=0 to the S =1 X X Z chain in the limit of λ→∞ . It is shown that there exist edge states at Δ=1 and λ=1 in relation to the four-fold degeneracy of the ground state and that an excitation gap opens nearly at λ=0 for 0.8≤Δ≤1. In addition, it is clarified that the transition between the Haldane phase and the antiferromagnetic phase occurs at λ=1.15 ±0.01 for Δ=1. A possible phase diagram in the λ-Δ plane is proposed, which is consistent with the scaling analysis based on the bosonization method.
We have obtained an analytic expression for the k-dependence of the excitation energy gap for an arbitrary double S = 1/2 spin chain by using the non-local unitary transformation and the variational method. We check that it explains the gap behaviour of various systems, which include the Haldane system and the dimer system in both extreme limits, and also the ladder model and the Majumdar - Ghosh model. The string order parameter, the dimer order parameter, and the local spin value are also calculated in the ground state. The ground-state energy exhibits a great stabilization by an antiferromagnetic bond dimerization, which might be realized in various new compounds. We also mention the relation of the convergence to the Haldane state with the spin-exchange symmetry of the model. The excited state has one domain wall of a local triplet type except in the vicinity of the Majumdar - Ghosh point, where a local triplet is decomposed into two S = 1/2 free spins moving among the singlet dimers.
Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.