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Physical review. B, Condensed matter

Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain

Kazuo Hida

Published Feb 1, 1992• 274 citations
Publisher

Abstract

The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.

DOI
10.1103/physrevb.45.2207

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The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.
Ground state properties, dispersion relations and scaling behaviour of spin gap of a bond alternating spin-1 2 anisotropic Heisenberg chain have been studied where the exchange interactions on alternate bonds are ferromagnetic (FM) and antiferromagnetic (AFM) in two separate cases.The resulting models separately represent nearest neighbour (NN) AFM-AFM and AFM-FM bond alternating chains.Ground state energy has been estimated analytically by using both bond operator and Jordan-Wigner representations and numerically by using exact diagonalization.Dispersion relations, spin gap and several ground state orders have been obtained.Dimer order and string orders are found to coexist in the ground state.Spin gap is found to develop as soon as the non-uniformity in alternating bond strength is introduced in the AFM-AFM chain which further remains non-zero for the AFM-FM chain.This spin gap along with the string orders attribute to the Haldane phase.The Haldane phase is found to exist in most of the anisotropic region similar to the isotropic point.
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.
The phase-dependent generalized string order introduced by Oshikawa in the VBS (valence bond solid)-type states of the integer spin chain is extended to the spin-1/2 J - J ′ alternating Heisenberg chain with antiferromagnetic J . The numerical diagonalization and the bosonization analysis show that the generalized string order remains finite in the ground state of the present model for -∞< J ′ < J , taking the maximum at J ′ =0. The characteristic feature of the phase dependence of the generalized string order also coincides with that of the spin-1 VBS state. This reconfirms the notion that the Haldane state of the spin-1 chain and the dimerized state of the alternating chain belong to a single phase characterized by the local singlet-type correlation.
We calculate the excitation spectrum and spectral weights of the alternating antiferromagnetic-ferromagnetic spin-half Heisenberg chain with exchange couplings JJJ and −∣λ∣J-|λ|J−∣λ∣J as a power series in λλλ. For small ∣λ∣|λ|∣λ∣, the gapped one-particle spectrum has a maximum at k=0k=0k=0 and there is a rich structure of bound (and anti-bound) states below (and above) the 2-particle continuum. As ∣λ∣|λ|∣λ∣ is increased past unity the spectrum crosses over to the Haldane regime, where the peak shifts away from k=0k=0k=0, the one particle states merge with the bottom of the continuum near k=0k=0k=0, and the spectral weights associated with the one-particle states become very small. Extrapolation of the spectrum to large ∣λ∣|λ|∣λ∣ confirms that the ground state energy and excitation gap map onto those of the spin-one chain.
The ground state of the ferromagnetic-antiferromagnetic alternating spin-1/2 Heisenberg chain is studied with Ising-type anisotropy on the ferromagnetic bond. This model tends to the spin-1 antiferromagnetic Heisenberg chain with on-site anisotropy in the limit of strong ferromagnetic bond. It is found that the Haldane-gap phase, the large-D phase, and the two types of N\'eel phases appear as the ground state of this model. The duality relations between these four phases are proved for several limiting cases. The string order parameter characterizing the large-D phase is introduced based on these duality relations. The ground-state phase diagram is determined by the numerical diagonalization of the finite-size system.
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.