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Journal of the Physical Society of Japan

String Order Parameters in the Haldane Gap Phase of the Spin-1/2 Alternating Heisenberg Chain –Numerical Diagonalization and Variational Study–

Kazuo Hida, Satoshi Takada

Published Jun 1, 1992• 24 citations
Publisher

Abstract

Two possible extensions of the string order parameter originally defined for the spin-1 Heisenberg chain to the spin-1/2 alternating Heisenberg chain are examined by comparing the numerical diagonalization results of finite size systems and the variational results. It is suggested that the naive extension in which the spin-1 operator is simply replaced by the sum of two spin-1/2 operators can be regarded as the valence bond solid (VBS) order parameter while another extension proposed by Hida corresponds to the measure of the strength of the localized singlet taking the full value for the completely localized singlet state.

DOI
10.1143/jpsj.61.1879

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Physical review. B, Condensed matter•Feb 1, 1992•274 citations
Two possible extensions of the string order parameter originally defined for the spin-1 Heisenberg chain to the spin-1/2 alternating Heisenberg chain are examined by comparing the numerical diagonalization results of finite size systems and the variational results. It is suggested that the naive extension in which the spin-1 operator is simply replaced by the sum of two spin-1/2 operators can be regarded as the valence bond solid (VBS) order parameter while another extension proposed by Hida corresponds to the measure of the strength of the localized singlet taking the full value for the completely localized singlet state.
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 propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the number of valence bonds (broken valence bonds) at the boundary for periodic (open) systems. This allows us to determine the phase transition point in between different VBS states. We demonstrate this theory in the successive dimerization transitions of the bond-alternating Heisenberg chains, using the quantum Monte Carlo method.
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.
We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin systems, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the configuration of the valence bonds. This allows us to determine the phase transition point in between different VBS states accurately. We demonstrate this theory in the bond-alternating Heisenberg chain and in the frustrated spin ladder.
Using the valence-bond-solid (VBS) approach and the Schwinger-boson mean-field approximation, we study the dependence of the Haldane gap of a spin-1 linear-chain Heisenberg antiferromagnet on impurity doping with different spins. The impurity spins affect the singlet pairing order parameter \ensuremath{\Delta} and the constraint factor \ensuremath{\lambda}. As a result, the Haldane gap is reduced by a factor \ensuremath{\sim}ni2/3{\mathit{n}}_{\mathit{i}}^{2/3}ni2/3​, with ni{\mathit{n}}_{\mathit{i}}ni​ as the impurity concentration, and eventually collapses at ni{\mathit{n}}_{\mathit{i}}ni​\ensuremath{\sim}1/\ensuremath{\xi} with \ensuremath{\xi} as the VBS correlation length. This theoretical prediction can be verified by neutron-scattering experiments.
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.