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Journal of the Physical Society of Japan

Numerical Study of the String Order inS=1 Antiferromagnetic Heisenberg Chain

Tôru Sakai, Minoru Takahashi

Published Feb 15, 1994• 5 citations
Publisher

Abstract

We study the spin-1 antiferromagnetic Heisenberg chain under the three boundary conditions, which are the free, periodic and antiperiodic boundary conditions, by numerical diagonalization up to 16 sites. We checked that the lowest excitation caused by the domain wall, which is imposed by the boundary condition and breaks the long-range string order, yields the same energy gap under all the boundary conditions in the infinite length limit. It is also found that the ground state and the first excited state have the same density (about 3%) of domain walls due to the quantum fluctuation which reduces the string order parameter about 10% smaller than that of the VBS model in the ground state. The analysis of the mean distance between two domain walls suggests that domain walls due to the quantum fluctuation form a bound state in pairs and does not break the long-range string order.

DOI
10.1143/jpsj.63.755

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We study the spin-1 antiferromagnetic Heisenberg chain under the three boundary conditions, which are the free, periodic and antiperiodic boundary conditions, by numerical diagonalization up to 16 sites. We checked that the lowest excitation caused by the domain wall, which is imposed by the boundary condition and breaks the long-range string order, yields the same energy gap under all the boundary conditions in the infinite length limit. It is also found that the ground state and the first excited state have the same density (about 3%) of domain walls due to the quantum fluctuation which reduces the string order parameter about 10% smaller than that of the VBS model in the ground state. The analysis of the mean distance between two domain walls suggests that domain walls due to the quantum fluctuation form a bound state in pairs and does not break the long-range string order.
The antiferromagnetic Heisenberg spin systems on the three-leg ladder are investigated. Periodic boundary condition is imposed in the rung direction. The system has an excitation gap for all antiferromagnetic inter-chain coupling (J⊥>0J_{\perp}>0J⊥​>0). The estimated gap for the strong coupling limit (J⊥/J1→∞J_{\perp}/J_1 \to \inftyJ⊥​/J1​→∞) is 0.28J1J_1J1​. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (J2J_2J2​), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (J2=0J_2=0J2​=0) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin 1/2 is studied by the density matrix renormalization group method. The ground state energy, the energy gap distribution and the string order parameter are calculated. Using the finite size scaling analysis, the dimerization dependence of the these quantities are obtained. The ground state energy gain due to dimerization behaves as u a with a > 2 where u denotes the degree of dimerization, suggesting the absence of spin-Peierls instability. It is explicitly shown that the string long range order survives even in the presence of randomness. The string order behaves as u 2β with β∼0.37 in agreement with the recent prediction of real space renormalization group theory (β=(3−5)/2≃0.382\beta =(3-\sqrt{5})/2 \simeq 0.382β=(3−5​)/2≃0.382). The physical picture of this behavior in this model is also discussed.
We investigate ground-state and low-lying excited states of the S = 1 antiferromagnetic Heisenberg chain with exchange anisotropy, single-ion anisotropy and biquadratic exchange and with free boundary conditions. We present an approximate mapping of the S = 1 model to an effective S = 1/2 chain in an external magnetic field, which can be solved exactly in several limiting cases. We obtain results for the phase diagram, correlation functions and the excitation spectrum. The transition to the Haldane phase is related to the condensation of the lowest soliton modes and the hidden order parameter of the Haldane phase is identified as transverse long-range order in the effective S = 1/2 chain.
The ground state properties of the Heisenberg antiferromagnetic spin- 1 / 2 chain with interactions varying as (distance) -p are investigated from finite cell calculations. The gap between the ground state and the first excited state as well as the end-to-end spin correlation function have been calculated for finite open chains up to 14 spins and 12 spins respectively. Aided by first-order perturbation calculations near p=0, the authors can extrapolate the gap in the thermodynamic limit, and they find that the gap vanishes for every p not=0. From the analysis of the spin correlation function they find that long-range order exists for p<1 and does not exist for p>2. The existence of long-range order in the range 1<p<2 remains an open question.
Low-lying excited states of the spin-1 antiferromagnetic Heisenberg chain with an impurity bond are investigated in terms of domain-wall excitations by means of analytical methods as well as a method of numerical diagonalization.It is shown that 1) the impurity bond brings about a massive triplet mode in the Haldane gap, 2) the triplet state comprises three of the four ground states of an open chain, 3) the excited states with the domain wall trapped at the impurity bond, including the massive triplet mode, exhibit a Néel-type configuration of the magnetic moment around the impurity bond.On the basis of the present results, we propose a new energy versus magnetic-field diagram.