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Journal of the Physical Society of Japan

S=1/2 Heisenberg Chain with Bond Alternation in theSz-SzCoupling

Kiyomi Okamoto

Published Oct 15, 1992• 4 citations
Publisher

Abstract

The effect of bond alternation in the S z - S z coupling ( z -alternation) on the dimerization in an S =1/2 Heisenberg chain is investigated both analytically and numerically. The spin Hamiltonian is transformed into the phase Hamiltonian, and the variational method is used. An analytical study shows that the z -alternation plays an important role and brings about the dimerization when there is no bond alternation in the S x - S x and S y - S y couplings. The results of numerical diagonalization for finite systems support the above analytical conclusion.

DOI
10.1143/jpsj.61.3488

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The effect of bond alternation in the S z - S z coupling ( z -alternation) on the dimerization in an S =1/2 Heisenberg chain is investigated both analytically and numerically. The spin Hamiltonian is transformed into the phase Hamiltonian, and the variational method is used. An analytical study shows that the z -alternation plays an important role and brings about the dimerization when there is no bond alternation in the S x - S x and S y - S y couplings. The results of numerical diagonalization for finite systems support the above analytical conclusion.
Isotropic spin-1 chains with bond alternation are studied. Analytic (both variational and perturbative) results are presented. A phase transition is found to separate the S=1 Haldane phase and the dimer phase. The critical point is numerically determined using the Binder parameter. The universality class is predicted to be given by the level-1 SU(2) Wess-Zumin-Witten model, and this prediction is supported by the present exact diagonalization study. The dimer phase is found to be connected to the "S=2" Haldane phase in a special limit. The change of the excitation spectra and the string order parameter with bond alternation is discussed.
Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and 0.0≲α≤1.00.0\lesssim\alpha \leq 1.00.0≲α≤1.0.
We study the phase diagram of S=3/2 and S=2 bond-alternating spin chains numerically. In previous papers, the phase diagram of S=1 XXZ spin chain with bond-alternation was shown to reflect the hidden Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry. But for the higher S Heisenberg spin chain, the successive dimerization transition occurs, and for anisotropic spin chains the phase structure will be more colorful than the S=1 case. Using recently developed methods, we show directly that the phase structure of the anisotropic spin chains relates to the Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry.
The magnetization plateaux in two dimensionally coupled S=1/2 dimerized zigzag Heisenberg chains are investigated by means of the bond operator mean field approximation. In the absence of the interchain coupling, this model is known to have a plateau at half of the saturation magnetization accompanied by the spontanuous translational symmetry breakdown. The parameter regime in which the plateau appears is reproduced well within the present approximation. In the presence of the interchain coupling, this plateau is shown to be suppressed. This result is also supported by the numerical diagonalization calculation.
We study the Heisenberg chain with frustration and alternation which is expressed by , both analytically and numerically. We focus on the case, where there is no marginal operator which brings about the logarithmic corrections in various quantities. By using the bosonization method, we calculate the energy gap, the change in the ground-state energy due to the alternation (so-called energy gain), the spin correlation and the string correlation (not only their exponents but also their amplitudes), and compare them with the results of the numerical diagonalization for finite systems. We point out the existence of the logarithmic correction in the energy gain despite the absence of the marginal operator. Taking into account this logarithmic correction, we can obtain a reasonable hyperscaling relation between the critical exponents of the energy gap and an energy gain from the numerical data.