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arXiv:cond-mat/9404017 · Journal of the Physical Society of Japan

Excitation Spectrum of the Spin-1/2 Ferromagnetic-Antiferromagnetic Alternating Heisenberg Chain

Kazuo Hida

Published Jul 15, 1994• 28 citations
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Abstract

The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum $k$ by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small $k$, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.

DOI
10.1143/jpsj.63.2514

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The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum kkk by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small kkk, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.
The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum k by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small k, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.
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.
We numerically investigate elementary excitations of the Heisenberg alternating-spin chains with two kinds of spins, 1 and 1/2,1/2,1/2, antiferromagnetically coupled to each other. Employing a recently developed efficient Monte Carlo technique as well as an exact-diagonalization method, we verify the spin-wave argument that the model exhibits two distinct excitations from the ground state which are gapless and gapped. The gapless branch shows a quadratic dispersion in the small-momentum region, which is of the ferromagnetic type. With the intention of elucidating the physical mechanism of both excitations, we make a perturbation approach from the decoupled-dimer limit. The gapless branch is directly related to spin 1's, while the gapped branch originates from cooperation of the two kinds of spins.
We study the properties of the ground state and the low-lying excited states of an alternating spin-1 and spin-12\frac{1}{2}21​ antiferromagnetic chain with ferromagnetic next-nearest-neighbor coupling with a variety of methods, namely, spin-wave analysis, density-matrix renormalization-group (DMRG) method, and exact-diagonalization method. The ground state of this model with 2N2N2N spins is ferrimagnetic with the total spin SG=N∕2{S}_{G}=N∕2SG​=N∕2 for all parameter values. In the long chain limit, a gapless excitation is found with characteristics of goldstone mode of the ferromagnetic order with total spin $S={S}_{G}\ensuremath{-}1$. The lowest gapped excitation, however, exists in the total spin S=SG+1S={S}_{G}+1S=SG​+1 sector. Interestingly, we find that with the increase in ferromagnetic coupling, spin-wave gapped excitation becomes lower than the massless mode for the large wave numbers. Correspondingly, from DMRG analysis we find that the excitation in the total spin S=SG+1S={S}_{G}+1S=SG​+1 is lower than that in the total spin $S={S}_{G}\ensuremath{-}1$ when the size of the chain is short and the ferromagnetic interaction is large. Characteristics of these excitations are also reflected in the low temperature thermodynamic quantities, where the low-energy properties are strongly affected by the system size and the coupling strength.
By using the method of exact diagonalization, the properties of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain are investigated. The spin-spin correlation, the sublattice magnetizations, the spin excitation spectrum and the magnetization curve of the model are calculated and compared with the corresponding properties of the mixed spin chain. The results show that the behaviors of the alternating Heisenberg chain are equivalent to those of the mixed spin chain when the ratio of antiferromagnetic to ferromagnetic coupling strength is very small. So, the properties of the mixed spin chain can be analyzed by using the spin-1/2 ferromagnetic-antiferromagnetic-antiferromagnetic alternating Heisenberg chain.
According to Haldane’s conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In an earlier publication (Hung and Gong 2005 Phys. Rev. B 71 054413), the Haldane gap and other properties of the spin-1 system are explained using a spin-1/2 alternating bond chain model. The ferromagnetic–ferromagnetic–antiferromagnetic Heisenberg spin-1/2 chain is expected to behave similarly to the uniform spin-3/2 Heisenberg chain for strong ferromagnetic coupling. And the general nonuniform spin-1/2 chain may be used to explain the properties of the integer and half-integer high spin- S chains, consistently. In this paper, we provide a detailed numerical verification for the above problems.