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Progress of Theoretical Physics

Excitation Spectra of the Singlet Bound States in the Antiferromagnetic Spin 1/2 Chain

K. Hashimoto

Published Oct 1, 1982• 13 citations
Publisher

Abstract

Excited singlet spectra in the antiferromagnetic Heisenberg chain of spin 1/2 are obtained by solving Bethe's equations containing a pair of complex wave numbers. The lowest spectrum is found to become degenerate with that of the triplet states derived by des Cloizeaux and Pearson in the limit of infinite number of spins.
DOI
10.1143/ptp.68.1122

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The restrictions on the excitation spectra of infinite linear magnetic chains due to the invariances of the Hamiltonian under time reversal and spin rotation have been studied by making use of the equivalence of a linear magnetic chain and a one-dimensional assembly of interacting Fermi particles. In particular, the spin-wave spectrum of an infinite antiferromagnetic chain is found to have a double periodicity of $\ensuremath{\pi}$ in the wave vector, in agreement with the numerical result of des Cloizeaux and Pearson.
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 excitation spectrum of the two-chain Heisenberg ladder with antiferromagnetic interactions is studied. Our approach is based on the description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet ( S=0S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0S=0) and triplet ( S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1) excitations are found as quasiparticle bound states. Thus it is demonstrated that the spectrum consists of one elementary triplet, one composite triplet, and one composite singlet.
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.