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Journal of Physics Condensed Matter

A bipartite spin 1/2 Heisenberg model with a gap to excitations

M W Long

Published Mar 25, 1991• 0 citations
Publisher

Abstract

The author considers the nearest neighbour quantum mechanical Heisenberg model acting on two simple one-dimensional chains of spin 1/2 atoms. By showing that these models can be mapped onto a chain of spin 1 composites, the author deduces that the spectrum has a gap, provided that the spin 1 chain has a gap. This result is in contrast to that found for the chain of spin 1/2 atoms and suggests that a distinction between integer and half-integer spins is restricted to the linear chain. The author presents a simple interpretation for the spin correlations of the low energy excitations and give new numerical evidence for a gap in the spectrum of the spin 1 chain. The lowest lying spin 1/2 chain excitation is a domain wall, but the lowest laying spin 1 chain excitation has previously been suggested to have spin-wave properties, this hypothesis is tested numerically.

DOI
10.1088/0953-8984/3/12/005

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The author considers the nearest neighbour quantum mechanical Heisenberg model acting on two simple one-dimensional chains of spin 1/2 atoms. By showing that these models can be mapped onto a chain of spin 1 composites, the author deduces that the spectrum has a gap, provided that the spin 1 chain has a gap. This result is in contrast to that found for the chain of spin 1/2 atoms and suggests that a distinction between integer and half-integer spins is restricted to the linear chain. The author presents a simple interpretation for the spin correlations of the low energy excitations and give new numerical evidence for a gap in the spectrum of the spin 1 chain. The lowest lying spin 1/2 chain excitation is a domain wall, but the lowest laying spin 1 chain excitation has previously been suggested to have spin-wave properties, this hypothesis is tested numerically.
The double chain Heisenberg antiferromagnet with anisotropy is studied using the mapping onto the coupled anisotropic nonlinear σ model. This model consists of two anisotropic antiferromagnetic quantum Heisenberg chains with half-integer spin coupled with each other via the ferromagnetic interchain coupling J '. For small J ', our analysis suggests that J ' is an irrelevant parameter and consequently the two chains behave independently and no gap is generated. On the other hand, for large enough J ', the two chains behave as a single Heisenberg chain with integer spin which has a gap in the excitation spectrum. The expected behavior in the isotropic limit is also discussed.
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.
In this paper we study the finite-size scaling for low-energy excitations of S=1 and S=2 Heisenberg chains, using the density matrix renormalization-group technique. A crossover from 1/L behavior (with L as the chain length) for medium chain length to 1/L2{\mathrm{L}}^{2}L2 scaling for long chain length is found for excitations in the continuum band as the length of the open chain increases. Topological spin S=1/2 excitations are shown to give rise to the two lowest energy states for both open and periodic S=1 chains. In periodic chains these two excitations are ``confined'' next to each other, while for open chains they are two free-edge 1/2 spins. The finite-size scaling of the two lowest energy excitations of open S=2 chains is determined by coupling the two free-edge S=1 spins. The gap and correlation length for S=2 open Heisenberg chains are shown to be 0.082 (in units of the exchange J) and 47, respectively.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
A family of one-dimensional magnetic Hamiltonians is introduced, where at each site there are nnn spin-SSS operators. It is shown that, for special couplings between spins and for S=12S=\frac{1}{2}S=21​, the model contains the complete spectrum of the Heisenberg chain with spins \textonehalf{}, 1, frac32;, etc., and the ground state is that of the corresponding Heisenberg chain. By the varying of a single parameter the model allows continuous transitions between chains with different spin. We map the spin-(S+SS+SS+S) model onto the nonlinear $\ensuremath{\sigma}$ model and discuss the possibility of a finite gap in the spin-(\textonehalf{}+\textonehalf{}) model.