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Physical review. B, Condensed matter

Anisotropic Heisenberg chain with composite spin

Jenő Sólyom, JUSSI TIMONEN

Published Jul 1, 1986• 31 citations
Publisher

Abstract

A family of one-dimensional magnetic Hamiltonians is introduced, where at each site there are $n$ spin-$S$ operators. It is shown that, for special couplings between spins and for $S=\frac{1}{2}$, 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-($S+S$) model onto the nonlinear $\ensuremath{\sigma}$ model and discuss the possibility of a finite gap in the spin-(\textonehalf{}+\textonehalf{}) model.

DOI
10.1103/physrevb.34.487

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ model and discuss the possibility of a finite gap in the spin-(\textonehalf{}+\textonehalf{}) model.
We consider in detail the ground-state properties of recently introduced generalized Heisenberg models which can have several spin operators at each site and which interpolate smoothly between Heisenberg chains of different spin lengths. We show that the mappings to field-theoretical models used to describe the critical properties of the Heisenberg model remain valid in the composite-spin model. In models which interpolate between the spin-(1/2 and the spin-1 behavior, these mappings predict an extended singlet phase around the isotropic antiferromagnetic point whenever the models move away from the spin-(1/2 point. Numerical calculations on finite chains seem to confirm the existence of this singlet phase. The phase boundaries are, however, found to be independent of the interpolation parameter, and thus in disagreement with the predictions of the continuum theory, indicating its limitation.
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.
Composite spin models are constructed such that, by varying two parameters, they interpolate between the spin-(1/2 antiferromagnetic Heisenberg chain and a number of spin-1 models. These include the usual Heisenberg model, the integrable spin-1 model, and the model with an exact valence-bond ground state. Finite-chain calculations are performed on the composite spin model to study its criticality, and to find if the integrable spin-1 model is a multicritical point with a finite gap generated away from it. We find indications for an extended gapless region.
One-dimensional spin chains are investigated by constructing a mapping of the anisotropic Heisenberg model in the limit Delta to 1 onto the non-linear Schrodinger model (a gas of bosons with delta function interactions). Three applications of this mapping are given. First it is shown that the Bethe ansatz solutions of the anisotropic spin-1/2 Heisenberg model go over to the continuous Bethe ansatz solutions of the non-linear Schrodinger model. Then bound-state energies for arbitrary spin are calculated and as a third application a nearly identical mapping of the anisotropic spin-1/2 Heisenberg model in the limit Delta to 0 onto a non-linear Schrodinger model for fermion fields is given. Finally the authors construct an anisotropic SU(N) model. The continuum limit of this model is shown to be the (N-1)-component non-linear Schrodinger model, which can be solved by means of the nested Bethe ansatz.
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.
Abstract This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.