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Physical Review

Pseudospin Approach to Linear Antiferromagnetic Chains

Z. G. Soos

Published Sep 9, 1966• 13 citations
Publisher

Abstract

The generalized linear antiferromagnetic chain with two spins per unit cell $J>0$, and $a, b, d, e>~0$, ${\mathcal{H}}_{G}=\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{j=1}J{a{{S}_{2j}}^{z}{{S}_{2j+1}}^{z}+b({{S}_{2j}}^{x}{{S}_{2j+1}}^{x}+{{S}_{2j}}^{y}{{S}_{2j+1}}^{y})+d{{S}_{2j}}^{z}{{S}_{2j\ensuremath{-}1}}^{z}+e({{S}_{2j}}^{x}{{S}_{2j\ensuremath{-}1}}^{x}+{{S}_{2j}}^{y}{{S}_{2j\ensuremath{-}1}}^{y})},$ is solved in the pseudospin approximation. The pseudospin method is compared with the exactly soluble special case of the Heisenberg-Ising antiferromagnet ($a=b=1+\ensuremath{\delta}$, $d=1\ensuremath{-}\ensuremath{\delta}$, $e=0$) and good agreement is obtained even for a sensitive property like the long-range order. It is suggested that the pseudospin approach provides a single, systematic approximation scheme, not only for describing the magnetic properties of a class of solid free radicals, but also as a check for widely used many-body techniques.

DOI
10.1103/physrev.149.330

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The exact solution is given for a linear chain of NNN atoms of spin \textonehalf{} coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{i=1}[{{S}_{i}}^{z}{{S}_{i+1}}^{z}+(1\ensuremath{-}\ensuremath{\alpha})({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\ensuremath{\alpha}$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\ensuremath{\Sigma}{i}^{}{{S}_{i}}^{z}{{S}_{i+1}}^{z}$, is calculated exactly from the variation of the ground-state energy with $\ensuremath{\alpha}$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\ensuremath{\partial}}^{2}E}{\ensuremath{\partial}{\ensuremath{\alpha}}^{2}}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.
The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.