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Physical Review

Linear Antiferromagnetic Chain with Anisotropic Coupling

R. Orbach

Published Oct 15, 1958• 532 citations
Publisher

Abstract

The exact solution is given for a linear chain of $N$ 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.

DOI
10.1103/physrev.112.309

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Linear chains (and rings) of S=12S=\frac{1}{2}S=21​ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for N=2to11N=2 \mathrm{to} 11N=2to11, $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region kT∣J∣> 0.5\frac{\mathrm{kT}}{|J|}>~0.5∣J∣kT​> 0.5 which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at T=0T=0T=0 are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.
The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of N=4,6,8,and10N=4,6,8, \mathrm{and} 10N=4,6,8,and10 spins, each of spin \textonehalf{}, for alternation parameter a=0.2,0.4,0.6,and0.8a=0.2,0.4,0.6, \mathrm{and} 0.8a=0.2,0.4,0.6,and0.8, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for a=0a=0a=0 and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for N=10N=10N=10 are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when kT∣J∣>a\frac{\mathrm{kT}}{|J|}>a∣J∣kT​>a. The magnetic-susceptibility data on the free radical 2,2-bis (ppp-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.
The long-range order in the ground state of the linear antiferromagnetic chain is investigated. A chain of NNN identical spin-\textonehalf{} particles coupled by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{j=1}{{S}_{j}}^{z}{{S}_{j+1}}^{z}+bJ\ensuremath{\Sigma}\stackrel{N}{j=1}({{S}_{j}}^{+}{{S}_{j+1}}^{\ensuremath{-}}+{{S}_{j}}^{\ensuremath{-}}{{S}_{j+1}}^{+})$ is considered, and the periodic boundary condition, ${S}_{i+N}\ensuremath{\equiv}{S}_{i}$, is imposed so that the chain may be considered a loop of particles. The operator M\mathfrak{M}M, which rotates the particles in the loop so that each of the particles moves to the next highest site number, is introduced. It is proved that simultaneous eigenstates of H\mathcal{H}H and M\mathfrak{M}M have no long-range order as measured by ${\ensuremath{\sigma}}^{*}$ of Kasteleijn. The ground state of the Hamiltonian for $b\ensuremath{\ne}0$, $N<\ensuremath{\infty}$ is shown to be nondegenerate. It is therefore necessarily an eigenstate of M\mathfrak{M}M and as such has no long-range order.
The Heisenberg-Ising Hamiltonian has been exactly solved us- ing the Lanczos method, for chain of up to N = 28 spins. In the whole anisotropic region, the mass gap displays a dominant exponential behavior with size. The extrapolated ground state is doubly degenerate, and presents long-range order with dominant components of the Neel type. Quantum fluctuations become more important as long as the anisotropy is reduced, leading to a disordered singlet ground state for the infinite system at the isotropic point.
We propose a coherent-state solution for the one-dimensional antiferromagnetic Heisenberg Hamiltonian in the quasi-Ising asymptotic regime. We discuss the nature of the spectrum and the eigenvectors as a function of the anisotropy parameter. Results for the energy of the ground state and the short-range-order correlation agree with exact numerical calculations in a wide range of variation of anisotropy. We also present a canonical formulation of the theory which interpolates continuously from the Ising limit to cases of lower anisotropies, thus yielding a novel view of antiferromagnetism. Finally we discuss possible connections with the theory of superconductivity of high-Tc{T}_{c}Tc​ ceramics.