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Physical Review

Exact Solution for a Linear Chain of Isotropically Interacting Classical Spins of Arbitrary Dimensionality

Eugene Stanley

Published Mar 10, 1969• 208 citations
Publisher

Abstract

The isotropic Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}=\ensuremath{-}J\ensuremath{\Sigma}{j=1}^{}^{N\ensuremath{-}1}\mathrm{S}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ is considered for an open linear chain of $N$ $\ensuremath{\nu}$-dimensional vector spins ${\mathrm{S}}_{j}$;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the $S=\frac{1}{2}$ Ising, planar, and Heisenberg models for $\ensuremath{\nu}=1,2,\mathrm{and} 3$. The thermodynamic properties (including the susceptibility) of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found for ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) exchange interactions for all temperatures $T$ and all spin dimensionalities $\ensuremath{\nu}$. The manner in which the various properties depend upon $T$ and $\ensuremath{\nu}$ is studied; in particular we find (a) that although the chain of spins does not display long-range order except at $T=0$ for any value of $\ensuremath{\nu}$ most of the properties vary monotonically with $\ensuremath{\nu}$ (in such a way that, e.g., the degree of "short-range order" decreases with increasing $\ensuremath{\nu}$; and (b) that as the spin dimensionality increases without limit, all of the calculated properties approach precisely those predicted by the Berlin-Kac spherical model.

DOI
10.1103/physrev.179.570

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Critical Indices for a System of Spins of Arbitrary Dimensionality Situated on a Lattice of Arbitrary Dimensionality

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A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].

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Results of exact numerical work on the spectra and thermodynamic properties are presented for finite clusters in one and two dimensions, for the isotropic Heisenberg Hamiltonian and the Heisenberg Hamiltonian with longitudinal anisotropy. Eigenvalues of the isotropic Heisenberg loops (that is, those with periodic boundary conditions) of $N\ensuremath{\le}12, 8, 6, 5$, 5 particles, each of spin \textonehalf{}, 1, $\frac{3}{2}$, 2 and $\frac{5}{2}$, respectively, are found. It is shown that the spin-$S$ linear chain ($N\ensuremath{\rightarrow}\ensuremath{\infty}$) will have ground-state energy per spin $\frac{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{s}}{2J}=\ensuremath{-}1.404\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{-}2.85\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{-}4.7\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{and} \ensuremath{-}7.1\ifmmode\pm\else\textpm\fi{}0.5$, for $S=1, \frac{3}{2}, 2, \mathrm{and} \frac{5}{2}$, respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-\textonehalf{} case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin $\frac{5}{2}$, the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-\textonehalf{} linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the $N=8$ spin-\textonehalf{} chain (with open ends), and find the difference to be small. Results on the 3\ifmmode\times\else\texttimes\fi{}4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.

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Exactly soluble model of interacting classical spins in one dimension with random interactions

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An exact calculation of the thermodynamic properties of a linear chain of classical spins with nearest-neighbor bilinear and biquadratic isotropic random exchange interactions is presented. Both open and closed chains are discussed. Only bond disorder is being considered here. It is shown that in contrast to the uniform system, "dipolar disorder points" as well as "quadrupolar disorder points" with qualitative and quantitative differences from the uniform case occur in this system. This is because the competing dipolar and quadrupolar interactions are further enriched by the random signs of these interactions. It is interesting to point out that much of the analysis can be made by means of a study of the eigenvalues of the "transfer kernel." The method can also be applied to obtain exact results for a linear chain of spins of arbitrary dimensionality interacting through an arbitrary isotropic nearest-neighbor Hamiltonian. In the case of the open chain, the Edwards-Anderson expedient is found to be exact in all the cases considered.

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Linear Magnetic Chains with Anisotropic Coupling

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Linear chains (and rings) of $S=\frac{1}{2}$ 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 $N=2 \mathrm{to} 11$, $\ensuremath{\gamma}=0 \mathrm{to} 1$ 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 $\frac{\mathrm{kT}}{|J|}>~0.5$ 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 $T=0$ 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 $\ensuremath{\gamma}$.

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Linear Antiferromagnetic Chain with Anisotropic Coupling

R. Orbach

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.

Physical Review•Oct 15, 1958•532 citations
The isotropic Hamiltonian {\mathcal{H}}^{(\ensuremath{\nu})}=\ensuremath{-}J\ensuremath{\Sigma}{j=1}^{}^{N\ensuremath{-}1}\mathrm{S}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1} is considered for an open linear chain of NNN \ensuremathν\ensuremath{\nu}\ensuremathν-dimensional vector spins Sj{\mathrm{S}}_{j}Sj​;H(\ensuremathν){\mathcal{H}}^{(\ensuremath{\nu})}H(\ensuremathν) reduces to the S=12S=\frac{1}{2}S=21​ Ising, planar, and Heisenberg models for \ensuremathν=1,2,and3\ensuremath{\nu}=1,2,\mathrm{and} 3\ensuremathν=1,2,and3. The thermodynamic properties (including the susceptibility) of H(\ensuremathν){\mathcal{H}}^{(\ensuremath{\nu})}H(\ensuremathν) are found for ferromagnetic (J>0J>0J>0) and antiferromagnetic (J<0J<0J<0) exchange interactions for all temperatures TTT and all spin dimensionalities \ensuremathν\ensuremath{\nu}\ensuremathν. The manner in which the various properties depend upon TTT and \ensuremathν\ensuremath{\nu}\ensuremathν is studied; in particular we find (a) that although the chain of spins does not display long-range order except at T=0T=0T=0 for any value of \ensuremathν\ensuremath{\nu}\ensuremathν most of the properties vary monotonically with \ensuremathν\ensuremath{\nu}\ensuremathν (in such a way that, e.g., the degree of "short-range order" decreases with increasing \ensuremathν\ensuremath{\nu}\ensuremathν; and (b) that as the spin dimensionality increases without limit, all of the calculated properties approach precisely those predicted by the Berlin-Kac spherical model.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
Some years ago Ashkin and Lamb noted that the phase transition in the two-dimensional Ising model with nearest-neighbor interaction was characterized mathematically by an asymptotic degeneracy of the largest eigenvalue of a linear operator. More recently Kac and Thompson showed this eigenvalue degeneracy also characterized the phase transition in the Kac model (with weak, long-range forces), suggesting that this ``mathematical mechanism'' is not restricted to systems with short-range forces. However both the Kac model and the Ising model consider the ``spins'' to be one-dimensional unit vectors assuming only the discrete values +1 and −1. We are therefore led to consider the nature of the phase transition in one of the few exactly soluble models in which the spins can assume an entire continuum of orientations-the closed linear chain of classical spins of arbitrary spin dimensionality interacting isotropically through the Hamiltonian Hν=− ∑ i=1N Ji,i+1Si(ν)·Si+1(ν),where the exchange constants Ji,i+1 are arbitrary numbers. We find for all spin dimensionalities ν that the two-spin correlation function may be expressed as ρN(ν) (γ) = (λ1(ν)/λ0(ν))r, where λ0(ν) and λ1(ν) are the largest and next-largest eigenvalues of a certain linear operator. Thus the onset of longrange order, limr→∞ρN(ν) (γ)≠0, is characterized by the degeneracy of λ0(ν) and λ1(ν). This onset of long-range order occurs at a critical temperature Tc(ν)=0, for all values of spin dimensionality ν.
Results of exact numerical work on the spectra and thermodynamic properties are presented for finite clusters in one and two dimensions, for the isotropic Heisenberg Hamiltonian and the Heisenberg Hamiltonian with longitudinal anisotropy. Eigenvalues of the isotropic Heisenberg loops (that is, those with periodic boundary conditions) of $N\ensuremath{\le}12, 8, 6, 5$, 5 particles, each of spin \textonehalf{}, 1, 32\frac{3}{2}23​, 2 and 52\frac{5}{2}25​, respectively, are found. It is shown that the spin-SSS linear chain ($N\ensuremath{\rightarrow}\ensuremath{\infty}$) will have ground-state energy per spin $\frac{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{s}}{2J}=\ensuremath{-}1.404\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{-}2.85\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{-}4.7\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{and} \ensuremath{-}7.1\ifmmode\pm\else\textpm\fi{}0.5$, for S=1,32,2,and52S=1, \frac{3}{2}, 2, \mathrm{and} \frac{5}{2}S=1,23​,2,and25​, respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-\textonehalf{} case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin 52\frac{5}{2}25​, the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-\textonehalf{} linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the N=8N=8N=8 spin-\textonehalf{} chain (with open ends), and find the difference to be small. Results on the 3\ifmmode\times\else\texttimes\fi{}4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.
An exact calculation of the thermodynamic properties of a linear chain of classical spins with nearest-neighbor bilinear and biquadratic isotropic random exchange interactions is presented. Both open and closed chains are discussed. Only bond disorder is being considered here. It is shown that in contrast to the uniform system, "dipolar disorder points" as well as "quadrupolar disorder points" with qualitative and quantitative differences from the uniform case occur in this system. This is because the competing dipolar and quadrupolar interactions are further enriched by the random signs of these interactions. It is interesting to point out that much of the analysis can be made by means of a study of the eigenvalues of the "transfer kernel." The method can also be applied to obtain exact results for a linear chain of spins of arbitrary dimensionality interacting through an arbitrary isotropic nearest-neighbor Hamiltonian. In the case of the open chain, the Edwards-Anderson expedient is found to be exact in all the cases considered.
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 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.