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.
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.
It is noted that the pair correlation matrix χ of the nearest neighbor Ising model on periodic three-dimensional (d = 3) kagome-like lattices of corner-sharing triangles can be calculated partially exactly.Specifically, a macroscopic number 1/3 N + 1 out of N eigenvalues of χ are degenerate at all temperatures T , and correspond to an eigenspace L -of χ, independent of T .Degeneracy of the eigenvalues, and L -are an exact result for a complex d = 3 statistical physical model.It is further noted that the eigenvalue degeneracy describing the same Lis exact at all T in an infinite spin dimensionality m limit of the isotropic m-vector approximation to the Ising models.A peculiar match of the opposite m = 1 and m → ∞ limits can be interpreted that the m → ∞ considerations are exact for m = 1.It is not clear whether the match is coincidental.It is then speculated that the exact eigenvalues degeneracy in L -in the opposite limits of m can imply their quasi-degeneracy for intermediate 1 m < ∞.For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from L -that for m 2 contribute to χ at low T .The m → ∞ formulae can be thus quantitatively correct in description of χ and clarifying the role of perturbations in kagome-like systems deep in the collective paramagnetic regime.An exception may be an interval of T , where the order-by-disorder mechanisms select sub-manifolds of L -.
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 Sj;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the S=21 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.
The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength JLR superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<2$ there is no phase transition in zero field, although there are classical critical points in a field.
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.].
The model is a linear chain in which each spin interacts with its 2r nearest neighbors, the interaction energy being proportional to 1/r. Using a method similar to that of Montroll, the partition function of the model in the thermodynamic limit is shown to be related to the largest eigenvalue of a certain matrix. The largest eigenvalue of the matrix is determined numerically for 3 < r < 12. Also, a correct method is demonstrated for evaluating an improper limit of the model, in which the interaction range is set to the chain length before the limit of an infinite number of spins is taken. Previously published works have performed this calculation incorrectly. Although the numerical results show some evidence of convergence to the improper limit, the results are inconclusive and furthermore raise doubts about the practicality of the numerical method in this context.
We introduce and analyze an exactly soluble one-dimensional Ising model with long range interactions that exhibits a mixed-order transition, namely a phase transition in which the order parameter is discontinuous as in first order transitions while the correlation length diverges as in second order transitions. Such transitions are known to appear in a diverse classes of models that are seemingly unrelated. The model we present serves as a link between two classes of models that exhibit a mixed-order transition in one dimension, namely, spin models with a coupling constant that decays as the inverse distance squared and models of depinning transitions, thus making a step towards a unifying framework.