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Journal of Physics C Solid State Physics

Spin-pair correlation function and susceptibility of the planar Ising model

Cyril Domb

Published May 1, 1970• 2 citations
Publisher

Abstract

Recent calculations of the form of the spin-pair correlation function of the Ising model for a simple quadratic lattice enable calculations to be undertaken of the form of the singularities of magnetic susceptibility chi 0 for T>T c . It is suggested that chi 0 has a Darboux-type singularity of the form (K c -K) -7/4 together with subsidiary singularities of the form ((K c -K) ln (K c -K)) t and ((K c +K) ln (K c +K)) t (t>or=1).

DOI
10.1088/0022-3719/3/5/001

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Journal of the Physical Society of Japan•Sep 15, 1997•0 citations
Recent calculations of the form of the spin-pair correlation function of the Ising model for a simple quadratic lattice enable calculations to be undertaken of the form of the singularities of magnetic susceptibility chi 0 for T>T c . It is suggested that chi 0 has a Darboux-type singularity of the form (K c -K) -7/4 together with subsidiary singularities of the form ((K c -K) ln (K c -K)) t and ((K c +K) ln (K c +K)) t (t>or=1).
For pt.I see ibid., vol.8, p.1920 of 1975. The authors have carried out a further analysis of the recently derived 16 term high temperature susceptibility series for the spin 1/2 Ising model on the tetrahedron lattice (crystobalite or B-site spinel lattice). They have found convincing evidence for the asymptotic form chi =C 0 (1-K/K c -5 4/+C 1 (1-K/K c -1 4/+X 0 +C 2 (1-K/K c ) 3 4/+... and have obtained estimates for the amplitudes C 0 , C 1 , C 2 and X 0 . They have also used recent lattice-lattice scaling hypotheses to calculate the specific heat and magnetization amplitudes.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ (1) d and χ (2) d of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution χ (5) d , but only modulo a given prime. We use these exact linear differential equations to show that not only the Russian-doll structure but also the direct sum structure on the linear differential operators for the n -particle contributions χ ( n ) d are quite directly inherited from the direct sum structure on the form factors f ( n ) . We show that the n th particle contributions χ ( n ) d have their singularities at roots of unity. These singularities become dense on the unit circle |sinh 2 E v / kT sinh 2 E h / kT | = 1 as n → ∞ .
The zero·field susceptibility x ( T) of the antiferromagnet in the Ising square lattice with the symmetric interaction was evaluated by carrying out the lattice sum of the short · range correlation function X ( m, n) (O~m, n~4), which is exactly and explicitly calculated on the basis of a "generalized Wronskian " form with dimension m. It is naturally derived from the singularity of X(m, n) that the highest singularity of x ( T) (per site) near the critical temperature Tc is logarithmic as kB Tz ( T) =A + B(l- T/Tc)lnI1- T/Tcl, with A =0.1596264029 and B=0.2009359779 in the best approximation of the present calculation. Both A and B are completely equal below and above the critical temperature.