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arXiv:math-ph/0703009 · Journal of Physics A Mathematical and Theoretical

The diagonal Ising susceptibility

S. Boukraa, S. Hassani, J.-M. Maillard, Barry M. McCoy, N Zenine

Published Jul 3, 2007• 35 citations
arXivHTMLPDF

Abstract

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 → ∞ .

DOI
10.1088/1751-8113/40/29/002

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In a previous paper ( 2004 J. Phys. A: Math. Gen. 37 9651–68 ) we have given the Fuchsian linear differential equation satisfied by χ (3) , the ‘three-particle’ contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the ( n − 1)-dimensional integrals representing the n -particle contribution to the isotropic square lattice Ising model susceptibility χ. The integration rules are straightforward due to remarkable formulae we derived for these variables. We obtain without any numerical approximation χ (3) as a fully integrated series in the variable w = s /2/(1 + s 2 ), where s = sh (2 K ), with K = J / kT the conventional Ising model coupling constant. We also give some perspectives and comments on these results.

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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 → ∞ .
In a previous paper ( 2004 J. Phys. A: Math. Gen. 37 9651–68 ) we have given the Fuchsian linear differential equation satisfied by χ (3) , the ‘three-particle’ contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the ( n − 1)-dimensional integrals representing the n -particle contribution to the isotropic square lattice Ising model susceptibility χ. The integration rules are straightforward due to remarkable formulae we derived for these variables. We obtain without any numerical approximation χ (3) as a fully integrated series in the variable w = s /2/(1 + s 2 ), where s = sh (2 K ), with K = J / kT the conventional Ising model coupling constant. We also give some perspectives and comments on these results.
The Ising model at magnetic field H=0 is one of the most important exactly solved models in statistical mechanics, solved on the square lattice in 1944 by Onsager, and later by others on the triangular lattice; its magnetic susceptibility at H=0 continues to be an unsolved aspect of the model, however. The susceptibility can either be viewed as a sum over all correlation functions of the integrable model at H=0, or else as a second derivative of the free energy of the non-integrable Ising model in a field. Therefore, its analytic properties, though derived from the non-integrable model, can be studied through series expansion of the integrable correlation functions. We begin this process by analyzing the first four terms in the form factor expansion of the diagonal correlation functions, and after summing over the diagonal form factor expansion, the first four terms of the diagonal susceptibility expansion. We have been able to reduce the form factor and susceptibility expansion terms, given as multi-dimensional integrals, to closed-form functions in all cases. Under the limit of H going to infinity with the interaction energy E going to minus infinity, the isotropic Ising model becomes a hard particle lattice gas model, with exclusion of nearest neighbor lattice sites. In this limit, the triangular lattice Ising model becomes the exactly solved hard hexagon model, while the square lattice becomes the non-integrable hard squares model. We study in detail these models for finite lattice sizes in order to understand the differences between integrable and non-integrable models. We consider the partition functions zeros and their density for different boundary conditions, and find notable differences in the density which is attributed to an extra factorization in the transfer matrices of hard hexagons which is absent in hard squares. We also study the special point at fugacity z=-1 of hard squares where all eigenvalues of the transfer matrix are equimodular and where the grand partition function's value depends on boundary conditions. | 300 pages
Square lattice Ising model susceptibility: Series expansion method and differential equation for chi(3)\\chi^{(3)}chi(3)
In a previous paper (J. Phys. A {\\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by chi(3)\\chi^{(3)}chi(3), the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the (n−1)(n-1)(n−1)-dimensional integrals representing the nnn-particle contribution to the isotropic square lattice Ising model susceptibility $\\chi $. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation chi(3)\\chi^{(3)}chi(3) as a fully integrated series in the variable w=s/2/(1+s2)w=s/2/(1+s^{2})w=s/2/(1+s2), where $ s =sh (2K)$, with K=J/kTK=J/kTK=J/kT the conventional Ising model coupling constant. We also give some perspectives and comments on these results.
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.
Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.