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Journal of Physics A Mathematical and General

On the susceptibility of the generalised square lattice Ising model

Héctor Giacomini

Published Jul 7, 1989• 0 citations
Publisher

Abstract

An exact functional relation is found for the susceptibility of the generalised square lattice Ising model. This result contains, as a particular case, a Fisher relation between the susceptibilities of the triangular and honeycomb lattices models. It is also shown that the closed-form expression for the susceptibility of the generalised square lattice, proposed by Syozi and Naya (1960), satisfies the functional relation presented in this letter.

DOI
10.1088/0305-4470/22/13/019

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An exact functional relation is found for the susceptibility of the generalised square lattice Ising model. This result contains, as a particular case, a Fisher relation between the susceptibilities of the triangular and honeycomb lattices models. It is also shown that the closed-form expression for the susceptibility of the generalised square lattice, proposed by Syozi and Naya (1960), satisfies the functional relation presented in this letter.
Fisher has found exactly for the square and honeycomb lattice the static initial perpendicular susceptibility of the quantum-mechanical version of the Ising model defined by the Hamiltonian H=−J ∑ nnσizσjz−mHx(t) ∑ iσix,where all symbols have their usual meanings. As the magnetization, Mx = mΣiσix, does not commute with the Hamiltonian, the system can relax and Kubo's linear response theory can be applied. We have been able to evaluate the response function exactly and hence obtain simply the exact frequency-dependent initial perpendicular susceptibility for the honeycomb lattice Jχ⊥(ω,T)/Nm2=¾[〈σ0zσ1z〉−〈σ0zσ1zσ2zσ3z〉]ω02/(ω02−ω2)+3[¾〈σ0zσ1z〉+¼〈σ0zσ1zσ2zσ3z〉]ω02/(9ω02−ω2),where ω0 = 2J/ℏ and 〈σ0zσ1z〉 is the nearest-neighbor correlation evaluated by Onsager. A similar expression is obtained for the square lattice. For the four-particle correlation 〈σ0zσ1zσ2zσ3z〉 the other three spins are nearest neighbors to spin zero, and we have obtained its value from Fisher's transformation theory. The behavior of χ⊥ as a function of T and ω is described and its physical significance is discussed.
Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
The Kubo linear response theory is applied to the quantum mechanical Ising model to obtain a general expression which is valid for all regular lattices for the perpendicular response function in terms of spin correlations. For the honeycomb and the square lattices, exact expressions are found for the frequency-dependent initial perpendicular susceptibility, [Formula: see text] valid for all temperatures. For the honeycomb lattice, the present result for zero frequency agrees with Fisher's static calculation, whereas there is disagreement in the case of the square lattice. The discrepancy is resolved by using an extended calculation which explicitly illustrates the difference between isothermal and adiabatic susceptibilities.
We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T −Tc | 9/4, though high-low temperature symmetry is still preserved. At terms of order |T −Tc | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − Tc) p (log |T − Tc|) q with p ≥ q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
The "star-triangle" and "decoration" transformations are generalized so as to apply to arbitrary mechanical systems coupled to the spins of a standard Ising net. This leads to exact solutions for further plane Ising nets and also for lattices in which the spins on alternate sites have a magnitude greater than S=12S=\frac{1}{2}S=21​. A general class of antiferromagnetic Ising models is constructed; exact closed expressions can be derived for all the thermodynamic and magnetic properties of these models in an arbitrary magnetic field.The magnetizations and susceptibilities of Ising nets in which different spins have different magnetic moments are investigated and a valuable relation between the susceptibilities of the honeycomb and triangular lattices is derived. It is shown how correlation functions involving a given spin can be expressed in terms of correlations involving the nearest-neighbor spins instead.