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Physical review. E

Application of the integral operator method for multispin correlation function calculations in the one-dimensional Ising model

T. Balcerzak

Published Feb 26, 2024• 1 citation
Publisher

Abstract

The one-dimensional (1D) Ising model is revisited. The integral operator method is used for the model to derive the general statistical identity for multispin correlation functions. As an application of this identity to generate various correlations, recursive formulas are derived for the correlations of pairs, threes, fours, and fives of spins. Both compact clusters and disjoint sets of spins are considered. It is shown that multispin correlations for the 1D Ising model can be exactly expressed in terms of pairwise correlations using recursive procedures. For clusters with different structures, numerical calculations of multispin correlations as a function of temperature and external magnetic field are performed. The results of these calculations are presented in the figures and discussed.

DOI
10.1103/physreve.109.024133

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The one-dimensional (1D) Ising model is revisited. The integral operator method is used for the model to derive the general statistical identity for multispin correlation functions. As an application of this identity to generate various correlations, recursive formulas are derived for the correlations of pairs, threes, fours, and fives of spins. Both compact clusters and disjoint sets of spins are considered. It is shown that multispin correlations for the 1D Ising model can be exactly expressed in terms of pairwise correlations using recursive procedures. For clusters with different structures, numerical calculations of multispin correlations as a function of temperature and external magnetic field are performed. The results of these calculations are presented in the figures and discussed.
We calculate the multispin correlation functions away from Tc{T}_{c}Tc​ of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.
The method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin Sz=±1S^z=\pm1Sz=±1) the expressions for pair and ternary correlation functions within two-particle approximation in { q }\set{q}{q}-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility χ({ q },E)χ(\set{q},E)χ({q},E) is found. In the 1D case and in the absence of external field the expression for χ({ q },E)χ(\set{q},E)χ({q},E) is exact. For the Emery-Blume-Griffiths model (Sz=−2,0,2S^z=-2,0,2Sz=−2,0,2) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility χ({ q })χ(\set{q})χ({q}) of KD2PO4KD_2PO_4KD2​PO4​ ferroelectrics.
Abstract We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .
A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.
We consider the one-dimensional random-field Ising model, where the spin-spin coupling J is ferromagnetic and the external field is chosen to be +h with probability p and -h with probability 1-p. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function 〈s0{\mathit{s}}_{0}s0​sn{\mathit{s}}_{\mathit{n}}sn​〉-〈s0{\mathit{s}}_{0}s0​〉〈sn{\mathit{s}}_{\mathit{n}}sn​〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈s0{\mathit{s}}_{0}s0​sn{\mathit{s}}_{\mathit{n}}sn​〉.
The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.