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Journal of Mathematical Physics

A novel approach to the exact calculation of correlation functions of a one-dimensional random Ising chain

A. K. Rajagopal, Gary S. Grest

Published May 1, 1974• 10 citations
Publisher

Abstract

An exact result for the partition function for a general one-dimensional Ising chain of N spin-1/2 particles described by the Hamiltonian HN=−∑i=1NJiσiσi+1−∑i=1NHiσi is given. For an open strand, JN = 0; for a closed chain, σN+1 = σ1, JN ≠ 0. The novelty of the trick used enables one to obtain the partition function and all the spin correlation functions for open and closed chains with equal ease. Special cases of this model have been discussed before to elucidate certain features of some biological systems. New expressions for parallel and perpendicular susceptibilities for this model are also derived. When Ji and Hi are treated as random variables, the above Hamiltonian describes a one-dimensional Ising spin glass. In this case some simple models and formal averaging procedures are discussed.

DOI
10.1063/1.1666688

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Journal of Mathematical Physics•May 1, 1974•12 citations
An exact result for the partition function for a general one-dimensional Ising chain of N spin-1/2 particles described by the Hamiltonian HN=−∑i=1NJiσiσi+1−∑i=1NHiσi is given. For an open strand, JN = 0; for a closed chain, σN+1 = σ1, JN ≠ 0. The novelty of the trick used enables one to obtain the partition function and all the spin correlation functions for open and closed chains with equal ease. Special cases of this model have been discussed before to elucidate certain features of some biological systems. New expressions for parallel and perpendicular susceptibilities for this model are also derived. When Ji and Hi are treated as random variables, the above Hamiltonian describes a one-dimensional Ising spin glass. In this case some simple models and formal averaging procedures are discussed.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.
As an illustration of a quasiclassical spin formalism developed in a recent paper a new method for solving the one-dimensional Ising model is presented. The method used is an extension to spin systems of the Wigner method familiar in computations involving position and momentum. The partition function, pair correlation function and the zero-field susceptibility for the one-dimensional Ising model with arbitrary spin are shown to be expressed in terms of the eigenfunctions and eigenvalues of a certain integral equation whose kernel approaches the classical kernel as the spin approaches infinity.
In this paper the exact solution and correlation functions for a double-chain Ising model with multi-spin interactions and symmetric Hamiltonian density are obtained. The study employs the transfer matrix method to derive fundamental thermodynamic characteristics of the system. The main results include exact expressions for the partition function, free energy, internal energy, specific heat capacity, magnetization, susceptibility, and entropy in a strip of finite length and in the thermodynamic limit. The work provides explicit formulas for the eigenvalues and shows structure of eigenvectors of the transfer matrix. The expression for magnetization in the thermodynamic limit using components of normalized eigenvector corresponding to the maximum eigenvalue is obtained. A detailed analysis is conducted for a special case of interactions involving all kinds of two- and four-spin interactions. This gives the simplified formula for free energy, it is calculated using the root of quadratic equation. Using special relation between parameters the solution for gonihedric model on the plane is obtained. The research reveals properties of the system, including specific features of ground states and phase diagram characteristics. Particular attention is given to the behavior of physical quantities near frustration points and the investigation of spin correlation functions. Plots of physical characteristics, including inverse correlation length, illustrating the obtained results are constructed.
We investigate the statistical mechanics of the periodic one-dimensional Ising chain when the number of positive spins is constrained to be either an even or an odd number. We calculate the partition function using a generalization of the transfer matrix method. On this basis, we derive the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. We show that in general the constraints substantially slow down convergence to the thermodynamic limit. By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. We demonstrate how these results solve a stochastic version of the one-dimensional voter model.
Exact, closed expressions are obtained for the perpendicular and parallel susceptibilities of the one-dimensional spin-1/2 Ising model (i) with nearest neighbor interactions in a uniform magnetic field and (ii) with next-nearest neighbor interactions but without the magnetic field, in the thermodynamic limit. Graphs of the susceptibilities as a function of temperature for various magnetic field strengths in (i), and as a function of different ratios of the interaction strengths in (ii), are discussed.