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1D ISINGOV MODEL I MATRICA TRANSFERA

Margareta Glumac

Published Sep 28, 2020• 0 citations
OpenAlex

Abstract

Ising’s one-dimensional model is a model whose constituent elements are magnetic moments of spin ±1/2. Spins interact with short-range forces limited only to the first neighbours. The partition function of this model is easy to calculate using transfer matrix (section 4.1). From the known partition function, free energy, magnetization and other thermodynamic functions can be easily obtained. It shows that the magnetization is analytical function of the temperature for all T > 0, which shows that in this Ising model, there is no phase transition for T > 0.

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Ising’s one-dimensional model is a model whose constituent elements are magnetic moments of spin ±1/2.
Spins interact with short-range forces limited only to the first neighbours. The partition function of this model is easy to calculate using transfer matrix (section 4.1). From the known partition function, free energy,
magnetization and other thermodynamic functions can be easily obtained. It shows that the magnetization
is analytical function of the temperature for all T > 0, which shows that in this Ising model, there is no
phase transition for T > 0.
In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-(s,(2t−1)/2) with nearest neighbors and the external magnetic field. We establish the partition function of the model by means of the transfer matrix. Under a null boundary condition, we compute certain thermodynamic quantities for the 1D-MSIM. We find some precise formulas to determine the model’s free energy, entropy, magnetization, and susceptibility. By examining the model’s associated iterative equations, we use the cavity approach to investigate the phase transition problem. We numerically determine the model’s periodicity.
This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.
Abstract The aim of this chapter is to describe how transfer matrices can be used to solve one-dimensional classical spin models. The idea is to write down the partition function in terms of a matrix, the transfer matrix. The thermodynamic properties of the model are then wholly described by the eigenspectrum of the matrix. In particular the free energy per spin in the thermodynamic limit depends only on the largest eigenvalue and the correlation length only on the two largest eigenvalues through simple formulae.
Abstract Having been linked to numerous applications, the one-dimensional Lenz–Ising model in a non-zero magnetic field is well-known in graduate- and senior-level statistical physics. The transfer matrix method is the standard way to solve the model. In this line, we reformulate the transfer matrix by subjecting the usual Pauli matrices to rotation. The transfer matrix is then shown as a ‘spin’-representative of a two-dimensional rotation just like the special case when the magnetic field is zero. Consequently, eigenvalues in exponential form are obtained, and the zeroes of the partition function lie on a unit circle in agreement with the Lee–Yang Circle Theorem. The zeroes pinching the positive side of the real axis give the known trivial transition temperature, T = 0 K. Finally, we discuss the significance of the results in teaching the model.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.