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Axioms

Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix

Hasan Akın

Published Sep 14, 2023• 7 citations
Publisher

Abstract

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.

DOI
10.3390/axioms12090880

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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.
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.
A magnetic property of the one dimensional spin 5/2 Ising model under the magnetic field has been investigated by means of transfer matrix method. Thermodynamic response functions are also obtained for varying values of scaling temperature and scaling magnetic field. Entropy and heat capacity of the system were calculated by benefiting from the temperature dependencies of Helmholtz free energy. Our simulation results demonstrate that as the strength of the magnetic field is increased, heat capacity tend to shift to the relatively higher temperature regions, and these findings are consistent with previous results for the low spin values in one dimensional Ising systems.
The thermodynamic properties of the two-dimensional Ising model with random (+or-J) nearest-neighbour interactions are determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The central result is that the spin-glass correlation length diverges as T- nu at low temperature T with nu =2.59+or-0.13. The heat capacity is similar to the Schottky anomaly. At zero temperature the entropy per spin is 0.0701+or-0.0005 and the internal energy per spin is -(1.4024+or-0.0012)J.
We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.
Abstract We introduce approximate expressions for the thermodynamics of the two-dimensional Ising model in an external magnetic field. The external field breaks the system’s symmetry, which complicates the exact calculation of the free energy. To address this, we write the model in its quaternion representation to apply a small angle approximation with respect to a reference frame. The approximation simplifies the transfer matrix to a centrosymmetric matrix for which the largest eigenvalue is known and depends on the external field strength. We therefore write an approximate expression for the system’s free energy and derive the magnetization, susceptibility, internal energy, and specific heat. These results are compared with numerical simulations performed with the BKL algorithm. We find that the approximate expressions and numerical simulations agree in regions where the approximation is valid.