The exact solution of the Axial Next-Nearest-Neighbor Ising model on width 2 spin chains

TL;DR

This paper derives exact solutions for the Axial Next-Nearest-Neighbor Ising model on a finite width 2 spin chain, including partition functions, free energy, and spectral properties, providing insights into its thermodynamic behavior.

Contribution

It presents the exact transfer matrix solution for the 2-width ANNNI model, including spectral analysis and proofs of key thermodynamic theorems, which was previously unresolved.

Findings

Exact partition function and free energy derived

Spectral decomposition of transfer matrix obtained

Heat capacity and physical characteristics analyzed

Abstract

Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.