Exact solution of generalized triple Ising chains with multi-spin interactions

TL;DR

This paper derives exact solutions for the generalized triple Ising chain with multi-spin interactions, calculating key physical properties and analyzing special cases, including the planar triangular and gonihedric models.

Contribution

It provides the first exact analytical expressions for the physical characteristics of the generalized triple Ising model with multi-spin interactions, including special cases and ground state structures.

Findings

Exact partition function and thermodynamic properties derived

Eigenvalues and eigenvectors of the transfer matrix obtained

Ground state structures and correlation lengths analyzed

Abstract

We obtain the exact physical characteristics of the triple-chain Ising model on a torus with all possible multispin interactions invariant with respect to rotation by the angle $2\pi / 3$. The exact value of the partition function in a finite cyclically closed strip of length $L$, as well as the free energy, internal energy, specific heat, magnetization, susceptibility, and entropy in the thermodynamic limit at $L \to \infty$ are found by the transfer-matrix method for the model. The spectrum of the transfer-matrix and the structure of its eigenvectors are found. For two special cases - for the model with multispin interactions of even number of spins and for the model with some interactions of two, three, four and six spins, simplified expressions of the mentioned physical characteristics are obtained; in the thermodynamic limit they are expressed through the logarithm of the root of the quadratic equation. For the model with multispin interactions of an even number of spins, a kind of pair correlations in the thermodynamic limit is found, and it is shown that the magnetization at zero magnetic field is equal to zero; the structure of the ground states of the system is found and examples of their projections of seven-dimensional space onto three-dimensional space and examples of configurations corresponding to these ground states are given. The correlation length is shown and its graphs are given. As special cases, we consider the planar triangular model with all possible interactions, including, perhaps, different triple interactions inside neighboring triangles, and the planar model with nearest-neighbor, next nearest-neighbor, and plaquette interactions. For them the main exact physical characteristics have been found. This allowed us to obtain them for the planar gonihedric model as well.