Exact solution and pair correlation functions for a generalized three-chain Ising tube with multispin interactions

TL;DR

This paper provides an exact solution for a generalized three-chain Ising tube with multispin interactions, deriving thermodynamic properties and correlation functions for various special cases and boundary conditions.

Contribution

It introduces an exact analytical approach to solve a complex three-chain Ising model with multiple interactions and boundary conditions, including explicit formulas for correlation functions.

Findings

Exact partition function derived for the generalized model.

Explicit formulas for pair correlation functions and magnetization.

Identified special cases with simplified spectra and entropy limits.

Abstract

We obtain an exact solution for a generalized three-chain Ising tube (TCGIT) of length $L$ with toroidal boundary conditions and the most general $C_3$-invariant Hamiltonian on an elementary prism, containing 20 independent coupling constants, including an external magnetic field. Using an $8\times 8$ transfer matrix, we derive the exact partition function of the finite system and obtain the free energy, internal energy, specific heat, magnetization, magnetic susceptibility, and entropy in the thermodynamic limit $L\to\infty$. In the general case, $\lambda_{\max}$ is determined by a quartic equation, whereas in the principal special case with even-spin interactions (PSC) the spectrum simplifies substantially: the characteristic polynomial factorizes, and $\lambda_{\max}$ is given by the root of a quadratic equation. For mirror-symmetric subfamilies, we derive explicit formulas for the pair correlation functions and express the magnetization in terms of the components of the eigenvector associated with $\lambda_{\max}$; in the even-spin case with $h=0$, the magnetization vanishes. Important special cases include the width-three planar model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, including the entropy limit $S(T\to0^+)=(\ln 2)/3$ for $k\ge 0$ and $S(T\to0^+)=0$ for $k<0$, as well as the width-three planar triangular model with distinct nearest-neighbor couplings, three-spin interactions involving neighboring triangles, and an external field.