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

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 $L\to \mathrm{\infty }$. 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 $S(T\to 0^{+})=(\ln 2)\mathrm{/}3$ for $k\ge 0$ and $S(T\to 0^{+})=0$ 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.