Some Simple Lattice-Spin Systems

A simple mathematical scheme is derived here. With this scheme, three problems of lattice‐spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz‐Ising model in a zero magnetic field with nearest, as well as next‐nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest‐neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1‐dimensional Heisenberg‐Dirac model, similar to the partially solved ``Ising‐Heisenberg'' model of Lieb‐Mattis‐Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as $$\begin{array}{c}{\epsilon }_1{\mathrm{(q)=[(A}}_q{\mathrm{+B}}_q{)}^2{\mathrm{+E}}_q^2{]}^{\frac{1}{2}},\\ {\epsilon }_2{\mathrm{(q)=[(A}}_q{\mathrm{-B}}_q{)}^2{\mathrm{+E}}_q^2{]}^{\frac{1}{2}},\end{array}$$ where A<sub>q</sub>, B<sub>q</sub>, and E<sub>q</sub> are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.