Phase transition of model interacting chains

Abstract

The partition function of $n$ interacting chains having a phase transition at ${T}_{c}(n)$ is reduced, by the transfer-matrix formalism, to the determination of an $n$-body bound state in one dimension. Subject to mild restrictions on the potentials, this $n$-body problem can be analyzed in terms of a two-body problem. It is shown for all $ng~2$ that ${T}_{c}(n)$ is an increasing function of $n$, bounded as follows: ${T}_{c}(2)l~{T}_{c}(n)l~{2}^{\frac{1}{2}}{T}_{c}(2)$. The important special case $n=2$ is studied explicitly, and a second-order phase transition (specific-heat discontinuity) is found to occur at a calculated ${T}_{c}(2)$.