The critical energy of the Ising double chain -- the explanation for melting

Abstract

The Ising double chain has been overlooked for more than 80 years, since one was mostly interested in phase transitions. We consider two types of degeneracy, spin occupation degeneracy and "bond" occupation degeneracy. We show that bond occupation on "open nets" is of particular importance for the understanding of melting. The number of open nets on a double chain is determined analytically. If only one bond site of a net is occupied, this already produces a definite number of bonds on sites, bonds that do not belong to the net. All nets on a given chain of length $n$ then yield a total number $i^{(n)}_{\rm tot}$ of bonds, allowing to calculate a characteristic energy $u^{(n)}_{ch}$, an approximation of the critical energy $u_c$ of the infinite double chain. We finally present an explanation for the critical energy $u^{\rm KG}_c$ of the kagom\'e lattice in terms of the critical energies $u^{\rm TR}_c$ and $u^{\rm HC}_c$ of the triangular and honeycomb lattices.