energies in the preceding work. The final section offers discussion and conclusions.

NNN SOS Model on a Square Lattice
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Including NNN interactions in the low-temperature expansion of the square-lattice Ising model lifts the remarkable degeneracy of the model with just NN bonds. In that simple case, the energy of a path depends solely on the number of NN links, independent of the arrangement of kinks along it; thus, the energy of the ground state is proportional to the number of NN links of the shortest path between two points, and the entropy is related to the number of combinations of horizontal and vertical links that can connect the points.[@rottman; @Cahn] Including NNN interactions causes the step energy to become a function of both the length of the step and the number of its kinks, eliminating the simple path-counting result.[@Cahn] It can then become energetically favorable for the step to lengthen rather than add another kink. This causes the NN energy levels to split in a non-trivial way, making it possible for a longer step to have a lower energy than a shorter step. A related complication is that the expansion itself depends on the relative strength of the NNN-interaction: 