Non-degenerated Ground States and Low-degenerated Excited States in the Antiferromagnetic Ising Model on Triangulations

We study the unexpected asymptotic behavior of the degeneracy of the first few energy levels in the antiferromagnetic Ising model on triangulations of closed Riemann surfaces. There are strong mathematical and physical reasons to expect that the number of ground states (i.e., degeneracy) of the antiferromagnetic Ising model on the triangulations of a fixed closed Riemann surface is exponential in the number of vertices. In the set of plane triangulations, the degeneracy equals the number of perfect matchings of the geometric duals, and thus it is exponential by a recent result of Chudnovsky and Seymour. From the physics point of view, antiferromagnetic triangulations are geometrically frustrated systems, and in such systems exponential degeneracy is predicted. We present results that contradict these predictions. We prove that for each closed Riemann surface S of positive genus, there are sequences of triangulations of S with exactly one ground state. One possible explanation of this phenomenon is that exponential degeneracy would be found in the excited states with energy close to the ground state energy. However, as our second result, we show the existence of a sequence of triangulations ${(\mathcal{T}_n)}$ of a closed Riemann surface of genus 10 with exactly one ground state such that the degeneracy of each of the 1^st, 2^nd, 3^rd and 4^th excited energy levels belongs to O(n), O(n ²), O(n ³) and O(n ⁴), respectively.