Small degeneracy of antiferromagnetic triangulations

Abstract

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., ground state degeneracy) of antiferromagnetic triangulations is exponential in the number of vertices. In the set of plane triangulations, this degeneracy equals the number of perfect matchings in the geometric dual, and thus it is exponential by 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 show that for each closed Riemann surface Ω of positive genus, there are sequences of triangulations of Ω 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 energy of the ground state. However, as our second result, we present a sequence of triangulations (Tn) of a closed Riemann surface of genus 10 with exactly one ground state, and such that the degeneracy of the first four energy levels is a polynomial in n, where n is the number of vertices of Tn.