Abstract

In this work, we continue our investigation on the antiferromagnetic Ising model on triangulations of closed Riemann surfaces. On the one hand, according to R. Moessner and A. P. Ramirez [11], the antiferromagnetic Ising model on triangulations exhibits geometrical frustration, a well-studied concept in condensed matter physics. Typical geometrically frustrated systems present an exponential ground state degeneracy. On the other hand, the dual graph of a triangulation of a closed Riemann surface is a cubic graph. Cubic bridgeless graphs have exponentially many perfect matchings [3, 5], which implies in the case of planar triangulations, an exponential ground state degeneracy. However, this phenomenon does not persist for triangulations of higher genus surfaces.