Abstract

In this work, we continue our studies on the antiferromagnetic Ising model on triangulations of closed Riemann surfaces. According to R. Moessner and A.P. Ramirez, the antiferromagnetic Ising model on triangulations exhibits geometrical frustration, a well-studied concept in condensed matter physics. Typical geometrically frustrated systems present an exponential groundstate degeneracy. A possible explanation for frustrated systems with a low groundstate degeneracy is that exponentially many states exist at a low enery level. In this work, we show that this is not the case for the antiferromagnetic Ising model on triangulations of closed Riemann surfaces. For each integer m≥1, we construct a collection of triangulations of a fixed closed Riemann surface with the property that for each such triangulation the degeneracy of each of its first m energy levels is a polynomial.