Non-degenerated groundstates in the antiferromagnetic Ising model on triangulations

TL;DR

This paper demonstrates that for any fixed closed Riemann surface, there exist triangulations of the surface with non-degenerate groundstates in the antiferromagnetic Ising model, including geometrically frustrated systems.

Contribution

It proves the existence of vertex-increasing sequences of triangulations with non-degenerate groundstates on any fixed closed Riemann surface.

Findings

Existence of non-degenerate groundstates in triangulations on all closed Riemann surfaces.

Construction of geometrically frustrated systems with unique groundstates.

Sequences of triangulations with increasing vertices maintaining non-degeneracy.

Abstract

A triangulation is an embedding of a graph into a closed Riemann surface so that each face boundary is a 3-cycle of the graph. In this work, groundstate degeneracy in the antiferromagnetic Ising model on triangulations is studied. We show that for every fixed closed Riemann surface S, there are vertex-increasing sequences of triangulations of S with a non-degenerated groundstate. In particular, we exhibit geometrically frustrated systems with a non-degenerated groundstate.