Fig. 2. Vertex types and string representation.

a Vertex moment configurations for different vertex coordination numbers, z = 2, 3, and 4, arranged in order of increasing energy, where the arrows indicate moment direction and each red circle denotes an excited state of the vertex (i.e., an unhappy vertex). The numbers in parentheses show the degeneracy for each given vertex type. The blue lines represent string segments “connecting” the plaquettes through the unhappy vertices. b Examples of moment and string configurations for peripheral (unshaded) and interior (yellow-shaded) plaquettes. Because of the SFI lattice geometry, a peripheral plaquette is connected by an even number of unhappy vertices, and an interior plaquette is connected by an odd number.