Topology of the space of periodic ground states in the antiferromagnetic Ising and Potts models in selected spatial structures

Symmetry induced compression of discrete phase space

“…[10]. From the density of different motifs, an anisotropy of the correlation function is found within three separate subsets (communities) of states.…”

“…Here we discuss the space of ground states, for lattices of two different numbers of nodes: N = 25 and N = 36. The triangular lattice was also one of the systems analysed in our previous paper [10]. There we have shown that it is possible to pass by one-spin flip from one ground state into another.…”

“…In our previous paper [10] it was shown that for the triangular lattice with N = 25 nodes the whole set of ground states can be divided into three equally sized subsets which were obtained by means of the breadth-first search (BFS) algorithm [19]. The action of this algorithm is limited to the transitions between states, mutually linked by one-spin flips.…”

“…The same procedure is applied for remaining classes (for details see Ref. [10]). The third column in Table 1 contains information about the number of states N c belonging to a given class.…”

“…This is equivalent to the classification according to the node degree. It was also shown [10] that in the case of a triangular lattice of size 25 the space of ground states can be divided into three equally sized subsets, within which it is possible to move through all states changing an orientation of only one spin between two subsequent ground states. This result indicates that for this system a kind of a constant of motion exists, which has three different values for those three subsets.…”