Subsystem Ginzburg-Landau and symmetry-protected topological order coexisting on a graph

Abstract

We write down and analyze a model demonstrating the co-existence of conventional symmetry-breaking order and symmetry-protected topological (SPT) order in the one-dimensional chain. When appropriately generalized to a model on a graph, the SPT and symmetry-breaking orders exist for each individual loop, or cycle, of the graph. It arises as a consequence of the kind of "global" symmetry operator responsible for SPT and the local-order parameter defining the Ginzburg-Landau order, both of which exist in our model and commute with the Hamiltonian. The anti-commuting character of these two-order parameters is responsible for the ground state degeneracy (GSD). As such operators and their anti-commuting relations can be defined for each independent loop, the GSD grows exponentially with the first Betti number for a graph.