Frustration-free Hamiltonian with Topological Order on Graphs

It is commonly believed that models defined on a closed one-dimensional manifold cannot give rise to topological order. Here we construct frustration-free Hamiltonians which possess both symmetry protected topological order (SPT) on the open chain {\it and} multiple ground state degeneracy (GSD) that is unrelated to global symmetry breaking on the closed chain. Instead of global symmetry breaking, there exists a {\it local} symmetry operator that commutes with the Hamiltonian and connects the multiple ground states, reminiscent of how the topologically distinct ground states of the toric code are connected by various winding operators. Our model solved on an open chain demonstrates symmetry fractionalization as an indication of SPT order and on a general graph the GSD can be shown to scale with the first Betti number - a topological invariant that counts the number of independent cycles or one dimensional holes of the graph.