Study of potential Hamiltonians for quantum graphity

In this work we show that the simple Hamiltonians used in Quantum Graphity models are highly degenerate, having multiple ground states that are not lattices. In order to assess the distance of the resulting graphs from a lattice graph, we propose a new measure of the equivalence of vertices in the graph. We then propose a Hamiltonian that has a rectangular lattice as a ground state that appears to be non-degenerate. The resulting graphs are close to being a rectangular lattice, and the defects from the perfect lattice seem to behave like particles of quantized mass that attract one another.