Study of potential Hamiltonians for quantum graphity

TL;DR

This paper investigates Hamiltonians in Quantum Graphity models, revealing degeneracy issues, and proposes a new Hamiltonian with a near-lattice ground state where defects behave like particles with quantized mass.

Contribution

The authors introduce a new measure of vertex equivalence and propose a Hamiltonian that yields a non-degenerate, lattice-like ground state in Quantum Graphity models.

Findings

Existing Hamiltonians are highly degenerate with non-lattice ground states.

A new Hamiltonian produces a near-lattice ground state with few defects.

Defects behave like particles with quantized mass and mutual attraction.

Abstract

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.