Energy gaps of Hamiltonians from graph Laplacians

Understand

The Cheeger inequalities give an upper and lower bound on the spectral gap of discrete Laplacians defined on a graph in terms of the geometric characteristics of the graph.

• We generalise this approach and we employ it to determine if a given discrete Hamiltonian with non-positive elements is gapped or not in the thermodynamic limit.
• First, we define the graph that corresponds to such a generic Hamiltonian.
• Then we present a suitable generalisation of the Cheeger inequalities that overcomes scaling deficiencies of the original version.