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The Cheeger inequalities give an upper and lower bound on the spectral gap ofdiscrete Laplacians defined on a graph in terms of the geometriccharacteristics of the graph. We generalise this approach and we employ it todetermine if a given discrete Hamiltonian with non-positive elements is gappedor not in the thermodynamic limit. First, we define the graph that correspondsto such a generic Hamiltonian. Then we present a suitable generalisation of theCheeger inequalities that overcomes scaling deficiencies of the originalversion. By employing simple examples we illustrate how the generalised Cheegerinequalities can successfully identify gapped or gapless phases and we commenton the computational complexity of this approach.

Energy gaps of Hamiltonians from graph Laplacians