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Quantum Thermodynamics

Energy gaps of Hamiltonians from graph Laplacians

arXiv
Authors: Abbas Al-Shimary, Jiannis K. Pachos

Year

2010

Paper ID

10777

Status

Preprint

Abstract Read

~2 min

Abstract Words

113

Citations

N/A

Abstract

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. By employing simple examples we illustrate how the generalised Cheeger inequalities can successfully identify gapped or gapless phases and we comment on the computational complexity of this approach.

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  • This paper contributes to the Quantum Thermodynamics research area in the Quantum Articles archive.
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  • 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.

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References & Citation Signals

[1] DOI https://doi.org/arXiv:1010.4130 [2] arXiv https://arxiv.org/abs/1010.4130 [3] Publisher https://arxiv.org/abs/1010.4130

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