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arXiv:2207.09890

Numerical methods for localization

Rudolf A. Römer

Published Jul 20, 2022• 0 citations
arXivHTMLPDF

Abstract

Anderson localization provides a challenge to numerical approaches due to the inherent randomness, and hence absence of simple symmetries, in its discrete Hamiltonian representation. Numerous algorithmic approaches have been developed or adopted from other fields and have been collected in this encyclopedia entry. In the discussions below, the emphasis is on the numerical algorithms for localization, while the discussion of the physics of localization is referred to in companion entries by Elgart and Oganesyan in this encyclopedia.

DOI
10.48550/arxiv.2207.09890

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Anderson localization provides a challenge to numerical approaches due to the inherent randomness, and hence absence of simple symmetries, in its discrete Hamiltonian representation. Numerous algorithmic approaches have been developed or adopted from other fields and have been collected in this encyclopedia entry. In the discussions below, the emphasis is on the numerical algorithms for localization, while the discussion of the physics of localization is referred to in companion entries by Elgart and Oganesyan in this encyclopedia.
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Obtaining the symmetries of a model is a critical step towards developing an understanding and ultimately analytically or numerically solving the model. However, finding symmetries is generally extremely complicated, often being the result of insightful thinking. In this work, we complement human ingenuity with an algorithm. We leverage the classically efficient Clifford group to find symmetries for arbitrary many-body Hamiltonians via a graph representation. We demonstrate our method on random and physical Hamiltonians, with instances of up to one thousand qubits and demonstrate how our approach can provide deeper understanding of the model.
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