Modelling equilibration of local many-body quantum systems by random graph ensembles

Daniel Nickelsen1,2,3 and Michael Kastner1,2

1National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa
2Institute of Theoretical Physics, Department of Physics, University of Stellenbosch, Stellenbosch 7600, South Africa
3African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa

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Abstract

We introduce structured random matrix ensembles, constructed to model many-body quantum systems with local interactions. These ensembles are employed to study equilibration of isolated many-body quantum systems, showing that rather complex matrix structures, well beyond Wigner's full or banded random matrices, are required to faithfully model equilibration times. Viewing the random matrices as connectivities of graphs, we analyse the resulting network of classical oscillators in Hilbert space with tools from network theory. One of these tools, called the maximum flow value, is found to be an excellent proxy for equilibration times. Since maximum flow values are less expensive to compute, they give access to approximate equilibration times for system sizes beyond those accessible by exact diagonalisation.

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[2] Lennart Dabelow, Patrick Vorndamme, and Peter Reimann, "Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure", Physical Review Research 2 3, 033210 (2020).

[3] Shoki Sugimoto, Ryusuke Hamazaki, and Masahito Ueda, "Test of Eigenstate Thermalization Hypothesis Based on Local Random Matrix Theory", arXiv:2005.06379.

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