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.
Highlights
By means of typicality arguments, equilibration of isolated quantum systems has been rigorously proved to occur under rather general conditions [1,2,3,4,5,6,7,8]
We introduced and studied various random graph ensembles, with the aim of modelling equilibration in a spin system with a given degree of locality
(b) Based on these observations we introduce the random graph ensembles EXA, BRF, BRC, BVF, and REG, which incorporate degree distribution and bandwidth to a certain extent
Summary
By means of typicality arguments, equilibration of isolated quantum systems has been rigorously proved to occur under rather general conditions [1,2,3,4,5,6,7,8] It remains a key open problem to identify which properties of quantum systems are responsible for ensuring that equilibration timescales are physically realistic [9,10,11,12,13,14,15,16,17,18]. As in virtually all other problems of many-body quantum physics, the exponential growth of Hilbert space dimension with system size renders direct studies infeasible already for moderate system sizes To circumvent this difficulty in models of nuclei of heavy atoms, Wigner proposed the use of random matrices as Hamiltonians that statistically share certain properties of the nuclei [19,20,21]. Gaussian random matrices are a convenient choice, as they allow for the analytic calculation of ensemble-averaged properties like limiting distributions of eigenvalue statistics, i.e. Wigner’s semi-circle
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