Determining the validity of cumulant expansions for central spin models

Fowler-Wright, Piper and Arnardóttir, Kristín B. and Kirton, Peter and Lovett, Brendon W. and Keeling, Jonathan (2023) Determining the validity of cumulant expansions for central spin models. Physical Review Research, 5 (3). 033148. ISSN 2643-1564 (https://doi.org/10.1103/PhysRevResearch.5.033148)

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Abstract

For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle N → ∞ limit, and that higher-order cumulant expansions of the Heisenberg equations converge to this same limit whilst providing improved approximations at finite N. Here we show that this is in fact not always the case. Instead, whether mean-field theory correctly describes the large-N limit depends on how the model parameters scale with N, and we show that convergence of cumulant expansions may be non-uniform across even and odd orders. Further, even when a higher-order cumulant expansion does recover the correct limit, the error is not monotonic with N and may exceed that of mean-field theory.

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