Determining the validity of cumulant expansions for central spin models

• URL: http://arxiv.org/abs/2303.04410v3
• Date: Fri, 1 Sep 2023 17:37:04 GMT
• Title: Determining the validity of cumulant expansions for central spin models
• Authors: Piper Fowler-Wright and Krist\'in B. Arnard\'ottir and Peter Kirton and Brendon W. Lovett and Jonathan Keeling
• Abstract summary: For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $Ntoinfty$ limit. Here we show that this is in fact not always the case. 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $N\to\infty$ 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 the 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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