Determining the validity of cumulant expansions for central spin models

TL;DR

This paper investigates the validity of cumulant expansions in central spin models, revealing that their convergence to mean-field theory depends on parameter scaling and can be non-uniform, challenging common assumptions.

Contribution

It demonstrates that cumulant expansions do not always converge to mean-field theory in large systems and that their accuracy varies with model parameters and order.

Findings

Mean-field theory may not always be exact in large-$N$ limits.

Cumulant expansion convergence depends on parameter scaling with $N$.

Higher-order cumulant errors can be non-monotonic and larger than mean-field errors.

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.