Determining the validity of cumulant expansions for central spin models

For a model with many-to-one connectivity it is widely expected that mean-field theory captures the 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 we show that convergence of cumulant expansions may be non-uniform across even and odd orders. Further, even when a given order of cumulant expansion does recover the correct limit, the error of higher-order cumulant expansions is not monotonic with $N$ and may exceed that of mean-field theory.