  Monte Carlo simulations are useful tools for modeling quantum systems, but in
some cases they suffer from a sign problem, leading to an exponential slow down
in their convergence to a value. While solving the sign problem is generically
NP-hard, many techniques exist for mitigating the sign problem in specific
cases; in particular, the technique of deforming the Monte Carlo simulation's
plane of integration onto Lefschetz thimbles (complex hypersurfaces of
stationary phase) has seen significant success in the context of quantum field
theories. We extend this methodology to spin systems by utilizing spin coherent
state path integrals to re-express the spin system's partition function in
terms of continuous variables. Using some toy systems, we demonstrate its
effectiveness at lessening the sign problem in this setting, despite the fact
that the initial mapping to spin coherent states introduces its own sign
problem. The standard formulation of the spin coherent path integral is known
to make use of uncontrolled approximations; despite this, for large spins they
are typically considered to yield accurate results, so it is somewhat
surprising that our results show significant systematic errors. Therefore,
possibly of independent interest, our use of Lefschetz thimbles to overcome the
intrinsic sign problem in spin coherent state path integral Monte Carlo enables
a novel numerical demonstration of a breakdown in the spin coherent path
integral.
