Deconvolving the components of the sign problem

Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field $h$ introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices $M_\sigma(h)$, where $\sigma$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\langle {\cal S} \rangle$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\langle {\cal S} \rangle$. Does $\langle {\cal S} \rangle \rightarrow 0$ because {\it randomly} chosen field configurations have ${\rm det}\big(M(h)\big) < 0$, or does the `sign problem' arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with $|{\rm det}\big(M(h)\big)|$ might steer the simulation to a region of configuration space of $h$ where positive and negative determinants are equally likely, even though randomly chosen $h$ would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of $\langle {\cal S} \rangle$ in quantum simulations.