Topological origin of the fermion sign problem

Monte Carlo simulations are a powerful tool for elucidating the properties of complex systems, but when applied to fermionic quantum systems, quantum Monte Carlo (QMC) algorithms suffer from the so-called negative sign problem, causing the computational effort to grow exponentially with problem size. Here we demonstrate that the fermion sign problem originates in topological properties of the configurations. Using the widely used determinantal approaches, that remove the trivial sign problem due to particle exchange, we prove that the negative sign of a configuration is a topological invariant-an imaginary time counterpart of the Aharonov-Anandan phase-and reduces to a Berry phase in the adiabatic limit. This provides an intriguing connection between QMC simulations and classification of topological states.