Fermionic Sign Problem Minimization by Constant Path Integral Contour Shifts

Abstract: The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hybrid Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases.