Sign Problem and the Worldvolume Hybrid Monte Carlo Method

Abstract

The sign problem has been a major obstacle to first-principles calculations of a variety of important physical systems. Typical examples are finite-density Quantum Chromodynamics, some statistical systems such as strongly correlated electron systems and frustrated spin systems, and the real-time dynamics of quantum many-body systems.

There have been proposed various approaches to the problem, among which the Lefschetz thimble method uses a continuous deformation of the integral surface into the complex space to tame the oscillatory behavior of integrals. The “tempered Lefschetz thimble method” (TLT method) implements the tempering algorithm with respect to the deformation parameter, and is the first algorithm that simultaneously solves the sign and ergodicity problems, which have been existing intrinsically in the Lefschetz thimble method.

The shortcomings of the TLT method regarding high numerical cost are resolved in the “Worldvolume Hybrid Monte Carlo method” (WV-HMC method), where Hybrid Monte Carlo updates are performed on a continuous accumulation of deformed surfaces (worldvolume). The WV-HMC algorithm may be a promising method towards solving the sign problem due to its versatility and reliability.