Phase Unwrapping and One-Dimensional Sign Problems

TL;DR

This paper investigates phase unwrapping techniques to address sign problems in path integrals within scalar field theory, combining them with cumulant expansion to improve signal quality but encountering sensitivity issues due to discretization artifacts.

Contribution

It introduces a novel approach combining phase unwrapping with cumulant expansion to approximate path integrals with sign problems, highlighting challenges with numerical stability.

Findings

Improved signal-to-noise ratios using unwrapped phases.

Cumulant expansion errors are sensitive to phase unwrapping parameters.

Discretization artifacts cause numerical instability near phase singularities.

Abstract

Sign problems in path integrals arise when different field configurations contribute with different signs or phases. Phase unwrapping describes a family of signal processing techniques in which phase differences between elements of a time series are integrated to construct non-compact unwrapped phase differences. By combining phase unwrapping with a cumulant expansion, path integrals with sign problems arising from phase fluctuations can be systematically approximated as linear combinations of path integrals without sign problems. This work explores phase unwrapping in zero-plus-one-dimensional complex scalar field theory. Results with improved signal-to-noise ratios for the spectrum of scalar field theory can be obtained from unwrapped phases, but the size of cumulant expansion truncation errors is found to be undesirably sensitive to the parameters of the phase unwrapping algorithm employed. It is argued that this numerical sensitivity arises from discretization artifacts that become large when phases fluctuate close to singularities of a complex logarithm in the definition of the unwrapped phase.