Structure of Lefschetz thimbles in simple fermionic systems

TL;DR

This paper explores the structure of Lefschetz thimbles in simple fermionic models, revealing their role in understanding phase transitions and solving sign problems, with implications for applying the method to QCD.

Contribution

It provides a detailed analysis of Lefschetz thimbles in fermionic systems, including phase diagrams and the behavior near chiral limits, advancing the application of Picard-Lefschetz theory to QCD-like models.

Findings

Thimble decomposition maps phase diagrams in fermionic models.

Anti-Stokes lines are linked to Lee-Yang zeros.

Multiple thimbles are necessary near the chiral limit due to cancellations.

Abstract

The Picard-Lefschetz theory offers a promising tool to solve the sign problem in QCD and other field theories with complex path-integral weight. In this paper the Lefschetz-thimble approach is examined in simple fermionic models which share some features with QCD. In zero-dimensional versions of the Gross-Neveu model and the Nambu-Jona-Lasinio model, we study the structure of Lefschetz thimbles and its variation across the chiral phase transition. We map out a phase diagram in the complex four-fermion coupling plane using a thimble decomposition of the path integral, and demonstrate an interesting link between anti-Stokes lines and Lee-Yang zeros. In the case of nonzero mass, it is shown that the approach to the chiral limit is singular because of intricate cancellation between competing thimbles, which implies the necessity to sum up multiple thimbles related by symmetry. We also consider a Chern-Simons theory with fermions in $0+1$-dimension and show how Lefschetz thimbles solve the complex phase problem caused by a topological term. These prototypical examples would aid future application of this framework to bona fide QCD.