Structure of Lefschetz thimbles in simple fermionic systems

TL;DR

Using the Picard-Lefschetz framework, the paper analyzes Lefschetz thimbles in zero- and one-dimensional fermionic toy models (Gross-Neveu-like, Nambu-Jona-Lasinio-like, and a Chern-Simons–like system) to illuminate how complex saddles, thimble decompositions, and Stokes/anti-Stokes structures encode chiral symmetry breaking and topological sign problems. It demonstrates that the number and composition of contributing thimbles can change across chiral transitions and that Lee-Yang zeros align with anti-Stokes lines, signaling phase boundaries in the complex-coupling plane. The study reveals singular behavior near the chiral limit due to intricate cancellations between competing thimbles, necessitating summation over multiple thimbles, and shows how a CS-like topological term also requires a multi-thimble treatment. Together, these prototypical models provide concrete guidance for applying Lefschetz-thimble methods to QCD-like theories and for understanding complex-action path integrals in fermionic systems.

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.

Figures (16)

• Figure 1: Lefschetz thimbles $\mathcal{J}$ (solid lines) and their duals $\mathcal{K}$ (dashed lines) for simple one-dimensional integrals. Orange blobs are critical points of the action. Hatched areas are "good" regions where the integrand tends to zero. In both panels, the origin is a singularity of the flow.
• Figure 2: Lefschetz thimbles (black lines) and upward flow lines (red dashed lines) on the complex $z$-plane for the GN-like model in the chiral limit. The three orange blobs at $z=0$ and $z_\pm=\pm\sqrt{G-1}\approx \pm 0.55i$ are the critical points of $S(z)$, while the two red blobs at $z=\pm i$ are the points where $S(z)$ diverges. The background color scale describes $\mathrm{Re}\,S(z)$.
• Figure 3: Same as Figure \ref{['fg:GN1']} but with $|G|=1.1$.
• Figure 4: Stokes lines for the GN-like model with $p=1$ and $m=0$ (blue lines). The global topology of Lefschetz thimbles and their duals changes across the Stokes lines: $\mathcal{J}(z_\pm)$ and $\mathcal{K}(0)$ jump across the horizontal line, while $\mathcal{J}(0)$ and $\mathcal{K}(z_\pm)$ jump across the round curve. In the shaded area, only $\mathcal{J}(0)$ contributes to $Z_N(G,0)$. Outside the shaded area, all thimbles contribute. (The points $G=0$ and $1$ are excluded because the action $S(z)$ is singular at $G=0$ and the critical points merge at $G=1$.)
• Figure 5: Anti-Stokes line for the GN-like model with $p=1$ and $m=0$ (green curve), overlaid with Lee-Yang zeros for $N=40$ (red bullets) and the Stokes line in Figure \ref{['fg:stokeslines']} (blue curve).