Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral

TL;DR

The paper addresses the fermion sign problem in mean-field approximations at finite density by employing a Picard–Lefschetz Lefschetz-thimble framework that complexifies field variables and organizes the integral into real contributions from CK-symmetric saddle points. A central result is a formal decomposition that ensures the partition function $Z$ is real at any order when saddles come in CK-symmetric pairs, with residual sign from the Jacobian and inter-saddle cancellations discussed. The Airy integral is used as a transparent demonstration, producing real asymptotics in both $a>0$ and $a<0$ regimes. The method is then applied to a Polyakov-loop effective model of dense QCD, showing that CK-symmetric saddle points in the complexified space yield real observables and provide a systematic improvement beyond MFA, with insights into the large-$\mu$ limit and potential limitations due to multiple saddles.

Abstract

The fermion sign problem appearing in the mean-field approximation is considered, and the systematic computational scheme of the free energy is devised by using the Lefschetz-thimble method. We show that the Lefschetz-thimble method respects the reflection symmetry, which makes physical quantities manifestly real at any order of approximations using complex saddle points. The formula is demonstrated through the Airy integral as an example, and its application to the Polyakov-loop effective model of dense QCD is discussed in detail.

Figures (1)

• Figure 1: Flows at $h=0.1$ and $\mu=2$ around the $\mathcal{CK}$-symmetric saddle point (the black blob) $z^*$ in the $\mathrm{Re}(z_1)$-$\mathrm{Im}(z_2)$ plane. The red solid and green dashed lines are the Lefschetz thimble $\mathfrak{J}_*$ and its dual $\mathfrak{K}_*$ of $z^*$, respectively. Since $\mathfrak{K}_*$ intersects with $\mathrm{Im}(z_2)=0$, the integral over $\mathfrak{J}_*$ is equal to that over $\mathfrak{C}$.