Complex saddle points and the sign problem in complex Langevin simulation

TL;DR

This paper analyzes why complex Langevin methods can converge to incorrect results when the path-integral weight has different phases across dominating complex saddle points. Through semiclassical analysis and connections to Lefschetz-thimble techniques and Dyson–Schwinger equations, it shows that multi-saddle interference prevents exact reconstruction of the original integral from complex Langevin dynamics. It then proposes a phase-reweighting prescription Theta, built from saddle-point data, to incorporate relative phases while preserving key equations in the semiclassical limit. Numerical tests on a one-site fermion model and a double-well potential model demonstrate improvements from the reweighting, though challenges remain, including potential sign problems for many-body systems and limitations for certain observables. The work highlights both a path to rehabilitate complex Langevin on multi-thimble problems and the cautions needed when applying it to dense QFT and many-body settings.

Abstract

We show that complex Langevin simulation converges to a wrong result, by relating it to the Lefschetz-thimble path integral, when the path-integral weight has different phases among dominant complex saddle points. Equilibrium solution of the complex Langevin equation forms local distributions around complex saddle points. Its ensemble average approximately becomes a direct sum of the average in each local distribution, where relative phases among them are dropped. We propose that by taking these phases into account through reweighting, we can solve the wrong convergence problem. However, this prescription may lead to a recurrence of the sign problem in the complex Langevin method for quantum many-body systems.

Figures (5)

• Figure 1: Fermion number $n$ as a function of chemical potential $\mu$ at $\beta U=30$ with and without reweighting.
• Figure 2: Average phase function $\langle \Theta(z_{\eta},\overline{z}_{\eta})\rangle_{\eta}$ as a function of chemical potential $\mu$ at $\beta U=30$.
• Figure 3: Partially integrated distribution $P_y$ and its moment $y^5P_y$ at $\alpha=1.4$.
• Figure 4: Expectation value of $\mathrm{i} z$ as a function of $\alpha$, with and without reweighting.
• Figure 5: Expectation value of $(\mathrm{i} z)^2$ as a function of $\alpha$, with and without reweighting.