Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

TL;DR

The paper tackles the Monte Carlo sign problem for complex actions by extending Lefschetz-thimble ideas to a continuum of complex manifolds $M(T_{ ext{flow}})$ obtained by flowing the tangent space at a critical point. Sampling is performed with the effective action $S_R- ext{log}|J|$, where $J$ is the Jacobian of the flow, and the method becomes exact as $T_{ ext{flow}} o ext{infty}$, yielding the full thimble contribution. In a 0+1D Thirring model, zero-flow testing on the main tangent space reproduces exact results in several regimes, while finite flow times can incorporate multiple thimbles without enumerating all critical points, albeit with multimodal sampling challenges at low temperature or strong sign problems. This approach links sign problems to multimodal sampling and offers a practical route to access continuum and low-temperature physics in fermionic theories, while highlighting the need for advanced sampling strategies in more challenging parameter regions.

Abstract

We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action. We describe a family of such manifolds that interpolate between the tangent space at one critical point, where the sign problem is milder compared to the real plane but in some cases still severe, and the union of relevant thimbles, where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling. We exemplify this approach using a simple 0 + 1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefshetz thimbles was elusive.

Figures (7)

• Figure 1: Average phase $\langle e^{-i S_I}\rangle_R$ in the reweighting method for $N=8, \hat{m}=1$ and $\hat{g}^2=1/6$ (blue) and $\hat{g}^2=1/2$ (red).
• Figure 2: Condensate value as a function of chemical potential $\hat{\mu}$ (left) and difference from the exact condensate value (right). The parameters are $N=8, \hat{m}=1, \hat{g}^2=1/6$. The blue points are the result of the integration over the tangent space while the red points were obtained in Alexandru:2015xva by using the contraction algorithm with flow time $T_{flow}=2$ and, essentially, correspond to the integration over the main thimble.
• Figure 3: Complex $\hat{\phi}=\frac{1}{N}\sum_t \hat{\phi}_t$ plane. The full (blue) circles are the critical points of $S$, the empty (blue) circles the log singularities of $S$ (points where the fermion determinant vanishes). The full (blue) line are five of the thimbles. The red dashed line is the main tangent space. The red, purple and violet lines are the result of flowing the tangent space by $T_{flow}=0.01, 0.05$ and $0.5$, respectively.
• Figure 4: Real (left panel) and imaginary (right panel) parts of the action along the curves shown in Fig. \ref{['fig:bigpicture']}. As the flow increases the imaginary part of the action becomes more (piecewise) constant while the barriers on the real part become more prominent.
• Figure 5: Condensate value as a function of chemical potential $\hat{\mu}$ (left) and difference from the exact condensate value (right). The parameters are $N=8, \hat{m}=1, \hat{g}^2=1/2$. The blue points are the result of the integration over the tangent space while the red points were obtained in Alexandru:2015xva by using the contraction algorithm with flow time $T_{flow}=2$ and, essentially, correspond to the integration over the main thimble.