[Paper Review] Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

Andrei Alexandru, Gökçe Başar+3 more|arXiv (Cornell University)|Dec 29, 2015

Theoretical and Computational Physics4 citations

TL;DR

This paper proposes using Monte Carlo simulations over complex manifolds other than Lefschetz thimbles—specifically, manifolds obtained by flowing the tangent space of a single critical point—to circumvent the sign problem in fermionic field theories. It demonstrates that sampling over the tangent space (zero flow) or moderately flowed manifolds enables accurate computation of partition functions and observables in a 0+1D fermionic model, including in the continuum and low-temperature limits where standard thimble methods fail.

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.

Motivation & Objective

• To address the sign problem in strongly correlated fermionic systems, particularly where Lefschetz thimbles are ineffective due to multiple contributing thimbles.
• To overcome the practical difficulty of identifying and sampling multiple critical points and their associated thimbles in realistic field theories.
• To explore alternative complex integration manifolds that preserve the topological equivalence to the original real integration domain.
• To demonstrate that the tangent space of a single critical point, or its flowed version, can effectively sample all relevant thimbles without explicit identification of all critical points.
• To provide a practical, computationally feasible alternative to thimble-based methods for models with severe sign problems, especially in the continuum and low-temperature limits.

Proposed method

• Proposes a family of complex integration manifolds parametrized by a flow time, interpolating between the tangent space at a single critical point and the full union of contributing thimbles.
• Uses the contraction algorithm (a type of complex flow) to deform the tangent space of one critical point into a manifold that captures contributions from multiple thimbles.
• Employs a Metropolis-Hastings Monte Carlo algorithm to sample field configurations on these flowed manifolds, with the action's imaginary part minimized to reduce sign oscillations.
• For the 0+1D fermionic model, the method uses the tangent space (zero flow) or moderate flow to achieve accurate results, even in the continuum and low-temperature limits.
• Demonstrates that the tangent space manifold is homologous to the original real integration domain, ensuring topological equivalence and correct partition function evaluation.
• Introduces a modified algorithm that starts from the real space ℝ^N instead of the tangent space, guaranteeing convergence to the correct thimble combination at the cost of increased flow time.

Experimental results

Research questions

• RQ1 Can Monte Carlo simulations over manifolds other than Lefschetz thimbles effectively solve the sign problem in fermionic models?
• RQ2 Is the tangent space of a single critical point sufficient to sample contributions from multiple thimbles in a complex field space?
• RQ3 Can a flow-based deformation of the tangent space capture the full partition function without identifying all critical points?
• RQ4 Does the method remain effective in the continuum and low-temperature limits, where sign problems are most severe?
• RQ5 How does the multimodal nature of the distribution on the tangent space affect sampling efficiency and convergence?

Key findings

• The zero-flow (tangent space) method successfully computes the exact partition function and fermion condensate in a 0+1D fermionic model across a wide parameter range, including in the continuum limit and at low temperatures.
• For the parameter set μ=1.688, the tangent space method reproduces the exact phase transition in the fermion condensate, matching the analytical result.
• At low temperatures where the tangent space method fails due to high sign oscillations and low acceptance rates (~15%), a moderate flow (T_flow=0.5) restores convergence and accuracy.
• The method samples multiple thimbles simultaneously, as evidenced by the histogram of the imaginary action showing contributions from several thimble regions despite high barriers.
• The flow-based approach avoids the need to identify and sample all critical points explicitly, offering a practical alternative to full thimble integration.
• The method’s success relies on the tangent space being in the same homology class as the original real domain, ensuring topological correctness of the path integral.