[Paper Review] Numerical methods for the sign problem in Lattice Field Theory

Lorenzo Bongiovanni|arXiv (Cornell University)|Mar 10, 2016

Theoretical and Computational Physics16 references3 citations

TL;DR

This PhD thesis investigates three leading approaches to overcome the sign problem in lattice field theory: complex Langevin dynamics with gauge cooling, Lefschetz thimbles, and the density of states method using the LLR algorithm. It demonstrates the viability of complex Langevin dynamics for SU(3) Yang-Mills theory with a θ-term and compares it with thimbles on toy models, while applying the density of states method to the relativistic Bose gas at finite chemical potential, showing good agreement with analytical results.

ABSTRACT

The great majority of algorithms employed in the study of lattice field theory are based on Monte Carlo's importance sampling method, i.e. on probability interpretation of the Boltzmann weight. Unfortunately in many theories of interest one cannot associated a real and positive weight to every configuration, that is because their action is explicitly complex or because the weight is multiplied by some non positive term. In this cases one says that the theory on the lattice is affected by the sign problem. An outstanding example of sign problem preventing a quantum field theory to be studied, is QCD at finite chemical potential. Whenever the sign problem is present, standard Monte Carlo methods are problematic to apply and, in general, new approaches are needed to explore the phase diagram of the complex theory. Here we will review three of the main candidate methods to deal with the sign problem, namely complex Langevin dynamics, Lefschetz thimbles and density of states method. We will first study complex Langevin dynamics, combined with the gauge cooling method, on the one-dimensional Polyakov line model, and then we will apply it to pure gauge Yang-Mills theory with a topological theta-term. It follows a comparison between complex Langevin dynamics and the Lefschetz thimbles method on three toy models, which are the quartic model, the U(1) one-link model with a mu dependent determinant, and the SU(2) non abelian one-link model with complex beta parameter. Lastly, we introduce the density of state method, based on the LLR algorithm, and we will employ it in the study of the relativistic Bose gas at finite chemical potential.

Motivation & Objective

• To address the sign problem in lattice field theory, which prevents standard Monte Carlo methods from being applied when the action is complex or the weight is not positive definite.
• To evaluate and compare three promising numerical methods—complex Langevin dynamics, Lefschetz thimbles, and density of states—for handling theories with a sign problem.
• To test the robustness and accuracy of complex Langevin dynamics with gauge cooling in non-Abelian gauge theories, particularly SU(3) Yang-Mills with a θ-term.
• To apply the density of states method via the LLR algorithm to the relativistic Bose gas at finite chemical potential, a system with a severe sign problem.
• To provide a systematic comparison of complex Langevin and Lefschetz thimbles on benchmark models, including the quartic model, U(1) one-link model, and SU(2) one-link model.

Proposed method

• Complex Langevin dynamics is employed by complexifying the field variables and evolving them stochastically according to a Fokker-Planck equation, with gauge cooling used to stabilize the dynamics in non-Abelian theories.
• Lefschetz thimbles are constructed by finding critical points of the action in the complexified manifold and integrating along manifolds where the imaginary part of the action is constant, ensuring a real and positive measure.
• The density of states method uses the LLR algorithm to compute the density of states from a sequence of constrained simulations, enabling the reconstruction of the partition function from highly oscillatory integrals.
• For the relativistic Bose gas, the method involves computing the free energy difference via the LLR algorithm by fitting the logarithmic derivative of the partition function with respect to a control parameter.
• In complex Langevin simulations, topological charge is monitored via gradient flow to extract the topological susceptibility and verify convergence.
• The comparison between complex Langevin and thimbles is performed by analyzing the distribution of field configurations and the contribution of individual thimbles to the path integral.

Experimental results

Research questions

• RQ1 Can complex Langevin dynamics with gauge cooling produce reliable results for SU(3) Yang-Mills theory with a θ-term, where the sign problem is severe?
• RQ2 How do the results of complex Langevin dynamics compare with those obtained using Lefschetz thimbles on the quartic model, the U(1) one-link model with μ-dependent determinant, and the SU(2) one-link model?
• RQ3 To what extent does the density of states method, based on the LLR algorithm, accurately compute the partition function and free energy in the relativistic Bose gas at finite chemical potential?
• RQ4 What is the role of gauge cooling in stabilizing complex Langevin dynamics in non-Abelian gauge theories, and how does it affect the convergence of topological charge distributions?
• RQ5 How do the contributions of different Lefschetz thimbles influence the path integral, and which fixed points dominate the measure in the presence of complex actions?

Key findings

• Complex Langevin dynamics with gauge cooling successfully reproduces the expected topological charge distribution in SU(3) Yang-Mills theory with a θ-term, showing agreement with HMC results for imaginary θ.
• The topological charge distribution for real and imaginary θ values exhibits opposite curvature, confirming the expected CP-odd behavior and validating the method’s consistency.
• In the quartic model, complex Langevin dynamics correctly captures the distribution of field configurations, with the stable thimbles matching the dominant contributions observed in the simulation.
• For the U(1) one-link model, the method identifies the correct contributing thimbles, and the complex Langevin results align with the thimble-based predictions, particularly in the presence of μ-dependent determinants.
• In the SU(2) one-link model with complex β, the complex Langevin simulation with gauge cooling converges to the correct distribution, and the thimble analysis confirms the dominance of a single stable thimble.
• The density of states method using the LLR algorithm accurately computes the free energy difference in the relativistic Bose gas, with results matching analytical mean field theory predictions within statistical errors.