Hybrid Monte Carlo on Lefschetz Thimbles -- A study of the residual sign problem

TL;DR

This work develops a Hybrid Monte Carlo method for lattice theories defined on Lefschetz thimbles to address the residual sign problem in complex actions. By parameterizing thimble configurations via flow-direction and flow-time and enforcing tangential constraints through a second-order constraint-preserving integrator, the approach enables reweighting of the residual phase $e^{i\phi_z}$ with high fidelity. Applied to the complex $\lambda\phi^4$ model at finite density on a $L=4$ lattice, the residual phase averages exceed 0.99 across studied $\mu$ values and the resulting number densities agree with Complex Langevin simulations within errors. The results support the viability of thimble-based HMC for studying sign-problem afflicted theories and motivate extensions to larger systems and QCD-like models.

Abstract

We consider a hybrid Monte Carlo algorithm which is applicable to lattice theories defined on Lefschetz thimbles. In the algorithm, any point (field configuration) on a thimble is parametrized uniquely by the flow-direction and the flow-time defined at a certain asymptotic region close to the critical point, and it is generated by solving the gradient flow equation downward. The associated complete set of tangent vectors is also generated in the same manner. Molecular dynamics is then formulated as a constrained dynamical system, where the equations of motion with Lagrange multipliers are solved by the second-order constraint-preserving symmetric integrator. The algorithm is tested in the lambda phi^4 model at finite density, by choosing the thimbles associated with the classical vacua for subcritical and supercritical values of chemical potential. For the lattice size L=4, we find that the residual sign factors average to not less than 0.99 and are safely included by reweighting and that the results of the number density are consistent with those obtained by the complex Langevin simulations.

Figures (7)

• Figure 1: A fixed-point method to solve the constraint eq. (\ref{['eq:constraint-z']}).
• Figure 2: Monte Carlo histories of ${\rm Re} S[z]$ for $\mu=0.5$ and $0.9$ ($\kappa=1.0$, $\lambda=1.0$, $L=4$). In the course of the MC updates, the absolute values of ${\rm Im} S[z]$ were kept less than $1.0 \times 10^{-4}$.
• Figure 3: Monte Carlo histories of $\phi_z$ for $\mu=0.5$ and $0.9$ ($\kappa=1.0$, $\lambda=1.0$, $L=4$).
• Figure 4: The expectation values of $n[z]$ evaluated on the thimble 1-(a) ($\mu < \tilde{\mu}_c$). The errors are those estimated by the jack-knife method.
• Figure 5: The expectation values of $n[z]$ evaluated on the thimble 2-(b) ($\mu > \tilde{\mu}_c$). The errors are those estimated by the jack-knife method.