Fast estimator of Jacobians in the Monte Carlo integration on Lefschetz thimbles

A solution to the sign problem is the so-called “Lefschetz thimble approach” where the domain of integration for field variables in the path integral is deformed from the real axis to a submanifold in the complex space. For properly chosen submanifolds (“thimbles”) the sign problem disappears or is drastically alleviated. The parametrization of the thimble by real coordinates requires the calculation of a Jacobian with a computational cost of order $\mathcal{O}(V^3)$, where $V$ is proportional to the spacetime volume. In this paper we propose two estimators for this Jacobian with a computational cost of order $\mathcal{O}(V)$. We discuss analytically the regimes where we expect the estimator to work and show numerical examples in two different models.

Figure 1

Statistical power for the parameter set $N=8$, $\widehat{m}=1$, $\widehat{\mu }=1$, ${\widehat{g}}^2=1/2$ as a function of flow time. The line indicates the average value of the last six points.

Figure 2

Statistical power of reweighting for $N=32$ in the low temperature limit (a) and the continuum limit (b).

Figure 3

Statistical power of the bosonic model on a $4^4$ lattice with parameters, $\widehat{m}=1$, $\widehat{\mu }=1.3$, $\lambda =1$ and $h=0.03+i0.003$ for the two estimators $W_1$ and $W_2$.