Reweighting complex Langevin trajectories

Although the complex Langevin method can solve the sign problem in simulations of theories with complex actions, the method will yield the wrong results if known validity conditions are not satisfied. We present a novel method to compute observables for a target ensemble by reweighting complex trajectories generated with the complex Langevin method for an auxiliary ensemble having itself a complex action. While it is imperative that the validity conditions be satisfied for the auxiliary ensemble, there are no such requirements for the target ensemble. This allows us to enlarge the applicability range of the complex Langevin method. We illustrate this at the hand of a one-dimensional partition function and two-dimensional strong-coupling QCD.

Figure 1

Results for $⟨x^2⟩$ as a function of $\alpha$ for the one-dimensional partition function (11). Comparison of CL (blue), RCL (red) and analytical (solid) results.

Figure 2

Comparison of RCL and PQR: Quark number density (top) and statistical error (bottom) in two-dimensional QCD as a function of $\mu$ for $\beta =0$ and $m=0.01$ on a $6\times 6$ lattice. The PQR data were computed using Markov chains with 2M configurations; the RCL results using 2M configurations evenly distributed along the auxiliary trajectory at ${\mu }_0=0.15$ and $m_0=0.1$ containing 10M configurations. For comparison, we also show the wrong CL results.