New insights into the problem with a singular drift term in the complex Langevin method

Understand

The complex Langevin method aims at performing path integral with a complex action numerically based on complexification of the original real dynamical variables.

• One of the poorly understood issues concerns occasional failure in the presence of logarithmic singularities in the action, which appear, for instance, from the fermion determinant in finite density QCD.
• We point out that the failure should be attributed to the breakdown of the relation between the complex weight that satisfies the Fokker-Planck equation and the probability distribution associated with the stochastic process.
• In fact, this problem can occur in general when the stochastic process involves a singular drift term.