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

Abstract

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. We show, however, in a simple example that there exists a parameter region in which the method works, although the standard reweighting method is hardly applicable.

Combining the complex Langevin method and the generalized Lefschetz-thimble method

“…As in the case of Monte Carlo methods, O(z) CLM can be replaced with a long-time average of the observable calculated for the generated configurations z(t) if ergodicity holds for the Langevin time-evolution. The derivation of the equality (2.4) uses integration by parts, which can be justified if the distribution of z falls off fast enough in the imaginary directions [7,8] as well as near the singularities of the drift term if they exist [9,40]. Recently [10], a subtlety in the use of time-evolved observables in the original argument [7,8] was pointed out, and the derivation of the equality (2.4) has been refined taking account of this subtlety.…”

“…Then, under certain conditions, one can show [7,8] that the expectation values of the observables calculated this way at some Langevin time are equal to the expectation values of the observables for the original real variables with a complex weight, which evolves with the Langevin time following the Fokker-Planck equation. If this is the case, one can obtain the desired expectation values in the long Langevin-time limit [9]. Recently, a subtlety in the use of time-evolved observables in the original argument [7,8] was recognized [10].…”

Toward solving the sign problem with path optimization method

“…It should be noted that the singular point in S at z = −iα does not cause any trouble. The factor of integrand (z + iα) p leads to the action term of −p log(z + iα) and causes the singular drift term, the drift term which diverges at z = −iα, in the complex Langevin method [8]. However, exp(−S) is analytic at this point as long as p is taken to be a positive integer, then it is not necessary to care in the path optimization method.…”

“…The complex Langevin method is based on the stochastic quantization and then we are free from the complex weight. Therefore, the sign problem does not appear, but it is well known that the complex Langevin method sometimes provides us wrong results when the drift term shows singular behavior in the Langevin-time evolution [8]. In comparison, the Lefschetz-thimble path-integral method is based on the Picard-Lefschetz theory [9] and thus it is within the standard path-integral formulation.…”

“…ExampleIn this article, we consider the following partition function as an example to demonstrate the path optimization method. The actual form of partition function [8] is…”