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We show that complex Langevin simulation converges to a wrong result, byrelating it to the Lefschetz-thimble path integral, when the path-integralweight has different phases among dominant complex saddle points. Equilibriumsolution of the complex Langevin equation forms local distributions aroundcomplex saddle points. Its ensemble average approximately becomes a direct sumof the average in each local distribution, where relative phases among them aredropped. We propose that by taking these phases into account throughreweighting, we can solve the wrong convergence problem. However, thisprescription may lead to a recurrence of the sign problem in the complexLangevin method for quantum many-body systems.

Complex saddle points and the sign problem in complex Langevin simulation