432 research outputs found
Unification of the complex Langevin method and the Lefschetz-thimble method
Recently there has been remarkable progress in solving the sign problem,
which occurs in investigating statistical systems with a complex weight. The
two promising methods, the complex Langevin method and the Lefschetz thimble
method, share the idea of complexifying the dynamical variables, but their
relationship has not been clear. Here we propose a unified formulation, in
which the sign problem is taken care of by both the Langevin dynamics and the
holomorphic gradient flow. We apply our formulation to a simple model in three
different ways and show that one of them interpolates the two methods by
changing the flow time.Comment: 8 pages, 12 figures, presented at the 35th International Symposium on
Lattice Field Theory (Lattice 2017), 18-24 June 2017, Granada, Spai
Complex Langevin simulation of QCD at finite density and low temperature using the deformation technique
We study QCD at finite density and low temperature by using the complex
Langevin method. We employ the gauge cooling to control the unitarity norm and
introduce a deformation parameter in the Dirac operator to avoid the
singular-drift problem. The reliability of the obtained results are judged by
the probability distribution of the magnitude of the drift term. By making
extrapolations with respect to the deformation parameter using only the
reliable results, we obtain results for the original system. We perform
simulations on a lattice and show that our method works well even
in the region where the reweighting method fails due to the severe sign
problem. As a result we observe a delayed onset of the baryon number density as
compared with the phase-quenched model, which is a clear sign of the Silver
Blaze phenomenon.Comment: 8 pages, 6 figures, presented at the 35th International Symposium on
Lattice Field Theory (Lattice 2017), 18-24 June 2017, Granada, Spai
Testing the criterion for correct convergence in the complex Langevin method
Recently the complex Langevin method (CLM) has been attracting attention as a
solution to the sign problem, which occurs in Monte Carlo calculations when the
effective Boltzmann weight is not real positive. An undesirable feature of the
method, however, was that it can happen in some parameter regions that the
method yields wrong results even if the Langevin process reaches equilibrium
without any problem. In our previous work, we proposed a practical criterion
for correct convergence based on the probability distribution of the drift term
that appears in the complex Langevin equation. Here we demonstrate the
usefulness of this criterion in two solvable theories with many dynamical
degrees of freedom, i.e., two-dimensional Yang-Mills theory with a complex
coupling constant and the chiral Random Matrix Theory for finite density QCD,
which were studied by the CLM before. Our criterion can indeed tell the
parameter regions in which the CLM gives correct results.Comment: 16 pages, 2 figures; (v2) reference and comment added; (v3) minor
revision; (v4) final version published in JHE
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