Justification of the complex Langevin method with the gauge cooling procedure

Abstract

Recently there has been remarkable progress in the complex Langevin method, which aims at solving the complex action problem by complexifying the dynamical variables in the original path integral. In particular, a new technique called the gauge cooling was introduced and the full QCD simulation at finite density has been made possible in the high temperature (deconfined) phase or with heavy quarks. Here we provide a rigorous justification of the complex Langevin method including the gauge cooling procedure. We first show that the gauge cooling can be formulated as an extra term in the complex Langevin equation involving a gauge transformation parameter, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified by this extra term. However, this modification is shown not to affect the Fokker-Planck equation for the corresponding complex weight as long as observables are restricted to gauge invariant ones. Thus we demonstrate explicitly that the gauge cooling can be used as a viable technique to satisfy the convergence conditions for the complex Langevin method. We also discuss the "gauge cooling" in 0-dimensional systems such as vector models or matrix models.

Complex Langevin for Lattice QCD at $T=0$ and $\mu \ge 0$.

“…This means that gauge-cooling is non-trivial. Reference [15] indicates why this is not expected to change the physics. After taking −iδ S(U, µ)/δU l , the cyclic properties of the trace are used to rearrange the fermion term so that it re mains real for µ = 0 even after replacing the trace by a stochastic estimator.…”

“…Even when the CLE converges to a limiting distribution, it is not guaranteed to produce correct values for the observables unless certain conditions are satisfied [13,14,15,16]. The reason one needs to check the validity of the CLE for QCD is to first check the requirement that the gauge fields evolve over a bounded region, which appears to be true.…”

Gauge cooling for the singular-drift problem in the complex Langevin method - an application to finite density QCD

“…Thus, the complex Langevin simulation for QCD was realized in the QGP phase [4,5] and in the heavy dense limit [6] up to quite large quark chemical potential, which goes far beyond the applicable limit of conventional approaches such as reweighting or the Taylor expansion. The validity of the gauge cooling was also proved explicitly [7] based on the argument for justification of the CLM. See also other contributions to this volume for recent studies on the CLM [8,9,10,11].…”

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

“…While the gauge cooling technique [27] (See refs. [10,28] for its justification) has enlarged this range of applicability to the extent that finite density QCD either with heavy quarks [29][30][31] or in the deconfined phase [32,33] can now be investigated, it is not yet clear whether one can investigate it even in the confined phase with light quarks [34][35][36][37][38][39][40][41].…”