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 far 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.

25 pages, no figures; (v2) reference and comments added, improved notations in sections 4,5 and appendix

References in corpus (6)

• Stochastic quantization at finite chemical potential
• New insights into the problem with a singular drift term in the complex Langevin method
• Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion
• Full simulation of chiral Random Matrix Theory at non-zero chemical potential by Complex Langevin
• Comparison of complex Langevin and mean field methods applied to effective Polyakov line models
• Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method

Cited by in corpus (25)

• Review on novel methods for lattice gauge theories
• The argument for justification of the complex Langevin method and the condition for correct convergence
• Complex Langevin and other approaches to the sign problem in quantum many-body physics
• Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model
• Calculating the EoS of the dense quark-gluon plasma using the Complex Langevin equation
• Applying Complex Langevin Simulations to Lattice QCD at Finite Density
• Progress in the numerical studies of the type IIB matrix model
• Complex Langevin calculations in finite density QCD at large $\mu /T$ with the deformation technique
• Testing the criterion for correct convergence in the complex Langevin method
• Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model
• Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel
• Complex Langevin Simulations of Zero-dimensional Supersymmetric Quantum Field Theories
• Complex spectrum of finite-density lattice QCD with static quarks at strong coupling
• On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region
• Complex Langevin study for polarons in an attractively interacting one-dimensional two-component Fermi gas
• Real-time correlators in 3+1D thermal lattice gauge theory
• Test for a universal behavior of Dirac eigenvalues in the complex Langevin method
• Lefschetz thimble-inspired weight regularizations for complex Langevin simulations
• Color superconductivity on the lattice -- analytic predictions from QCD in a small box
• Complex Langevin Dynamics and Supersymmetric Quantum Mechanics
• Alternating Descent Method for Gauge Cooling of Complex Langevin Simulations
• On the validity of the complex Langevin method near the deconfining phase transition in QCD at finite density
• A projected complex Langevin sampling method for bosons in the canonical and microcanonical ensembles
• Correctness criteria for complex Langevin