Controlling complex Langevin dynamics at finite density

Abstract

At nonzero chemical potential the numerical sign problem in lattice field theory limits the use of standard algorithms based on importance sampling. Complex Langevin dynamics provides a possible solution, but it has to be applied with care. In this review, we first summarise our current understanding of the approach, combining analytical and numerical insight. In the second part we study SL(N,C) gauge cooling, which was introduced recently as a tool to control complex Langevin dynamics in nonabelian gauge theories. We present new results in Polyakov chain models and in QCD with heavy quarks and compare various adaptive cooling implementations.

invited mini-review (with new results) to appear in a special issue of EPJA, 15 pages; v2: clarification added at various places

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Cited by in corpus (100)

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• Complex Langevin Dynamics for chiral Random Matrix Theory
• Critical line of 2+1 flavor QCD: Toward the continuum limit
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• Complex Langevin and other approaches to the sign problem in quantum many-body physics
• Complex Langevin dynamics and zeroes of the fermion determinant
• Exploring QCD matter in extreme conditions with Machine Learning
• Holographic QCD in the Veneziano limit at finite Magnetic Field and Chemical Potential
• Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density
• Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion
• Parallel tempering algorithm for integration over Lefschetz thimbles
• Complex Langevin dynamics for dynamical QCD at nonzero chemical potential: a comparison with multi-parameter reweighting
• Lefschetz thimbles and stochastic quantisation: Complex actions in the complex plane
• Localised distributions and criteria for correctness in complex Langevin dynamics
• Thimble regularization at work: from toy models to chiral random matrix theories
• Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice
• The QCD phase diagram in the limit of heavy quarks using complex Langevin dynamics
• Justification of the complex Langevin method with the gauge cooling procedure
• Applying the tempered Lefschetz thimble method to the Hubbard model away from half-filling
• Calculating the EoS of the dense quark-gluon plasma using the Complex Langevin equation
• Applying Complex Langevin Simulations to Lattice QCD at Finite Density
• Heavy dense QCD and nuclear matter from an effective lattice theory
• Complex Langevin simulations and the QCD phase diagram: Recent developments
• Lattice QCD Equation of State for Nonvanishing Chemical Potential by Resumming Taylor Expansion
• Thermodynamics of strong-interaction matter from Lattice QCD
• Dynamical stabilisation of complex Langevin simulations of QCD
• Path integral contour deformations for noisy observables
• Gauge cooling for the singular-drift problem in the complex Langevin method --a test in Random Matrix Theory for finite density QCD
• Combining the complex Langevin method and the generalized Lefschetz-thimble method
• Worldvolume approach to the tempered Lefschetz thimble method
• Bose-Einstein condensation and Silver Blaze property from the two-loop $\Phi$-derivable approximation
• Understanding of QCD at high density from Z3-symmetric QCD-like theory
• Quark number densities at imaginary chemical potential in $N_f=2$ lattice QCD with Wilson fermions and its model analyses
• Analytical structure of the equation of state at finite density: Resummation versus expansion in a low energy model
• Complex Langevin calculations in finite density QCD at large $\mu /T$ with the deformation technique
• A complex path around the sign problem
• Progress in complex Langevin simulations of full QCD at nonzero density
• Deconfinement transition line with the Complex Langevin equation up to $\mu /T\sim 5$
• Determination of the strength of the vector-type four-quark interaction in the entanglement Polyakov-extended Nambu-Jona-Lasino model
• Complex Langevin simulation of quantum vortices in a Bose-Einstein condensate
• Reweighting complex Langevin trajectories
• Complex Langevin calculations in QCD at finite density
• Phase structure of two-color QCD at real and imaginary chemical potentials; lattice simulations and model analyses
• New Way to Resum the Lattice QCD Taylor Series Equation of State at Finite Chemical Potential
• Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel
• Complex Langevin method applied to the 2D $SU(2)$ Yang-Mills theory
• Stable solvers for real-time Complex Langevin
• On the gauge invariant path-integral measure for the overlap Weyl fermions in $\underline{16}$ of SO(10)
• Non-Abelian chiral instabilities at high temperature on the lattice
• Why is the mission impossible? -- Decoupling the mirror Ginsparg-Wilson fermions in the lattice models for two-dimensional abelian chiral gauge theories
• Complex spectrum of finite-density lattice QCD with static quarks at strong coupling
• Strong Coupling Lattice QCD in the Continuous Time Limit
• Equation of state for cold and dense heavy QCD
• Interplay between sign problem and Z_3 symmetry in three-dimensional Potts model
• Cooling Stochastic Quantization with colored noise
• Lattice QCD: bulk and transport properties of QCD matter
• Restricted phase-space approximation in real-time stochastic quantization
• Dual Formulation and Phase Diagram of Lattice QCD in the Strong Coupling Regime
• Extended Mean Field study of complex ${\varphi }^4$-theory at finite density and temperature
• Thermal properties of $\pi$ and $\rho$ meson
• Improved convergence of Complex Langevin simulations
• Distance between configurations in Markov chain Monte Carlo simulations
• Sign problem in $Z_3$-symmetric effective Polyakov-line model
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region
• Simple QED- and QCD-like Models at Finite Density
• Overview of the QCD phase diagram -- Recent progress from the lattice
• Complex Langevin Simulations of QCD at Finite Density -- Progress Report
• Complex Langevin for Lattice QCD at $T=0$ and $\mu \ge 0$
• Perturbative aspects of the deconfinement transition -- Physics beyond the Faddeev-Popov model
• The equation of state in two-, three-, and four-color QCD at non-zero temperature and density
• Stochastic quantization and diffusion models
• Classification of sign-problem-free relativistic fermions on the basis of the Majorana positivity
• Developments in lattice QCD for matter at high temperature and density
• Getting even with CLE
• Lefschetz thimble-inspired weight regularizations for complex Langevin simulations
• QCD at nonzero chemical potential: recent progress on the lattice
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• Fermion Bag Approach for Massive Thirring Model at Finite Density
• Alternating Descent Method for Gauge Cooling of Complex Langevin Simulations
• Extending complex Langevin simulations to full QCD at nonzero density
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models
• A hadron-quark hybrid model reliable for the EoS in ${\mu }_B\le 400$ MeV
• A new way of determining the Lattice QCD equation of state at a finite chemical potential
• Exploring the phase diagram of finite density QCD at low temperature by the complex Langevin method
• Cuprates and center vortices: A QCD confinement mechanism in a high-Tc context
• Performance of Complex Langevin Simulation in 0+1 dimensional massive Thirring model at finite density
• Solving the sign problem of two flavor scalar electrodynamics at finite chemical potential
• Probing the finite density equation of state of QCD via resummed perturbation theory
• Extrapolation for meson screening masses from imaginary to real chemical potential
• Nonanalyticity, sign problem and Polyakov line in Z3-symmetric heavy quark model at low temperature: Phenomenological model analyses
• Lattice String Field Theory: The linear dilaton in one dimension
• Comparison between Fermion Bag Approach and Complex Langevin Dynamics for Massive Thirring Model at Finite Density in 0 + 1 Dimensions
• Exploring the QCD phase diagram at finite density by the complex Langevin method on a $16^3\times 32$ lattice
• On the validity of complex Langevin method for path integral computations
• From pQCD to neutron stars: matching equations of state to constrain global star properties
• Complex Langevin for Lattice QCD
• Taming the pion condensation in QCD at finite baryon density