The Complex Langevin method: When can it be trusted?

Abstract

We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.

14 pages, including several eps figures and tables; clarification and minor corrections added, to appear in PRD

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• Can stochastic quantization evade the sign problem? -- the relativistic Bose gas at finite chemical potential
• Lattice simulations of real-time quantum fields
• Stochastic quantization at finite chemical potential
• Real-time gauge theory simulations from stochastic quantization with optimized updating
• Complex Langevin Equations and Schwinger-Dyson Equations
• Effective Potential for Complex Langevin Equations

Cited by in corpus (161)

• The phase diagram of dense QCD
• Properties of hot and dense matter from relativistic heavy ion collisions
• Simulating full QCD at nonzero density using the complex Langevin equation
• Complex Langevin: Etiology and Diagnostics of its Main Problem
• Review on novel methods for lattice gauge theories
• Gauge cooling in complex Langevin for QCD with heavy quarks
• Finite-density lattice QCD and sign problem: current status and open problems
• Controlling complex Langevin dynamics at finite density
• The argument for justification of the complex Langevin method and the condition for correct convergence
• Introductory lectures on lattice QCD at nonzero baryon number
• Complex Langevin Dynamics for chiral Random Matrix Theory
• Critical line of 2+1 flavor QCD: Toward the continuum limit
• Complex Langevin dynamics in the SU(3) spin model at nonzero chemical potential revisited
• Vacuum structure of bifundamental gauge theories at finite topological angles
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• Toward solving the sign problem with path optimization method
• Complex Langevin and other approaches to the sign problem in quantum many-body physics
• Onset Transition to Cold Nuclear Matter from Lattice QCD with Heavy Quarks
• Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model
• Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density
• Stability of complex Langevin dynamics in effective models
• The Hot QCD White Paper: Exploring the Phases of QCD at RHIC and the LHC
• Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion
• 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
• Complex Langevin dynamics and other approaches at finite chemical potential
• Complex Langevin and boundary terms
• Localised distributions and criteria for correctness in complex Langevin dynamics
• Full simulation of chiral Random Matrix Theory at non-zero chemical potential by Complex Langevin
• Path Integral Molecular Dynamics for Fermions: Alleviating the Sign Problem with the Bogoliubov Inequality
• Toward Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States
• Status of Complex Langevin
• Application of neural network to sign problem via path optimization method
• Justification of the complex Langevin method with the gauge cooling procedure
• Controlling Complex Langevin simulations of lattice models by boundary term analysis
• Does the complex Langevin method give unbiased results?
• New approach to lattice QCD at finite density; results for the critical end point on coarse lattices
• Complex Langevin Simulation of a Random Matrix Model at Nonzero Chemical Potential
• Complex saddle points and the sign problem in complex Langevin simulation
• Calculating the EoS of the dense quark-gluon plasma using the Complex Langevin equation
• Applying Complex Langevin Simulations to Lattice QCD at Finite Density
• Surmounting the sign problem in non-relativistic calculations: a case study with mass-imbalanced fermions
• Finite-temperature equation of state of polarized fermions at unitarity
• Complex Langevin simulations and the QCD phase diagram: Recent developments
• Control the model sign problem via path optimization method: Monte-Carlo approach to QCD effective model with Polyakov loop
• Complex Langevin analysis of the spontaneous symmetry breaking in dimensionally reduced super Yang-Mills models
• Third-order perturbative lattice and complex Langevin analyses of the finite-temperature equation of state of non-relativistic fermions in one dimension
• Thirring model at finite density in 0+1 dimensions with stochastic quantization: Crosscheck with an exact solution
• The QCD Critical Point : marching towards continuum
• An improvement in complex Langevin dynamics from a view point of Lefschetz thimbles
• Worldvolume approach to the tempered Lefschetz thimble method
• Progress in the numerical studies of the type IIB matrix model
• Diffusion Models as Stochastic Quantization in Lattice Field Theory
• 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
• Baryons in the Gross-Neveu model in 1+1 dimensions at finite number of flavors
• How to obtain thermostatted ring polymer molecular dynamics from exact quantum dynamics and when to use it
• Thermal quantum time-correlation functions from classical-like dynamics
• Thirring model at finite density in 2+1 dimensions with stochastic quantization
• New developments for dual methods in lattice field theory at non-zero density
• Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations
• Application of Bootstrap to $\theta$-term
• Complex Langevin calculations in QCD at finite density
• Polarized fermions in one dimension: density and polarization from complex Langevin calculations, perturbation theory, and the virial expansion
• Thermodynamics of spin-orbit coupled bosons in two dimensions from complex Langevin
• Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model
• QCD Critical Point: The Race is On
• Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel
• Complex Langevin method applied to the 2D $SU(2)$ Yang-Mills theory
• Kenneth Wilson and lattice QCD
• On the gauge invariant path-integral measure for the overlap Weyl fermions in $\underline{16}$ of SO(10)
• Pairing patterns in one-dimensional spin- and mass-imbalanced Fermi gases
• Complex Langevin: Boundary terms at poles
• Modifying partition functions: a way to solve the sign problem
• Lefschetz thimbles in fermionic effective models with repulsive vector-field
• Why is the mission impossible? -- Decoupling the mirror Ginsparg-Wilson fermions in the lattice models for two-dimensional abelian chiral gauge theories
• Phase diagram of the quantum O(2)-model in 2+1 dimensions
• Complex Langevin Simulations of Zero-dimensional Supersymmetric Quantum Field Theories
• Complex Langevin Dynamics in Large $N$ Unitary Matrix Models
• Exponential reduction of the sign problem at finite density in the 2+1D XY model via contour deformations
• Complex Langevin simulations of a finite density matrix model for QCD
• Non-Hermitian BCS-BEC crossover of Dirac fermions
• Restricted phase-space approximation in real-time stochastic quantization
• On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model
• Simulating Yang-Mills theories with a complex coupling
• Net baryon number fluctuations across the chiral phase transition at finite density in the strong coupling lattice QCD
• Exploring the properties of the phases of QCD matter - research opportunities and priorities for the next decade
• Temporal breakdown and Borel resummation in the complex Langevin method
• Lattice QCD at nonzero temperature and density
• Holographic realization of large-Nc orbifold equivalence with non-zero chemical potential
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region
• Few-body physics on a space-time lattice in the worldline approach
• Collective effects in emission of quantum dots strongly coupled to a microcavity photon
• A subset solution to the sign problem in random matrix simulations
• Improving efficiency of the path optimization method for a gauge theory
• QCD at nonzero temperature and density
• Complex Langevin studies of the emergent space-time in the type IIB matrix model
• Schwinger-Dyson equations and line integrals
• Complex Langevin approach to interacting Bose gases
• QCD thermodynamics on the lattice
• Adaptive gauge cooling for complex Langevin dynamics
• Complex Langevin for Lattice QCD at $T=0$ and $\mu \ge 0$
• Real-time correlators in 3+1D thermal lattice gauge theory
• Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential
• Test for a universal behavior of Dirac eigenvalues in the complex Langevin method
• Contribution to understanding the phase structure of strong interaction matter: Lee-Yang edge singularities from lattice QCD
• Complex Langevin study for polarons in an attractively interacting one-dimensional two-component Fermi gas
• Auxiliary field Monte-Carlo simulation of strong coupling lattice QCD for QCD phase diagram
• QCD at nonzero chemical potential: recent progress on the lattice
• Lefschetz thimble-inspired weight regularizations for complex Langevin simulations
• Testing a generalized cooling procedure in the complex Langevin simulation of chiral Random Matrix Theory
• Getting even with CLE
• Developments in lattice QCD for matter at high temperature and density
• Optimisation of complex integration contours at higher order
• Effect of momentum anisotropy on quark matter in the quark-meson model
• Fermion Bag Approach for Massive Thirring Model at Finite Density
• Complex Langevin Dynamics and Supersymmetric Quantum Mechanics
• Testing dynamic stabilisation in complex Langevin simulations
• Color superconductivity on the lattice -- analytic predictions from QCD in a small box
• Complex Langevin dynamics for SU(3) gauge theory in the presence of a theta term
• Towards sampling complex actions
• Regularization of Complex Langevin Method
• Complex Langevin analysis of 2D U(1) gauge theory on a torus with a $\theta$ term
• Beyond complex Langevin equations I: two simple examples
• Alternating Descent Method for Gauge Cooling of Complex Langevin Simulations
• Localised distributions in complex Langevin dynamics
• Hopping parameter expansion to all orders using the complex Langevin equation
• Equation of state of non-relativistic matter from automated perturbation theory and complex Langevin
• Comparative studies of the deformation techniques for the singular-drift problem in the complex Langevin method
• Extending complex Langevin simulations to full QCD at nonzero density
• Advances in non-relativistic matter via complex Langevin approaches
• The density of states approach for the simulation of finite density quantum field theories
• Complex Langevin Method on Rotating Matrix Quantum Mechanics at Thermal Equilibrium
• Positive representations of complex distributions on groups
• Simulating the Berezinskii-Kosterlitz-Thouless Transition with Complex Langevin
• Necessary and sufficient conditions for correctness of complex Langevin
• Performance of Complex Langevin Simulation in 0+1 dimensional massive Thirring model at finite density
• Towards the heavy dense QCD phase diagram using Complex Langevin simulations
• The Role of Integration Cycles in Complex Langevin Simulations
• Real-time Quantum evolution in the Classical approximation and beyond
• Holographic de Sitter Spacetime and Quantum Corrections to The Cosmological Constant
• Exploring the phase diagram of finite density QCD at low temperature by the complex Langevin method
• On complex Langevin dynamics and zeroes of the measure I: Formal proof and simple models
• Exploring the QCD phase diagram at finite density by the complex Langevin method on a $16^3\times 32$ lattice
• Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models
• On the validity of the complex Langevin method near the deconfining phase transition in QCD at finite density
• Resummed lattice QCD equation of state at finite baryon density: strangeness neutrality and beyond
• Classical Statistical simulation of Quantum Field Theory
• On the validity of complex Langevin method for path integral computations
• Can the complex Langevin method see the deconfinement phase transition in QCD at finite density?
• Combining complex Langevin dynamics with score-based and energy-based diffusion models
• Correctness criteria for complex Langevin
• Dynamical Compactification of Extra Dimensions in the Euclidean IKKT Matrix Model via Spontaneous Symmetry Breaking
• Loss of ergodicity in a quantum hopping model of a dense many body system with repulsive interactions
• The path optimization for the sign problem of low dimensional QCD
• The XY model at finite chemical potential using complex Langevin dynamics
• Dynamical compactification of extra dimensions in the Euclidean type IIB matrix model: A numerical study using the complex Langevin method
• Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation
• On Complex Langevin Dynamics and the Evaluation of Observables
• Complex Langevin for Lattice QCD