Complex Langevin and boundary terms

Abstract

As is well known the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit. We identified one reason for this long ago: insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, we analyze the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. We also show how some simple modification stabilizes the CL process in such a way that it can produce results agreeing with direct integration. Besides explicitly demonstrating the connection between boundary terms and correct convergence our analysis also suggests a correctness criterion which could be applied in realistic lattice simulations.

arXiv comment15 pages, 13 figures v2: changed format to two columns for journal submission, added references and a few comments on an old criterion for correctness v3: Added some plots and discussion on the criterion from citation [13]

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Cited by in corpus (31)

• Review on novel methods for lattice gauge theories
• Complex Langevin and other approaches to the sign problem in quantum many-body physics
• Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice
• Controlling Complex Langevin simulations of lattice models by boundary term analysis
• Calculating the EoS of the dense quark-gluon plasma using the Complex Langevin equation
• Complex Langevin simulations and the QCD phase diagram: Recent developments
• Diffusion Models as Stochastic Quantization in Lattice Field Theory
• Deconfinement transition line with the Complex Langevin equation up to $\mu /T\sim 5$
• Thermodynamics of spin-orbit coupled bosons in two dimensions from complex Langevin
• Complex Langevin calculations in QCD at finite density
• Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel
• Towards learning optimized kernels for complex Langevin
• Stable solvers for real-time Complex Langevin
• Complex Langevin: Boundary terms at poles
• Exponential reduction of the sign problem at finite density in the 2+1D XY model via contour deformations
• Overview of the QCD phase diagram -- Recent progress from the lattice
• Complex Langevin approach to interacting Bose gases
• Real-time correlators in 3+1D thermal lattice gauge theory
• Complex Langevin study for polarons in an attractively interacting one-dimensional two-component Fermi gas
• Lefschetz thimble-inspired weight regularizations for complex Langevin simulations
• Alternating Descent Method for Gauge Cooling of Complex Langevin Simulations
• Regularization of Complex Langevin Method
• Towards sampling complex actions
• Complex Langevin analysis of 2D U(1) gauge theory on a torus with a $\theta$ term
• Necessary and sufficient conditions for correctness of complex Langevin
• The Role of Integration Cycles in Complex Langevin Simulations
• On the validity of the complex Langevin method near the deconfining phase transition in QCD at finite density
• On the validity of complex Langevin method for path integral computations
• Classical Statistical simulation of Quantum Field Theory
• Correctness criteria for complex Langevin
• Combining complex Langevin dynamics with score-based and energy-based diffusion models