Optimisation of complex integration contours at higher order

Abstract

We continue our study of contour deformation as a practical tool for dealing with the sign problem using the $d$-dimensional Bose gas with non-zero chemical potential as a toy model. We derive explicit expressions for contours up to the second order with respect to a natural small parameter and generalise these contours to an ansatz for which the evaluation of the Jacobian is fast ($O(1)$). We examine the behaviour of the various proposed contours as a function of space-time dimensionality, the chemical potential, and lattice size and geometry and use the mean phase factor as a measure of the severity of the sign problem. In turns out that this method leads to a substantial reduction of the sign problem and that it becomes more efficient as space-time dimensionality is increased. Correlations among contributions to $\mathrm{Im}⟨S⟩$ play a key role in determining the mean phase factor and we examine these correlations in detail.

30 pages, 17 figures

References in corpus (18)

• The QCD phase diagram for small densities from imaginary chemical potential
• Finite density QCD via imaginary chemical potential
• High density QCD on a Lefschetz thimble?
• Analytic Continuation Of Chern-Simons Theory
• The Complex Langevin method: When can it be trusted?
• Pressure and non-linear susceptibilities in QCD at finite chemical potentials
• Can stochastic quantization evade the sign problem? -- the relativistic Bose gas at finite chemical potential
• The argument for justification of the complex Langevin method and the condition for correct convergence
• A Monte Carlo algorithm for simulating fermions on Lefschetz thimbles
• Toward solving the sign problem with path optimization method
• Finite-Density Monte Carlo Calculations on Sign-Optimized Manifolds
• Deep Learning Beyond Lefschetz Thimbles
• Complex Langevin dynamics at finite chemical potential: mean field analysis in the relativistic Bose gas
• Fast Estimator of jacobians in Monte Carlo Integration on Lefschetz Thimbles
• Control the model sign problem via path optimization method: Monte-Carlo approach to QCD effective model with Polyakov loop
• A simple approach towards the sign problem using path optimisation
• Path optimization in 0+1 dimensional QCD at finite density
• Perturbative Removal of a Sign Problem

Cited by in corpus (3)

• Exponential reduction of the sign problem at finite density in the 2+1D XY model via contour deformations
• Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models
• Correctness criteria for complex Langevin