A simple approach towards the sign problem using path optimisation

Abstract

We suggest an approach for simulating theories with a sign problem that relies on optimisation of complex integration contours that are not restricted to lie along Lefschetz thimbles. To that end we consider the toy model of a one-dimensional Bose gas with chemical potential. We identify the main contribution to the sign problem in this case as coming from a nearest neighbour interaction and approximately cancel it by an explicit deformation of the integration contour. We extend the obtained expressions to more general ones, depending on a small set of parameters. We find the optimal values of these parameters on a small lattice and study their range of validity. We also identify precursors for the onset of the sign problem. A fast method of evaluating the Jacobian related to the contour deformation is proposed and its numerical stability is examined. For a particular choice of lattice parameters, we find that our approach increases the lattice size at which the sign problem becomes serious from $L\approx 32$ to $L\approx 700$. The efficient evaluation of the Jacobian ($O(L)$ for a sweep) results in running times that are of the order of a few minutes on a standard laptop.

V1: 25 pages, 8 figures; V2: 28 pages, 8 figures, the methods used for finding the contour parameters are clarified, further discussion added, typos corrected, refs added

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
• Monte Carlo calculations of the finite density Thirring model
• Fermions at Finite Density in (2+1)d with Sign-Optimized Manifolds
• Application of neural network to sign problem via path optimization method
• A complex path around the sign problem
• Lattice String Field Theory: The linear dilaton in one dimension

Cited by in corpus (20)

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• Review on novel methods for lattice gauge theories
• Exploring QCD matter in extreme conditions with Machine Learning
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• Control the model sign problem via path optimization method: Monte-Carlo approach to QCD effective model with Polyakov loop
• Application of the path optimization method to the sign problem in an effective model of QCD with a repulsive vector-type interaction
• Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations
• Path optimization in 0+1 dimensional QCD at finite density
• Path optimization for $U(1)$ gauge theory with complexified parameters
• Exponential reduction of the sign problem at finite density in the 2+1D XY model via contour deformations
• Simulating Yang-Mills theories with a complex coupling
• Gauge invariant input to neural network for path optimization method
• Improving efficiency of the path optimization method for a gauge theory
• Deep Learning of Fermion Sign Fluctuations
• Optimisation of complex integration contours at higher order
• Application of the path optimization method to a discrete spin system
• Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models
• Path optimization method for the sign problem caused by fermion determinant
• Correctness criteria for complex Langevin
• Complex path simulations of geometrically frustrated ladders