Parallel tempering algorithm for integration over Lefschetz thimbles

Abstract

The algorithm based on integration over Lefschetz thimbles is a promising method to resolve the sign problem for complex actions. However, this algorithm often meets a difficulty in actual Monte Carlo calculations because the configuration space is not easily explored due to the infinitely high potential barriers between different thimbles. In this paper, we propose to use the flow time of the antiholomorphic gradient flow as an auxiliary variable for the highly multimodal distribution. To illustrate this, we implement the parallel tempering method by taking the flow time as a tempering parameter. In this algorithm, we can take the maximum flow time to be sufficiently large such that the sign problem disappears there, and two separate modes are connected through configurations at small flow times. To exemplify that this algorithm does work, we investigate the (0+1)-dimensional massive Thirring model at finite density and show that our algorithm correctly reproduces the analytic results for large flow times such as T=2.

14 pages, 5 figures. v2: typos corrected, figures updated. v3: title changed, references added, typos corrected

References in corpus (6)

• Structure of Lefschetz thimbles in simple fermionic systems
• Monte Carlo calculations of the finite density Thirring model
• Does the complex Langevin method give unbiased results?
• Reweighting complex Langevin trajectories
• Complex Langevin simulations of a finite density matrix model for QCD
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region

Cited by in corpus (39)

• Review on novel methods for lattice gauge theories
• Toward solving the sign problem with path optimization method
• Complex Langevin and other approaches to the sign problem in quantum many-body physics
• Deep Learning Beyond Lefschetz Thimbles
• Tempered transitions between thimbles
• Applying the tempered Lefschetz thimble method to the Hubbard model away from half-filling
• Machine learning for molecular dynamics with strongly correlated electrons
• Finite Density $QE{D}_{1+1}$ Near Lefschetz Thimbles
• Control the model sign problem via path optimization method: Monte-Carlo approach to QCD effective model with Polyakov loop
• Worldvolume approach to the tempered Lefschetz thimble method
• Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles
• Implementation of the HMC algorithm on the tempered Lefschetz thimble method
• Existence of real time quantum path integrals
• Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects
• Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model
• On the gauge invariant path-integral measure for the overlap Weyl fermions in $\underline{16}$ of SO(10)
• 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
• Path optimization for $U(1)$ gauge theory with complexified parameters
• Emergence of AdS geometry in the simulated tempering algorithm
• Distance between configurations in Markov chain Monte Carlo simulations
• Improving efficiency of the path optimization method for a gauge theory
• Truly Lorentzian quantum cosmology
• Settling an old story: solution of the Thirring model in thimble regularization
• Lefschetz Thimble Quantum Monte Carlo for Spin Systems
• Color superconductivity on the lattice -- analytic predictions from QCD in a small box
• Statistical analysis method for the worldvolume hybrid Monte Carlo algorithm
• Complex Langevin analysis of 2D U(1) gauge theory on a torus with a $\theta$ term
• Real-time quantum dynamics, path integrals and the method of thimbles
• Application of the path optimization method to a discrete spin system
• A quantum Monte Carlo method on asymptotic Lefschetz thimbles for quantum spin systems: An application to the Kitaev model in a magnetic field
• Holographic de Sitter Spacetime and Quantum Corrections to The Cosmological Constant
• Tensor network approach to 2D Yang-Mills theories
• Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models
• Distance between configurations in MCMC simulations and the geometrical optimization of the tempering algorithms
• Complex path simulations of geometrically frustrated ladders
• Path optimization method for the sign problem caused by fermion determinant
• Emergent quantum geometry from stochastic random matrices
• Tempered Lefschetz thimble method and its application to the Hubbard model away from half filling