Abstract
Abstract As a solution towards the numerical sign problem, we propose a novel hybrid Monte Carlo algorithm, in which molecular dynamics is performed on a continuum set of integration surfaces foliated by the antiholomorphic gradient flow (“the worldvolume of an integration surface”). This is an extension of the tempered Lefschetz thimble method (TLTM) and solves the sign and multimodal problems simultaneously, as the original TLTM does. Furthermore, in this new algorithm, one no longer needs to compute the Jacobian of the gradient flow in generating a configuration, and only needs to evaluate its phase upon measurement. To demonstrate that this algorithm works correctly, we apply the algorithm to a chiral random matrix model, for which the complex Langevin method is known not to work.
Highlights
The sign problem is one of the major obstacles to numerical computation in various areas of physics, including finite density QCD [1], quantum Monte Carlo simulations of statistical systems [2], and the numerical simulations of real-time quantum field theories.There have been proposed many Monte Carlo algorithms towards solving the sign problem, such as those based on the complex Langevin equation [3–9] and those on Lefschetz thimbles [11–21], each of which has its own advantages and disadvantages
Since the lattice size is small, we adopt the direct method in the HMC algorithm; we compute J by integrating the flow equation (12) and use the LU decomposition in the inversion processes
We have proposed an HMC algorithm on the worldvolume R of an integration surface t, where the flow time t changes in the course of molecular dynamics, and the multimodal problem is resolved without introducing replicas
Summary
The sign problem is one of the major obstacles to numerical computation in various areas of physics, including finite density QCD [1], quantum Monte Carlo simulations of statistical systems [2], and the numerical simulations of real-time quantum field theories.There have been proposed many Monte Carlo algorithms towards solving the sign problem, such as those based on the complex Langevin equation [3–9] (see, e.g., Ref. [10] for a review) and those on Lefschetz thimbles [11–21], each of which has its own advantages and disadvantages. There have been proposed many Monte Carlo algorithms towards solving the sign problem, such as those based on the complex Langevin equation [3–9] The advantage of using the complex Langevin equation is its cheap computational cost, but such algorithms are known to suffer from a notorious problem called the “wrong convergence problem” (giving incorrect results with small statistical errors) for physically important ranges of parameters [6,7,9]. Computationally expensive, the algorithms based on Lefschetz thimbles are basically free from the wrong convergence. This is the case when and only when a single. The TLTM has proved effective and versatile when applied to various models, including the (0 + 1)-dimensional massive Thirring model [17] and the 2D Hubbard model away from half filling [20,21].1 The disadvantage of the original TLTM is its computational cost, the cost coming from the computation of the Jacobian and from the additional cost due to the introduction of replicas
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