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

Distance between configurations in Markov chain Monte Carlo simulations

“…This gives a flat and translation invariant metric in the entire configuration space M = R. 8 Note that the distance decreases exponentially as d 2 t ∼ e −ωt , from which we find that the relaxation time of the system is given by ∼ 1/ω. 9 Since the manner of relaxation is almost the same for unimodal distributions, we see that the distance rapidly decreases to zero when the action gives a unimodal distribution.…”

“…We thus expect that q satisfies the inequality 0 < q ≤ 1. Our on-going work with numerical calculation [8] shows that q is actually in this range, excluding a logarithmic growth of g(β).…”

“…We denote by X n the set of sequences of n processes in M and by X n (x 1 , x 2 ) the subset of X n that consists of sequences which start from x 2 and end at x 1 . We then define f n (x 1 , x 2 ) to be the ratio of the sizes of two sets: 8) which can be expressed as…”

Combining the complex Langevin method and the generalized Lefschetz-thimble method

“…(See refs. [11][12][13][14][15][16][17][18][19][20][21][22] for related work.) The phase of the complex weight becomes constant on each thimble, which makes this limit attractable at first sight.…”

“…The GLTM [4] avoids this problem of the original method by choosing a large but finite flow time. Recently, it has been pointed out [21,22] that the flow time can be made as large as one wishes without the ergodicity problem if one uses the parallel tempering algorithm.…”