Monte Carlo algorithm for simulating fermions on Lefschetz thimbles

Abstract

A possible solution of the notorious sign problem preventing direct Monte Carlo calculations for systems with non-zero chemical potential is to deform the integration region in the complex plane to a Lefschetz thimble. We investigate this approach for a simple fermionic model. We introduce an easy to implement Monte Carlo algorithm to sample the dominant thimble. Our algorithm relies only on the integration of the gradient flow in the numerically stable direction, which gives it a distinct advantage over the other proposed algorithms. We demonstrate the stability and e ciency of the algorithm by applying it to an exactly solvable fermionic model and compare our results with the analytical ones. We report a very good agreement for a certain region in the parameter space where the dominant contribution comes from a single thimble, including a region where standard methods su↵er from a severe sign problem. However, we find that there are also regions in the parameter space where the contribution from multiple thimbles is important, even in the continuum limit.2

Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

“…Furthermore, a semiclassical estimate of the contribution from other thimbles matched, in order of magnitude, the size of the discrepancy. This result suggests that the algorithm proposed in [14] properly samples one thimble and that the discrepancies result from neglecting the other thimbles' contributions.…”

“…Furthermore, as an optimization we can use as a starting point for the flow another manifold that has the same integral as the original contour. We exemplify these observations by showing that for a 0+1 model we considered previously we can obtain the exact results both in the continuum limit and up to very low temperatures -something that eluded us in [14] -by simply integrating over the tangent space of the main thimble. In the case of the lowest temperatures, where the integration over the tangent space fails, the calculation is still possible if a different manifold, obtained by flowing by a moderate amount, is used.…”

“…1 The need for a multi-thimble integration, at least in some models, was evidenced in [14] where we considered a simple soluble 0 + 1 dimensional fermionic model previously used as a toy model for the sign problem [15,16]. We showed that an algorithm designed to compute the contribution of one thimble gave the exact answer as long as the coupling constant was small and the temperature and lattice spacing were not too small.…”

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.…”

Toward solving the sign problem with path optimization method

“…In the path optimization method, we can resolve the residual sign problem, but not the global sign problem. This problem also exists in the ordinary and generalized Lefschetz thimble methods [12]. Figure 4 shows the expectation value of x 2 in the hybrid Monte-Carlo method on the modified integration path.…”