Fast estimator of Jacobians in the Monte Carlo integration on Lefschetz thimbles

Abstract

A solution to the sign problem is the so-called "Lefschetz thimble approach" where the domain of integration for field variables in the path integral is deformed from the real axis to a sub-manifold in the complex space. For properly chosen sub-manifolds ("thimbles") the sign problem disappears or is drastically alleviated. The parametrization of the thimble by real coordinates require the calculation of a jacobian with a computational cost of order O(V 3 ), where V is proportional to the spacetime volume. In this note we propose two estimators for this jacobian with a computational cost of order O(V ). We discuss analytically the regimes where we expect the estimator to work and show numerical examples in two different models.

New techniques and results for worldline simulations of lattice field theories

“…Certainly one may improve the determination of W by considering a full correlation matrix of 4-point correlators, but we found that the simple correlator (22) entirely dominates the signal. We implemented the determination of W based on (22) for both the 2-d and the 4-d cases for the same values of N s as studied for the condensation thresholds.…”

“…The first successful dual representation was discussed by Endres in [6,7] and since then the complex φ 4 field has been used as a model system for many important developments of worldline techniques [8][9][10][11][12][13][14][15][16][17][18], as well as as a reference system for other approaches to overcome the complex action problem (see, e.g., [19][20][21][22][23][24] in this context).…”

Unification of the complex Langevin method and the Lefschetzthimble method

“…In the GLTM [2], we deform the integration contour from R n to an n-dimensional real manifold in C n by using the so-called holomorphic gradient flow, which makes the sign problem milder. The holomorphic gradient flow is defined by the differential equation…”

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

“…(See, however, ref. [26].) It should also be noted that one can include fermions in the CLM with the cost of O(V ) for the system size V , whereas in the GLTM, it requires O(V 3 ).…”