Abstract
The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.
Highlights
One of the main difficulties for the Path Integral Monte Carlo (PIMC) simulation of the quantum systems of particles is the so called “sign problem”
The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action
Discrepancy between exact values of the Airy function and related values obtained by proposed procedure can be explained by approximations used in transitions from initial integral to its Lefschetz thimbles representation accounting for only the main contribution to the contour integrals
Summary
One of the main difficulties for the Path Integral Monte Carlo (PIMC) simulation of the quantum systems of particles is the so called “sign problem”. The “sign problem” arises in the simulations of the Wigner and Feynman path integrals, describing quantum systems and the finite density quantum chromodynamics due to the fast oscillations of the integrand defined by the complex-valued action. This integrand does not give a real and positive Boltzmann-like weight (for example, by the Wick rotation) to resort to the traditional Monte Carlo methods. Larkin the action becomes very large, because one needs to take a sample from a configuration space, where the weights of nearby configurations have almost the same amplitudes but very different phases
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