Lefschetz Thimbles and Wigner Functions
Lefschetz Thimbles and Wigner Functions作者机构：Joint Institute for High Temperatures Russian Academy of Sciences Moscow Russia
出 版 物：《Journal of Applied Mathematics and Physics》 (应用数学与应用物理（英文）)
年 卷 期：2020年第8卷第7期
页 面：1278-1290页
学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学]
主 题：Wigner Function Path Integral Quantum Monte Carlo Picard-Lefschetz Theory Morse Theory
摘 要：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.
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