Research Support
SPIRITS

Development and application of fundamental algorithms solving the sign problem in Monte Carlo calculations

Project Gist

Developing a versatile solution to the sign problem in Monte Carlo calculations

Keywords

Monte Carlo method, sign problem, Lefschetz thimble, tempering method, quantum chromodynamics at finite density

Background and Purpose

The sign problem in Monte Carlo calculations appears in various fields of natural science, not limited to physics, and a versatile solution has long been coveted. The “tempered Lefschetz thimble method (TLT method)” developed by us (Fukuma and Umeda) has been found to be useful in terms of both versatility and high reliability [1]. The purpose of this research project is to further improve this algorithm and apply it to various problems in physics, in order to activate those fields at one time whose progress has been stagnant due to the sign problem.
[1] M. Fukuma and N. Umeda, “Parallel tempering algorithm for integration over Lefschetz thimbles”,
PTEP 2017 (2017) 073B01.

Project Achievements

We applied the TLT method to simplified models of strongly correlated electron systems and finite density quantum chromodynamics, and showed that the TLT method gives correct results in all cases. We have also succeeded in improving the calculation algorithm of the TLT method. With these achievements, our TLT method is beginning to be recognized as one of the most powerful methods for the sign problem. The research results were published as five treatises, and were presented at a total of six international conferences (although the number declined in 2020 due to the new corona virus). We also established a network with researchers in various fields other than particle physics.

Future Prospects

In order to establish the status as an originative work from Japan, the TLT method needs to be easy to use for many researchers. In the near future, while further developing the algorithm, I would like to expand the scope of application at one time and create a large academic flow centered on solving the sign problem.

Figure

Three Lefschetz thimbles obtained by deforming the original integration surface. Each thimble is a constant-phase surface and thus has no sign problem, but there exists an ergodicity problem due to indefinitely high potential barriers between different thimbles. In the TLT method, transitions between different thimbles are prompted by the tempering, where configurations can pass through the integration surfaces in the middle of deformation.
The chiral condensate in the (0+1)-dimensional Thirring model [1]. In the TLT method, the results agree with the exact values because the sign and ergodicity problems are solved simultaneously by the tempering.

Principal Investigator

FUKUMA Masafumi

FUKUMA Masafumi
Graduate School of Science
Prof. Fukuma received his PhD from the University of Tokyo. He has been working on theoretical high energy physics, especially on the clarification of the dynamics of quantum field theories and the construction of quantum gravity theory. He is trying to explain the fundamental laws and initial conditions of the Universe by randomness.