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arXiv:2205.10896 · SIAM Journal on Scientific Computing

The Bold-Thin-Bold Diagrammatic Monte Carlo Method for Open Quantum Systems

Zhenning Cai, Geshuo Wang, Siyao Yang

Published Jul 19, 2023• 8 citations
arXivHTMLPDF

Abstract

Abstract. We present two diagrammatic Monte Carlo methods for quantum systems coupled with harmonic baths, whose dynamics are described by integro-differential equations. The first approach can be considered as a reformulation of the Dyson series, and the second one, called “bold-thin-bold diagrammatic Monte Carlo,” is based on resummation of the diagrams in the Dyson series to accelerate its convergence. The underlying mechanism of the governing equations associated with the two methods lies in the recurrence relation of the path integrals, which is the most costly part in the numerical methods. The proposed algorithms give an extension to the work of Cai, Lu, and Yang [ Comput. Phys. Commun. 278 (2022), 108417], where the algorithms are designed based on reusing the previous calculations of bath influence functionals. Compared with the algorithms therein, our methods further include the reuse of system-associated functionals and show better performance in terms of computational efficiency and memory cost. We demonstrate the two methods in the framework of spin-boson model, and numerical experiments are carried out to verify the validity of the methods.

DOI
10.1137/22m1499297

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We study the dynamical simulation of an open quantum spin chain with nearest neighboring coupling, where each spin in the chain is associated with a harmonic bath. This is an extension of our previous work (Wang, G.; Cai, Z. J. Chem. Theory Comput. 2023, 19, 8523–8540) by generalizing the application of the inchworm method and the technique of modular path integrals from diagonally coupled cases to off-diagonally coupled cases. Additionally, to reduce computational and memory cost in long time simulation, we apply tensor-train representation to efficiently represent the reduced density matrix of the spin chains, and employ the transfer tensor method (TTM) to avoid exponential growth of computational cost with respect to time. Abundant numerical experiments are performed to validate our method.
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