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arXiv:1610.09566 · The Journal of Chemical Physics

Inchworm Monte Carlo for exact non-adiabatic dynamics. I. Theory and algorithms

Hsing-Ta Chen, Guy Cohen, David R. Reichman

Published Feb 1, 2017• 82 citations
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Abstract

In this paper, we provide a detailed description of the inchworm Monte Carlo formalism for the exact study of real-time non-adiabatic dynamics. This method optimally recycles Monte Carlo information from earlier times to greatly suppress the dynamical sign problem. Using the example of the spin-boson model, we formulate the inchworm expansion in two distinct ways: The first with respect to an expansion in the system-bath coupling and the second as an expansion in the diabatic coupling. The latter approach motivates the development of a cumulant version of the inchworm Monte Carlo method, which has the benefit of improved scaling. This paper deals completely with methodology, while Paper II provides a comprehensive comparison of the performance of the inchworm Monte Carlo algorithms to other exact methodologies as well as a discussion of the relative advantages and disadvantages of each.

DOI
10.1063/1.4974328

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In this paper, we provide a detailed description of the inchworm Monte Carlo formalism for the exact study of real-time non-adiabatic dynamics. This method optimally recycles Monte Carlo information from earlier times to greatly suppress the dynamical sign problem. Using the example of the spin-boson model, we formulate the inchworm expansion in two distinct ways: The first with respect to an expansion in the system-bath coupling and the second as an expansion in the diabatic coupling. The latter approach motivates the development of a cumulant version of the inchworm Monte Carlo method, which has the benefit of improved scaling. This paper deals completely with methodology, while Paper II provides a comprehensive comparison of the performance of the inchworm Monte Carlo algorithms to other exact methodologies as well as a discussion of the relative advantages and disadvantages of each.
In this second paper of a two part series, we present extensive benchmark results for two different inchworm Monte Carlo expansions for the spin-boson model. Our results are compared to previously developed numerically exact approaches for this problem. A detailed discussion of convergence and error propagation is presented. Our results and analysis allow for an understanding of the benefits and drawbacks of inchworm Monte Carlo compared to other approaches for exact real-time non-adiabatic quantum dynamics.
In this thesis, I explore dissipative quantum dynamics of several prototypical model systems via various approaches, ranging from approximate to numerically exact schemes. In particular, in the realm of the approximate I explore the accuracy of Padé–resummed master equations and the fewest switches surface hopping (FSSH) algorithm for the spin–boson model, and non-crossing approximations (NCA) for the Anderson–Holstein model. Next, I develop new and exact Monte Carlo approaches and test them on the spin–boson model. I propose well–defined criteria for assessing the accuracy of Padé-resummed quantum master equations, which correctly demarcate the regions of parameter space where the Padé approximation is reliable. I continue the investigation of spin–boson dynamics by benchmark comparisons of the semiclassical FSSH algorithm to exact dynamics over a wide range of parameters. Despite small deviations from golden-rule scaling in the Marcus regime, standard surface hopping algorithm is found to be accurate over a large portion of parameter space. The inclusion of decoherence corrections via the augmented FSSH algorithm improves the accuracy of dynamical behavior compared to exact simulations, but the effects are generally not dramatic for the cases I consider. Next, I introduce new methods for numerically exact real-time simulation based on real-time diagrammatic Quantum Monte Carlo (dQMC) and the inchworm algorithm. These methods optimally recycle Monte Carlo information from earlier times to greatly suppress the dynamical sign problem. In the context of the spin–boson model, I formulate the inchworm expansion in two distinct ways: the first with respect to an expansion in the system–bath coupling and the second as an expansion in the diabatic coupling. In addition, a cumulant version of the inchworm Monte Carlo method is motivated by the latter expansion, which allows for further suppression of the growth of the sign error. I provide a comprehensive comparison of the performance of the inchworm Monte Carlo algorithms to other exact methodologies as well as a discussion of the relative advantages and disadvantages of each. Finally, I investigate the dynamical interplay between the electron–electron interaction and the electron–phonon coupling within the Anderson–Holstein model via two complementary NCAs: the first is constructed around the weak-coupling limit and the second around the polaron limit. The influence of phonons on spectral and transport properties is explored in equilibrium, for non-equilibrium steady state and for transient dynamics after a quench. I find the two NCAs disagree in nontrivial ways, indicating that more reliable approaches to the problem are needed. The complementary frameworks used here pave the way for numerically exact methods based on inchworm dQMC algorithms capable of treating open systems simultaneously coupled to multiple fermionic and bosonic baths.
We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are performed to validate our method.
The inchworm expansion is a promising approach to solving strongly correlated quantum impurity models due to its reduction of the sign problem in real and imaginary time. However, inchworm Monte Carlo is computationally expensive, converging as 1 / N where N is the number of samples. We show that the imaginary-time integration is amenable to quasi Monte Carlo, with parametrically better 1 / N convergence, by mapping the Sobol low-discrepancy sequence from the hypercube to the simplex with the so-called Root transform. This extends the applicability of the inchworm method to, e.g., multiorbital Anderson impurity models with off-diagonal hybridization, relevant for materials simulation, where continuous-time hybridization expansion Monte Carlo has a severe sign problem. Published by the American Physical Society 2024
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchworm Monte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.
Current nonequilibrium Monte Carlo methods suffer from a dynamical sign problem that makes simulating real-time dynamics for long times exponentially hard. We propose a new "inchworm algorithm," based on iteratively reusing information obtained in previous steps to extend the propagation to longer times. The algorithm largely overcomes the dynamical sign problem, changing the scaling from exponential to quadratic. We use the method to solve the Anderson impurity model in the Kondo and mixed valence regimes, obtaining results both for quenches and for spin dynamics in the presence of an oscillatory magnetic field.