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arXiv:cond-mat/0702555 · Physical Review Letters

Bold Diagrammatic Monte Carlo Technique: When the Sign Problem Is Welcome

Nikolay V. Prokof'ev, Boris V. Svistunov

Published Dec 18, 2007• 147 citations
arXivHTMLPDF

Abstract

We introduce a Monte Carlo scheme for sampling a bold-line diagrammatic series specifying an unknown function in terms of itself. The range of convergence of this bold(-line) diagrammatic Monte Carlo (BMC) technique is significantly broader than that of a simple iterative scheme for solving integral equations. With the BMC technique, a moderate "sign problem" turns out to be an advantage in terms of the convergence of the process. For an illustrative purpose, we solve the one-particle s-scattering problem. As an important application, we obtain the T matrix for a Fermi polaron (one spin-down particle interacting with the spin-up fermionic sea).

DOI
10.1103/physrevlett.99.250201

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We introduce a Monte Carlo scheme for sampling a bold-line diagrammatic series specifying an unknown function in terms of itself. The range of convergence of this bold(-line) diagrammatic Monte Carlo (BMC) technique is significantly broader than that of a simple iterative scheme for solving integral equations. With the BMC technique, a moderate "sign problem" turns out to be an advantage in terms of the convergence of the process. For an illustrative purpose, we solve the one-particle s-scattering problem. As an important application, we obtain the T matrix for a Fermi polaron (one spin-down particle interacting with the spin-up fermionic sea).
We develop a Monte Carlo scheme for sampling series of Feynman diagrams for the proper self-energy which are self-consistently expressed in terms of renormalized particle propagators. This approach is used to solve the problem of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles. Though the original series based on bare propagators are sign-alternating and divergent one can still determine the answer behind them by using two strategies (separately or together): (i) using proper series re-summation techniques, and (ii) introducing renormalized propagators which are defined in terms of the simulated proper self-energy, i.e. making the entire scheme self-consistent. Our solution is important for understanding the phase diagram and properties of the BCS-BEC crossover in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem tolerant method for exact numerical solution of polaron-type models, and, possibly, of the interacting many-body Hamiltonians.
We develop a Monte Carlo scheme for sampling series of Feynman diagrams for the proper self-energy, which are self-consistently expressed in terms of renormalized particle propagators. This approach is used to solve the problem of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles. Though the original series based on bare propagators are sign alternating and divergent, one can still determine the answer behind them by using two strategies (separately or together): (i) using proper series resummation techniques and (ii) introducing renormalized propagators which are defined in terms of the simulated proper self-energy, i.e., making the entire scheme self-consistent. Our solution is important for understanding the phase diagram and properties of the Bardeen-Cooper-Schrieffer--Bose-Einstein Condensation crossover in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem tolerant method for exact numerical solution of polaron-type models and, possibly, of the interacting many-body Hamiltonians.
We use the diagrammatic Monte Carlo approach to solve the problem of a single spin-down fermion resonantly interacting with a Fermi gas of spin-up particles. Our solution is important for understanding the phase diagram and properties of the crossover from the BCS regime to the Bose-Einstein condensate in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem-tolerant method for exact numerical solution of polaron-type models. This is a characteristic example of how Monte Carlo methods can be used to simulate divergent sign-alternating diagrammatic series.
These notes are intended as a detailed discussion on how to implement the diagrammatic Monte Carlo method for a physical system which is technically simple and where it works extremely well, namely the Fröhlich polaron problem. Sampling schemes for the Green function as well as the self-energy in the bare and skeleton (bold) expansion are disclosed in full detail. We discuss the Monte Carlo updates, possible implementations in terms of common data structures, as well as techniques on how to perform the Fourier transforms for functions with discontinuities. Control over the variety of parameters, especially in the bold scheme, is demonstrated. Sample codes are made available online along with extensive documentation. Towards the end, we discuss various extensions of the method and their applications. After working through these notes, the reader will be well equipped to explore the richness of the diagrammatic Monte Carlo method for quantum many-body systems.
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.
We present a general framework for performing ``bold-line'' diagrammatic Monte Carlo calculations using an analytical partial resummation as a starting point for a stochastic summation of all diagrams. As a stringent test case we assess the accuracy of the method by solving the equations of single-site dynamical mean-field theory, using the noncrossing approximation as a starting point. We establish the validity of the starting approximations and show that the bold method provides a very accurate treatment of the Mott-insulating phase.