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Bold diagrammatic Monte Carlo for the resonant Fermi gas

Kris Van Houcke, Félix Werner, Nikolay V. Prokof'ev, Boris V. Svistunov

Published May 16, 2013• 2 citations
OpenAlex

Abstract

We provide a comprehensive description of the Bold Diagrammatic Monte Carlo algorithm for the normal resonant Fermi gas reported and used in [Nature Phys. 8, 366 (2012)] and [arXiv:1303.6245]. Details are given on all key aspects of the scheme: diagrammatic framework, Monte Carlo moves, incorporation of ultraviolet asymptotics, and resummation techniques.

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Bold diagrammatic Monte Carlo: A generic sign-problem tolerant technique for polaron models and possibly interacting many-body problems

Nikolay V. Prokof'ev, Boris V. Svistunov

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.

Physical Review B•Mar 3, 2008•202 citations
We provide a comprehensive description of the Bold Diagrammatic Monte Carlo algorithm for the normal resonant Fermi gas reported and used in [Nature Phys. 8, 366 (2012)] and [arXiv:1303.6245]. Details are given on all key aspects of the scheme: diagrammatic framework, Monte Carlo moves, incorporation of ultraviolet asymptotics, and resummation techniques.
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.
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.
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.