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arXiv:1807.00290 · Physical review. B./Physical review. B

Inclusion-exclusion principle for many-body diagrammatics

Aviel Boag, Emanuel Gull, Guy Cohen

Published Sep 26, 2018• 38 citations
arXivHTMLPDF

Abstract

Diagrammatic resummation techniques have recently enabled new classes of quantum Monte Carlo algorithms, but are restricted by the need to sum over factorially large sets of diagrams. Here, the authors present a fast algorithm for summing over the diagrams appearing in Inchworm hybridization expansions. The method relies on the inclusion-exclusion principle to reduce the scaling from factorial to exponential. For the Inchworm hybridization expansion, when compared to the previous state of the art, the new algorithm is shown to provide practical performance gains of approximately two orders of magnitude.

DOI
10.1103/physrevb.98.115152

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Diagrammatic resummation techniques have recently enabled new classes of quantum Monte Carlo algorithms, but are restricted by the need to sum over factorially large sets of diagrams. Here, the authors present a fast algorithm for summing over the diagrams appearing in Inchworm hybridization expansions. The method relies on the inclusion-exclusion principle to reduce the scaling from factorial to exponential. For the Inchworm hybridization expansion, when compared to the previous state of the art, the new algorithm is shown to provide practical performance gains of approximately two orders of magnitude.
Abstract We present two fast algorithms which apply inclusion–exclusion principle to sum over the bosonic diagrams in bare diagrammatic quantum Monte Carlo and inchworm Monte Carlo method, respectively. In the case of inchworm Monte Carlo, the proposed fast algorithm gives an extension to the work [2018 Inclusion–exclusion principle for many-body diagrammatics Phys. Rev. B 98 115152] from fermionic to bosonic systems. We prove that the proposed fast algorithms reduce the computational complexity from double factorial to exponential. Numerical experiments are carried out to verify the theoretical results and to compare the efficiency of the methods.
Diagrammatic expansions are a paradigmatic and powerful tool of quantum many-body theory. Their evaluation to high order, e.g., by the Diagrammatic Monte Carlo technique, can provide unbiased results in strongly correlated and challenging regimes. However, calculating a factorial number of terms to acceptable precision remains very costly even for state-of-the-art methods. We achieve a dramatic acceleration of evaluating Feynman's diagrammatic series by use of specialised hardware architecture within the recently introduced combinatorial summation (CoS) framework. We present how exploiting the massive parallelism and concurrency available from GPUs leads to orders of magnitude improvement in computation time even on consumer-grade hardware. This provides a platform for making probes of novel phenomena of strong correlations much more accessible.
We present a simple trick that allows us to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models, such that the thermodynamic limit can be taken analytically. With our approach one can achieve superior performance compared to conventional diagrammatic Monte Carlo algorithm, while rendering the algorithmic part dramatically simpler. By considering the sum of all connected diagrams at once, we allow for massive cancellations between different diagrams, greatly reducing the sign problem. In the end, the computational effort increases only exponentially with the order of the expansion, which should be contrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model.
Multiorbital quantum impurity models with general interaction and hybridization terms appear in a wide range of applications, including embedding, quantum transport, and nanoscience. However, most quantum impurity solvers are restricted to a few impurity orbitals, discretized baths, diagonal hybridizations, or density-density interactions. Here, we generalize the inchworm quantum Monte Carlo method to the interaction expansion, and we explore its application to typical single- and multiorbital problems encountered in investigations of impurity and lattice models. Our implementation generically outperforms bare and bold-line quantum Monte Carlo algorithms in the interaction expansion. For the systems studied here, our implementation remains inferior to the more specialized hybridization expansion and auxiliary field algorithms. The problem of convergence to unphysical fixed points, which hampers so-called bold-line methods, is not encountered in inchworm Monte Carlo.
High-order virtual excitations play an important role in microscopic models of nuclear reactions at intermediate energies. However, the factorial growth of their complexity has prevented their consistent inclusion in ab initio many-body calculations. For infinite systems at finite temperature, such drawbacks can be overcome using diagrammatic Monte Carlo techniques to resum entire series of Feynman diagrams. We present a diagrammatic Monte Carlo algorithm that can be applied to self-bound systems with discrete energy levels at zero temperature, and demonstrate its potential for the Richardson model of nuclear pairing. We show that sampling the topological space of diagrams allows the inclusion of high-order excitations that are neglected in state-of-the-art approximations used in nuclear physics and quantum chemistry. We propose that sampling the diagrammatic space can overcome the long-standing gap between our microscopic understanding of structure and reactions in nuclear physics.
High order perturbation theory has seen an unexpected recent revival for controlled calculations of quantum many-body systems, even at strong coupling. We adapt integration methods using low-discrepancy sequences to this problem. They greatly outperform state-of-the-art diagrammatic Monte Carlo simulations. In practical applications, we show speed-ups of several orders of magnitude with scaling as fast as 1/N in sample number N; parametrically faster than 1/sqrt[N] in Monte Carlo simulations. We illustrate our technique with a solution of the Kondo ridge in quantum dots, where it allows large parameter sweeps.