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arXiv:2507.12349

CoS++: Towards More General and Explicit Implementations for Sampling High-Order Feynman Diagrammatic Series

Boyuan Shi

Published Jul 16, 2025• 0 citations
arXivHTMLPDF

Abstract

We extend combinatorial summation (CoS) of connected Feynman diagrams to Hubbard mod- els with flavor-dependent propagators and counterterm shifts. The construction combines an interaction-occupancy mask with labels for the open fermion cycle, allowing equivalent endpoint his- tories to be accumulated while retaining the flavor and sublattice information needed by subsequent contractions. In the flavor-dependent version, denoted symmetry-broken CoS (SB-CoS), normal heads carry the cycle flavor and counterterm heads inherit the flavor at their interaction partner. We prove the correctness of the resulting recurrence and introduce terminal readouts that return all nonempty interaction-subset values from a shared circuit, as required by a many-configuration Markov-chain sampler. For three flavors, a second formulation uses determinant polynomials and a local interaction tensor, with connected coefficients extracted by subset recursion or Fourier interpo- lation and a support sieve. Applications to the two-spin honeycomb and three-flavor square-lattice Hubbard models illustrate the treatment of sublattice structure and non-uniform reference shifts.

DOI
10.48550/arxiv.2507.12349

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Feynman's diagrammatic series is a common language for a formally exact theoretical description of systems of infinitely-many interacting quantum particles, as well as a foundation for precision computational techniques. Here we introduce a universal framework for efficient summation of connected or skeleton Feynman diagrams for generic quantum many-body systems. It is based on an explicit combinatorial construction of the sum of the integrands by dynamic programming, at a computational cost that can be made only exponential in the diagram order on a classical computer and potentially polynomial on a quantum computer. We illustrate the technique by an unbiased diagrammatic Monte Carlo calculation of the equation of state of the 2D SU(N) Hubbard model in an experimentally relevant regime, which has remained challenging for state-of-the-art numerical methods.
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We present a technique that enables the evaluation of perturbative expansions based on one-loop-renormalized vertices up to large expansion orders. Specifically, we show how to compute large-order corrections to the random phase approximation in either the particle-hole or particle-particle channels. The algorithm's efficiency is achieved by the summation over contributions of all symmetrized Feynman diagram topologies using determinants, and by integrating out analytically the two-body long-range interactions in order to yield an effective zero-range interaction. Notably, the exponential scaling of the algorithm as a function of perturbation order leads to a polynomial scaling of the approximation error with computational time for a convergent series. To assess the performance of our approach, we apply it to the nonperturbative regime of the square-lattice fermionic Hubbard model away from half-filling and report, as compared to the bare interaction expansion algorithm, significant improvements of the Monte Carlo variance as well as the convergence properties of the resulting perturbative series.
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