Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2504.06060

Fast summation of fermionic Feynman diagrams beyond Bravais Lattices and on-site Hubbard interactions

Boyuan Shi

Published Apr 8, 2025• 0 citations
arXivHTMLPDF

Abstract

We designed new algorithms for summing bold-line Feynman diagrams in arbitrary channels, where it can be readily modified for bare interaction series as well. When applied to magnetic channel bold-line series with on-site Hubbard interactions, the algorithm achieves competitive performance compared with the state-of-art RPADet. We then generalize it beyond square lattice and on-site Hubbard interactions and achieve better scaling in the number of sites within a unit cell, while there is substantial increase when there are more types of interactions. This work paves the way of diagrammatic Monte Carlo simulations for real materials, holding the premise for a robust replacement of state-of-art simulation tools in the thermodynamical limit.

DOI
10.48550/arxiv.2504.06060

Related papers

Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

Riccardo Rossi

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.

arXiv:1612.05184•Physical Review Letters•Jul 25, 2017•110 citations

Bold-line diagrammatic Monte Carlo method: General formulation and application to expansion around the noncrossing approximation

Emanuel Gull, David R. Reichman, Andrew J. Millis

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.

arXiv:1004.0724•Physical Review B•Aug 6, 2010•81 citations

Efficient one-loop-renormalized vertex expansions with connected determinant diagrammatic Monte Carlo

Fedor Šimkovic, Riccardo Rossi, Michel Ferrero

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.

arXiv:2008.07377•Physical review. B./Physical review. B•Nov 13, 2020•15 citations

New Developments in Diagrammatic Monte Carlo for Interacting Lattice Fermions: Efficient Algorithms and Applications

Boyuan Shi

This thesis develops diagrammatic methods for thermodynamic and subsystem quantities of interacting lattice fermions, providing reference calculations for quantum simulators and many-body approximations. I formulate an inverse-flavour expansion about a self-consistent Hartree saddle, enumerate non-Gaussian diagrams, and use a discrete Lehmann representation to integrate selected internal vertices analytically. Density results cover 31 state points of the six-flavour Hubbard model on a $70\times70$ lattice at $T/t=0.15$. I generalize combinatorial summation to flavour-dependent propagators and expansion shifts. CoS++ combines partial contractions to obtain connected sums for interaction subsets. A determinant-polynomial sieve evaluates the same sums through local coefficient filters; at ninth order it is 24--25 times faster than the specified CoS baselines in three-flavour dense-random-matrix benchmarks. Thermodynamic applications include a 2048-site honeycomb lattice and a three-flavour square lattice with 1024 sites. For the second Renyi entropy, I derive a fermionic replica path integral and a stable evaluation of its two-time Green functions. The connected-determinant calculation reaches tenth order on a $3\times3$ cluster and ninth order on larger lattices. On $50\times50$ sites at $(U/t,μ/t,βt)=(3,1,8)$, the ninth-order continuation at shift $α=1.3$ gives $S_2/N_s=0.06695$ with propagated Monte Carlo uncertainty $3.0\times10^{-5}$. Comparing two expansion shifts at their respective retained orders gives an endpoint difference of $2.6\times10^{-4}$.

arXiv:2610.37505•Oct 27, 2026•0 citations

Semideterministic and stochastic sampling of Feynman diagrams with 1 / N f expansions

Boyuan Shi

We introduce a family of (semi)-bold-line series, assisted with 1 / N f expansions, with N f being the number of fermion flavors. If there is no additional N f cut, the series reduces to the random phase approximation series in the density-density channel, complementary to the particle-hole and particle-particle channels introduced in . To address the very localized integrands in diagrammatic Monte Carlo, we introduced an innovative VEGAS-RG-MCMC sampling method, where we found a significant decrease of autocorrelation time without the usage of the state-of-the-art many-configuration MCMC (MCMCMC) method while the combination of both is also straightforward. We performed extensive benchmarks for density, energy, and pressure with the t − t ′ SU ( N f ) Hubbard model on a square lattice and honeycomb lattices over a wide range of numerical methods. For benchmark purposes, we also implement bare- U symmetry-broken perturbation series for the two-dimensional SU(2) Hubbard model on the honeycomb lattice, where we found encouraging results from weak to intermediate couplings.

arXiv:2409.02032•Physical Review Research•May 22, 2025•0 citations

Combinatorial summation of Feynman diagrams

Evgeny Kozik

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.

arXiv:2309.13774•Nature Communications•Sep 10, 2024•6 citations
We designed new algorithms for summing bold-line Feynman diagrams in arbitrary channels, where it can be readily modified for bare interaction series as well. When applied to magnetic channel bold-line series with on-site Hubbard interactions, the algorithm achieves competitive performance compared with the state-of-art RPADet. We then generalize it beyond square lattice and on-site Hubbard interactions and achieve better scaling in the number of sites within a unit cell, while there is substantial increase when there are more types of interactions. This work paves the way of diagrammatic Monte Carlo simulations for real materials, holding the premise for a robust replacement of state-of-art simulation tools in the thermodynamical limit.
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.
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 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.
This thesis develops diagrammatic methods for thermodynamic and subsystem quantities of interacting lattice fermions, providing reference calculations for quantum simulators and many-body approximations. I formulate an inverse-flavour expansion about a self-consistent Hartree saddle, enumerate non-Gaussian diagrams, and use a discrete Lehmann representation to integrate selected internal vertices analytically. Density results cover 31 state points of the six-flavour Hubbard model on a 70×7070\times7070×70 lattice at T/t=0.15T/t=0.15T/t=0.15. I generalize combinatorial summation to flavour-dependent propagators and expansion shifts. CoS++ combines partial contractions to obtain connected sums for interaction subsets. A determinant-polynomial sieve evaluates the same sums through local coefficient filters; at ninth order it is 24--25 times faster than the specified CoS baselines in three-flavour dense-random-matrix benchmarks. Thermodynamic applications include a 2048-site honeycomb lattice and a three-flavour square lattice with 1024 sites. For the second Renyi entropy, I derive a fermionic replica path integral and a stable evaluation of its two-time Green functions. The connected-determinant calculation reaches tenth order on a 3×33\times33×3 cluster and ninth order on larger lattices. On 50×5050\times5050×50 sites at (U/t,μ/t,βt)=(3,1,8)(U/t,μ/t,βt)=(3,1,8)(U/t,μ/t,βt)=(3,1,8), the ninth-order continuation at shift α=1.3α=1.3α=1.3 gives S2/Ns=0.06695S_2/N_s=0.06695S2​/Ns​=0.06695 with propagated Monte Carlo uncertainty 3.0×10−53.0\times10^{-5}3.0×10−5. Comparing two expansion shifts at their respective retained orders gives an endpoint difference of 2.6×10−42.6\times10^{-4}2.6×10−4.
We introduce a family of (semi)-bold-line series, assisted with 1 / N f expansions, with N f being the number of fermion flavors. If there is no additional N f cut, the series reduces to the random phase approximation series in the density-density channel, complementary to the particle-hole and particle-particle channels introduced in . To address the very localized integrands in diagrammatic Monte Carlo, we introduced an innovative VEGAS-RG-MCMC sampling method, where we found a significant decrease of autocorrelation time without the usage of the state-of-the-art many-configuration MCMC (MCMCMC) method while the combination of both is also straightforward. We performed extensive benchmarks for density, energy, and pressure with the t − t ′ SU ( N f ) Hubbard model on a square lattice and honeycomb lattices over a wide range of numerical methods. For benchmark purposes, we also implement bare- U symmetry-broken perturbation series for the two-dimensional SU(2) Hubbard model on the honeycomb lattice, where we found encouraging results from weak to intermediate couplings.
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.