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arXiv:2409.02032 · Physical Review Research

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

Boyuan Shi

Published May 22, 2025• 0 citations
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Abstract

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.

DOI
10.1103/physrevresearch.7.023177

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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.
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.
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.
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.
The dual fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work, we compute the dual fermion expansion for the two-dimensional Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained self-energy against numerically exact diagrammatic determinant Monte Carlo simulations to systematically assess convergence of the dual fermion series and the validity of these approximations. We observe that, from high temperatures down to the vicinity of the DMFT N\'eel transition, the dual fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered ($4\ensuremath{\le}U/t\ensuremath{\le}12$), implying that contributions from higher-order vertices are small. As the temperature is lowered further, we observe slower series convergence, convergence to incorrect solutions, and ultimately divergence. This happens in a regime where magnetic correlations become significant. We find, however, that the self-consistent particle-hole ladder approximation yields reasonable and often even highly accurate results in this regime.
We introduce a numerical algorithm to stochastically sample the dual fermion perturbation series around the dynamical mean field theory, generating all topologies of two-particle interaction vertices. We show results in the weak and strong coupling regime of the half-filled Hubbard model in two dimensions, illustrating that the method converges quickly where dynamical mean field theory is a good approximation, and show that corrections are large in the strong correlation regime at intermediate interaction. The fast convergence of dual corrections to dynamical mean field results illustrates the power of the approach and opens a practical avenue towards the systematic inclusion of nonlocal correlations in correlated materials simulations. An analysis of the frequency scale shows that only low-frequency propagators contribute substantially to the diagrams, putting the inclusion of higher order vertices within reach.
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.