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arXiv:2008.07377 · Physical review. B./Physical review. B

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

Fedor Šimkovic, Riccardo Rossi, Michel Ferrero

Published Nov 13, 2020• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.102.195122

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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.
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.
Nonlocal effective interactions are inherent to non-relativistic quantum many-body systems, but their systematic resummation poses a significant challenge known as the ``vertex problem" in many-body perturbation theory. We introduce a renormalization scheme based on a projection-based renormalization condition that selectively resums the most essential nonlocal contributions to the effective interaction vertex, avoiding the computational complexity of the full vertex function. This enables us to derive a renormalized Feynman diagrammatic series with large parameters canceled by counter-diagrams, efficiently generated using a perturbative expansion of the parquet equations and computed using a diagrammatic Monte Carlo algorithm. Applying our approach to a 3D Yukawa Fermi liquid, we demonstrate that the renormalized perturbation theory remains predictive even in the strongly correlated regime and uncover significant sign cancellations between different channels contributing to the scattering amplitude. Our work establishes a novel framework for investigating strong correlations in quantum many-body systems, offering a systematic approach to explore nonlocal theories for challenging systems like the electron liquid in material science.
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 develop a numerically exact method for the summation of irreducible Feynman diagrams for fermionic self-energy in the thermodynamic limit. The technique, based on the diagrammatic determinant Monte Carlo and its recent extension to connected diagrams, allows us to reach high $(\ensuremath{\sim}10)$ orders of the weak-coupling expansion for the self-energy of the two-dimensional Hubbard model. Access to high orders reveals a nontrivial analytic structure of the self-energy and enables its controlled reconstruction with arbitrary momentum resolution in the nonperturbative regime of essentially strong correlations, which has recently been reached with ultracold atoms in optical lattices.
We systematically generate the perturbative expansion for the two-particle spin susceptibility in the Feynman diagrammatic formalism and apply this expansion to a model system---the single-band Hubbard model on a square lattice. We make use of algorithmic Matsubara integration (AMI) [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 99, 035120 (2019)] to analytically evaluate Matsubara frequency summations, allowing us to symbolically impose analytic continuation to the real-frequency axis. We minimize our computational expense by applying graph invariant transformations [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 101, 125109 (2020)]. We highlight extensions of the random-phase approximation and T-matrix methods that, due to AMI, become tractable. We present results for weak interaction strength where the direct perturbative expansion is convergent, and verify our results on the Matsubara axis by comparison to other numerical methods. By examining the spin susceptibility as a function of real frequency via an order-by-order expansion, we can identify precisely what role higher-order corrections play on spin susceptibility and demonstrate the utility and limitations of our approach.
We propose a randomized algorithm to compute the log-partition function of weakly interacting fermions with polynomial runtime in both the system size and precision. Although weakly interacting fermionic systems are considered tractable for many computational methods such as the diagrammatic quantum Monte Carlo, a mathematically rigorous proof of polynomial runtime has been lacking. In this work we first extend the proof techniques developed in previous works for proving the convergence of the cumulant expansion in periodic systems to the non-periodic case. A key equation used to analyze the sum of connected Feynman diagrams, which we call the tree-determinant expansion, reveals an underlying tree structure in the summation. This enables us to design a new randomized algorithm to compute the log-partition function through importance sampling augmented by belief propagation. This approach differs from the traditional method based on Markov chain Monte Carlo, whose efficiency is hard to guarantee, and enables us to obtain a algorithm with provable polynomial runtime.