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arXiv:1808.01759 · Physical review. E

Counting Feynman diagrams via many-body relations

Fabian B. Kugler

Published Aug 7, 2018• 12 citations
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Abstract

We present an iterative algorithm to count Feynman diagrams via many-body relations. The algorithm allows us to count the number of diagrams of the exact solution for the general fermionic many-body problem at each order in the interaction. Further, we apply it to different parquet-type approximations and consider spin-resolved diagrams in the Hubbard model. Low-order results and asymptotics are explicitly discussed for various vertex functions and different two-particle channels. The algorithm can easily be implemented and generalized to many-body relations of different forms and levels of approximation.

DOI
10.1103/physreve.98.023303

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We present an iterative algorithm to count Feynman diagrams via many-body relations. The algorithm allows us to count the number of diagrams of the exact solution for the general fermionic many-body problem at each order in the interaction. Further, we apply it to different parquet-type approximations and consider spin-resolved diagrams in the Hubbard model. Low-order results and asymptotics are explicitly discussed for various vertex functions and different two-particle channels. The algorithm can easily be implemented and generalized to many-body relations of different forms and levels of approximation.
Abstract The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here, we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a 16×1616\times 1616×16 16 × 16 lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated-unity approximation. Graphical abstract
Hedin's equations are solved perturbatively in zero dimension to count Feynman graphs for self-energy, polarization, propagator, effective potential and vertex function in a many-body theory of fermions with two-body interaction. Counting numbers are also obtained in the GWGWGW approximation.
We present a simple method for summing so-called parquet diagrams of fermionic many-body systems with competing instabilities using the functional renormalization group. Our method is based on partial bosonization of the interaction using multi-channel Hubbard–Stratonovich transformations. A simple truncation of the resulting flow equations, retaining only the frequency-independent parts of the two-point and three-point vertices amounts to solving coupled Bethe–Salpeter equations for the effective interaction to leading logarithmic order. We apply our method by revisiting the X-ray problem and deriving the singular frequency dependence of the X-ray response function and the particle–particle susceptibility. Our method is quite general and should be useful in many-body problems involving strong fluctuations in several scattering channels.
We present and implement a parquet approximation within the dual-fermion formalism based on a partial bosonization of the dual vertex function which substantially reduces the computational cost of the calculation. The method relies on splitting the vertex exactly into single-boson exchange contributions and a residual four-fermion vertex, which physically embody, respectively, long- and short-range spatial correlations. After recasting the parquet equations in terms of the residual vertex, these are solved using the truncated-unity method of Eckhardt et al. [Phys. Rev. B 101, 155104 (2020)], which allows for a rapid convergence with the number of form factors in different regimes. While our numerical treatment of the parquet equations can be restricted to only a few Matsubara frequencies, reminiscent of Astretsov et al. [Phys. Rev. B 101, 075109 (2020)], the one- and two-particle spectral information is fully retained. In applications to the two-dimensional Hubbard model the method agrees quantitatively with a stochastic summation of diagrams over a wide range of parameters.
One of the fundamental approaches to microscopic many-body theory is through the use of perturbation theory. This paper presents a clear derivation of the equations that sum the two-body and three-body reducible diagrams that are generated from some input set of irreducible diagrams (typically the bare interaction) in a crossing-symmetric form. The diagrammatic structure is useful for both bosons and fermions at zero and finite temperature. The essential features of the procedure for the two-body case are the following. (i) All constructions involve only direct diagrams. (ii) Diagrams can be joined together by five different ladder, chain, and vertex correction operations. (iii) The sequences in which these operations are applied satisfy certain associativity relations. (iv) These associativity relations are used to insure that each permitted diagram is generated exactly once. (v) The ladder-and-chain operations are used to construct reducible vertex sums ${\mathrm{\ensuremath{\Gamma}}}_{0}$ which have no external vertex corrections. (vi) The full direct vertex \ensuremath{\Gamma} consists of the vertex ${\mathrm{\ensuremath{\Gamma}}}_{0}$ plus ${\mathrm{\ensuremath{\Gamma}}}_{0}$ dressed on either or both sides by the full direct vertex \ensuremath{\Gamma}. (vii) Finally, the full vertex is obtained by adding the exchange of the direct vertex. The extension to the three-body formalism proceeds in much the same way.