Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>M</a:mi></a:math>th-order diagram at inverse temperature <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>β</c:mi></c:math> and spectral width <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>ω</e:mi><e:mi>max</e:mi></e:msub></e:math> from <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi mathvariant="script">O</g:mi><g:mo mathvariant="bold" stretchy="false">(</g:mo><g:mrow><g:mo stretchy="false">(</g:mo><g:mi>β</g:mi><g:msub><g:mrow><g:mi>ω</g:mi></g:mrow><g:mrow><g:mi>max</g:mi></g:mrow></g:msub><g:msup><g:mrow><g:mo stretchy="false">)</g:mo></g:mrow><g:mrow><g:mn>2</g:mn><g:mi>M</g:mi><g:mo>−</g:mo><g:mn>1</g:mn></g:mrow></g:msup></g:mrow><g:mo mathvariant="bold" stretchy="false">)</g:mo></g:mrow></g:math> for a direct quadrature to <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mrow><p:mi mathvariant="script">O</p:mi><p:mo mathvariant="bold" stretchy="false">(</p:mo><p:mrow><p:mi>M</p:mi><p:mo stretchy="false">(</p:mo><p:mi>log</p:mi><p:mo stretchy="false">(</p:mo><p:mi>β</p:mi><p:msub><p:mrow><p:mi>ω</p:mi></p:mrow><p:mrow><p:mi>max</p:mi></p:mrow></p:msub><p:mo stretchy="false">)</p:mo><p:msup><p:mrow><p:mo stretchy="false">)</p:mo></p:mrow><p:mrow><p:mi>M</p:mi><p:mo>+</p:mo><p:mn>1</p:mn></p:mrow></p:msup></p:mrow><p:mo mathvariant="bold" stretchy="false">)</p:mo></p:mrow></p:math>, with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mrow><ab:msub><ab:mrow><ab:mi>Ca</ab:mi></ab:mrow><ab:mrow><ab:mn>2</ab:mn></ab:mrow></ab:msub></ab:mrow><ab:mrow><ab:msub><ab:mrow><ab:mi>RuO</ab:mi></ab:mrow><ab:mrow><ab:mn>4</ab:mn></ab:mrow></ab:msub></ab:mrow></ab:math>, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024