Jason Kaye, Zhen Huang, Hugo U. R. Strand, Denis Golež
{"title":"使用可分离基函数分解虚时费曼图:安德森杂质模型强耦合展开","authors":"Jason Kaye, Zhen Huang, Hugo U. R. Strand, Denis Golež","doi":"10.1103/physrevx.14.031034","DOIUrl":null,"url":null,"abstract":"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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>th-order diagram at inverse temperature <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>β</mi></math> and spectral width <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mi>ω</mi><mi>max</mi></msub></math> from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi mathvariant=\"script\">O</mi><mo mathvariant=\"bold\" stretchy=\"false\">(</mo><mrow><mo stretchy=\"false\">(</mo><mi>β</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>max</mi></mrow></msub><msup><mrow><mo stretchy=\"false\">)</mo></mrow><mrow><mn>2</mn><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo mathvariant=\"bold\" stretchy=\"false\">)</mo></mrow></math> for a direct quadrature to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi mathvariant=\"script\">O</mi><mo mathvariant=\"bold\" stretchy=\"false\">(</mo><mrow><mi>M</mi><mo stretchy=\"false\">(</mo><mi>log</mi><mo stretchy=\"false\">(</mo><mi>β</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>max</mi></mrow></msub><mo stretchy=\"false\">)</mo><msup><mrow><mo stretchy=\"false\">)</mo></mrow><mrow><mi>M</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo mathvariant=\"bold\" stretchy=\"false\">)</mo></mrow></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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mrow><mi>Ca</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>RuO</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></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.","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":null,"pages":null},"PeriodicalIF":11.6000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion\",\"authors\":\"Jason Kaye, Zhen Huang, Hugo U. R. Strand, Denis Golež\",\"doi\":\"10.1103/physrevx.14.031034\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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 <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>M</mi></math>th-order diagram at inverse temperature <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>β</mi></math> and spectral width <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msub><mi>ω</mi><mi>max</mi></msub></math> from <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mrow><mi mathvariant=\\\"script\\\">O</mi><mo mathvariant=\\\"bold\\\" stretchy=\\\"false\\\">(</mo><mrow><mo stretchy=\\\"false\\\">(</mo><mi>β</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>max</mi></mrow></msub><msup><mrow><mo stretchy=\\\"false\\\">)</mo></mrow><mrow><mn>2</mn><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo mathvariant=\\\"bold\\\" stretchy=\\\"false\\\">)</mo></mrow></math> for a direct quadrature to <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mrow><mi mathvariant=\\\"script\\\">O</mi><mo mathvariant=\\\"bold\\\" stretchy=\\\"false\\\">(</mo><mrow><mi>M</mi><mo stretchy=\\\"false\\\">(</mo><mi>log</mi><mo stretchy=\\\"false\\\">(</mo><mi>β</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>max</mi></mrow></msub><mo stretchy=\\\"false\\\">)</mo><msup><mrow><mo stretchy=\\\"false\\\">)</mo></mrow><mrow><mi>M</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo mathvariant=\\\"bold\\\" stretchy=\\\"false\\\">)</mo></mrow></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 <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mrow><msub><mrow><mi>Ca</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>RuO</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></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.\",\"PeriodicalId\":20161,\"journal\":{\"name\":\"Physical Review X\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":11.6000,\"publicationDate\":\"2024-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Physical Review X\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1103/physrevx.14.031034\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review X","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevx.14.031034","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
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 th-order diagram at inverse temperature and spectral width from for a direct quadrature to , 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 , 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.
期刊介绍:
Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.