Diagrammatic expansion of a Φ4theory and lattice models with local interactions up to eighth order

Z. Gulácsi, Miklós Gulácsi

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6 Citations (Scopus)

Abstract

Four-degree-vertices-type diagrams are the most frequently used in solid-state and statistical physics. These are applicable in the following fields: (i) standard perturbation expansion of the Green functions; (ii) variational description of models with local interaction; (iii) Hugenholtz-type diagrams for two-body potentials; (iv) renormalization group; (v) standard Φ4theory, including four point and two point functions, which is applied to localized spin systems, amplitude functions and e expansions; (vi) exact solution of the non-interacting, two-dimensional (sixteen, thirty-two, one hundred and twenty-eight) vertex models. We present the complete topology of the four-degree-vertices diagrams up to eighth order, in a condensed manner, characterizing a total number of contributing graphs of order 109, and we determine the topologically different contributions for every order. The result is a simplification in calculating the contributions of the different orders, e.g. in eighth order instead of calculating 1·62 x 109diagrams we deal only with 179. The method by which these diagrams were obtained is described in detail and can be easily applied to determine the topology of higher-order diagrams. We present a procedure that allows the complete set of diagrams with four degree vertices and any even number of external lines, to be determined. The graph theory characteristics and structure of the various high-order diagrams that we analyse are also presented. Application in the field of diagrammatic expansions of a standard Φ4theory and lattice models with local interactions are discussed.

Original languageEnglish
Pages (from-to)437-460
Number of pages24
JournalPhilosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties
Volume69
Issue number3
DOIs
Publication statusPublished - 1994

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diagrams
expansion
Topology
apexes
Graph theory
interactions
Green's function
Physics
topology
graph theory
simplification
Green's functions
solid state
perturbation
physics

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Chemical Engineering(all)

Cite this

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title = "Diagrammatic expansion of a Φ4theory and lattice models with local interactions up to eighth order",
abstract = "Four-degree-vertices-type diagrams are the most frequently used in solid-state and statistical physics. These are applicable in the following fields: (i) standard perturbation expansion of the Green functions; (ii) variational description of models with local interaction; (iii) Hugenholtz-type diagrams for two-body potentials; (iv) renormalization group; (v) standard Φ4theory, including four point and two point functions, which is applied to localized spin systems, amplitude functions and e expansions; (vi) exact solution of the non-interacting, two-dimensional (sixteen, thirty-two, one hundred and twenty-eight) vertex models. We present the complete topology of the four-degree-vertices diagrams up to eighth order, in a condensed manner, characterizing a total number of contributing graphs of order 109, and we determine the topologically different contributions for every order. The result is a simplification in calculating the contributions of the different orders, e.g. in eighth order instead of calculating 1·62 x 109diagrams we deal only with 179. The method by which these diagrams were obtained is described in detail and can be easily applied to determine the topology of higher-order diagrams. We present a procedure that allows the complete set of diagrams with four degree vertices and any even number of external lines, to be determined. The graph theory characteristics and structure of the various high-order diagrams that we analyse are also presented. Application in the field of diagrammatic expansions of a standard Φ4theory and lattice models with local interactions are discussed.",
author = "Z. Gul{\'a}csi and Mikl{\'o}s Gul{\'a}csi",
year = "1994",
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AU - Gulácsi, Z.

AU - Gulácsi, Miklós

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N2 - Four-degree-vertices-type diagrams are the most frequently used in solid-state and statistical physics. These are applicable in the following fields: (i) standard perturbation expansion of the Green functions; (ii) variational description of models with local interaction; (iii) Hugenholtz-type diagrams for two-body potentials; (iv) renormalization group; (v) standard Φ4theory, including four point and two point functions, which is applied to localized spin systems, amplitude functions and e expansions; (vi) exact solution of the non-interacting, two-dimensional (sixteen, thirty-two, one hundred and twenty-eight) vertex models. We present the complete topology of the four-degree-vertices diagrams up to eighth order, in a condensed manner, characterizing a total number of contributing graphs of order 109, and we determine the topologically different contributions for every order. The result is a simplification in calculating the contributions of the different orders, e.g. in eighth order instead of calculating 1·62 x 109diagrams we deal only with 179. The method by which these diagrams were obtained is described in detail and can be easily applied to determine the topology of higher-order diagrams. We present a procedure that allows the complete set of diagrams with four degree vertices and any even number of external lines, to be determined. The graph theory characteristics and structure of the various high-order diagrams that we analyse are also presented. Application in the field of diagrammatic expansions of a standard Φ4theory and lattice models with local interactions are discussed.

AB - Four-degree-vertices-type diagrams are the most frequently used in solid-state and statistical physics. These are applicable in the following fields: (i) standard perturbation expansion of the Green functions; (ii) variational description of models with local interaction; (iii) Hugenholtz-type diagrams for two-body potentials; (iv) renormalization group; (v) standard Φ4theory, including four point and two point functions, which is applied to localized spin systems, amplitude functions and e expansions; (vi) exact solution of the non-interacting, two-dimensional (sixteen, thirty-two, one hundred and twenty-eight) vertex models. We present the complete topology of the four-degree-vertices diagrams up to eighth order, in a condensed manner, characterizing a total number of contributing graphs of order 109, and we determine the topologically different contributions for every order. The result is a simplification in calculating the contributions of the different orders, e.g. in eighth order instead of calculating 1·62 x 109diagrams we deal only with 179. The method by which these diagrams were obtained is described in detail and can be easily applied to determine the topology of higher-order diagrams. We present a procedure that allows the complete set of diagrams with four degree vertices and any even number of external lines, to be determined. The graph theory characteristics and structure of the various high-order diagrams that we analyse are also presented. Application in the field of diagrammatic expansions of a standard Φ4theory and lattice models with local interactions are discussed.

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