### 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 Φ^{4}theory, 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 10^{9}, 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 10^{9}diagrams 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 Φ^{4}theory and lattice models with local interactions are discussed.

Original language | English |
---|---|

Pages (from-to) | 437-460 |

Number of pages | 24 |

Journal | Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties |

Volume | 69 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

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

### Cite this

^{4}theory and lattice models with local interactions up to eighth order.

*Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties*,

*69*(3), 437-460. https://doi.org/10.1080/01418639408240120

**Diagrammatic expansion of a Φ ^{4}theory and lattice models with local interactions up to eighth order.** / Gulácsi, Z.; Gulácsi, Miklós.

Research output: Contribution to journal › Article

^{4}theory and lattice models with local interactions up to eighth order',

*Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties*, vol. 69, no. 3, pp. 437-460. https://doi.org/10.1080/01418639408240120

^{4}theory and lattice models with local interactions up to eighth order. Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties. 1994;69(3):437-460. https://doi.org/10.1080/01418639408240120

}

TY - JOUR

T1 - Diagrammatic expansion of a Φ4theory and lattice models with local interactions up to eighth order

AU - Gulácsi, Z.

AU - Gulácsi, Miklós

PY - 1994

Y1 - 1994

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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U2 - 10.1080/01418639408240120

DO - 10.1080/01418639408240120

M3 - Article

VL - 69

SP - 437

EP - 460

JO - Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties

JF - Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties

SN - 1364-2812

IS - 3

ER -