DIAGRAMMATIC EXPANSION OF A PHI-4 THEORY AND LATTICE MODELS WITH LOCAL INTERACTIONS UP TO 8TH ORDER

Four-degree-vertices-type diagrams are the most frequently used in sol id-state and statistical physics. These are applicable in the followin g fields: (i) standard perturbation expansion of the Green functions; (ii) variational description of models with local interaction; (iii) H ugenholtz-type diagrams for two-body potentials; (iv) renormalization group; (v) standard PHI4 theory, including four point and two point fu nctions, which is applied to localized spin systems, amplitude functio ns and epsilon expansions; (vi) exact solution of the non-interacting, two-dimensional (sixteen, thirty-two, one hundred and twenty-eight) v ertex models. We present the complete topology of the four-degree-vert ices diagrams up to eighth order, in a condensed manner, characterizin g a total number of contributing graphs of order 10(9), and we determi ne the topologically different contributions for every order. The resu lt is a simplification in calculating the contributions of the differe nt orders, e.g. in eighth order instead of calculating 1.62 x 10(9) di agrams 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 al lows the complete set of diagrams with four degree vertices and any ev en number of external lines, to be determined. The graph theory charac teristics and structure of the various high-order diagrams that we ana lyse are also presented. Application in the field of diagrammatic expa nsions of a standard PHI4 theory and lattice models with local interac tions are discussed.