Diagram Technique for Quantum Models with Internal Lie-Group Dynamics, pp. 141-188
Authors: (L.V. Lutsev, Research Institute "Ferrite-Domen", Chernigovskaya 8, Saint Petersburg, Russia)
Abstract: Quantum systems with strong electron interactions and nanosystems can possess a more complicated internal Lie-group dynamics in comparison with the Lie-group dynamics of Bose and Fermi systems described by the Heisenberg algebra and superalgebra, respectively. In order to investigate properties of such quantum systems, we represent operators of quantum systems by differential operators over the commutative algebra A of regular functionals. Taking into account this differential representation, we construct a new diagram technique based on the expansion of the generating functional for the temperature Green functions. The generating functional is determined by differential functional equations. Solutions of the differential functional equations belong to a module over the algebraA and are found in the form of series. Each term of the series corresponds to a diagram. This method of the construction of the diagram expansion is more general than the methods based on the Wick theorem and on the expansion of functional integrals. The differential representation makes it possible to generalize functional equations and the diagram technique for the case of quantum systems on topologically nontrivial manifolds by the substitution of the generating functional on a sheaf of function rings on a nontrivial manifold for the generating functional of a constant sheaf of functions. Nontrivial cohomologies of the manifold, on which the quantum system is acted, lead to the existence of additional excitations. The self-consistent-field approximation and the approximation of effective Green functions and interactions are considered. Poles of the matrix of effective interactions and Green functions (P-matrix) determine quasi-particle excitations of the quantum system. For special cases of models the diagram expansion is simplified. In particular, if the internal dynamics is determined by the Heisenberg algebra (superalgebra), the diagram expansion reduces to Feynman’s diagrams for Bose (Fermi) quantum systems. We carry out detailed consideration of the diagram technique for the Heisenberg model of the spin system described by the Lie group Spin(3) and find the self-consistent field, spin excitations and relaxation of spin wave modes. The reduction of the developed diagram technique and excitations for the case of the spin system with an uniaxial anisotropy and the diagram technique for the Hubbard model are considered.