Many-body systems are those in which particle interactions cannot be neglected. In particular, when they are strong, they cause critical fluctuations and may lead to qualitative changes and phase transitions in extended systems. The critical behavior must be treated self-consistently. Even simple models of correlated elementary objects are, however, unsolvable. We hence must resort to approximations, apart from a few limiting cases where exact partial solutions exist. We gain the approximations either from numerical simulations or from semi-analytic calculations. The numerical approach tends to offer unbiased approximations with all degrees of freedom left in play, while the analytic approximations are based on a reduction of complexity of interaction effects. The numerical calculations offer good quantitative predictions that can set trends in the dependence of the solution on the model input parameters. The latter schemes aspire to reproduce qualitative features of the exact solution. They, unlike the numerical methods, can address and control singularities directly.
The basic and first analytic estimates of the model solutions are mean-field theories. A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the two-particle vertices, determining the response of the interacting systems on external perturbations, based on the parquet equations. We explain in detail how to reduce the full set of analytically unsolvable parquet equations not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable in the critical regions of their singularities. We connect the self-energy with the two-particle vertices to satisfy the thermodynamic Ward identity and the dynamical Schwinger-Dyson equation and discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. Finally, we demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model. We qualitatively reproduced the behavior of the exact algebraic solution in the Kondo limit.
Kontaktní osoba: Václav Janiš