Renormalization group flow of the Luttinger-Ward functional: conserving   approximations and application to the Anderson impurity model
  We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle irreducible vertex functions, given a cut-off in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to non-conserving approximations introduced in an earlier publication ctuation exchange approximation as well as to nonconserving approximations introducen [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle irreducible functional renormalization group schemes. However, we identify a simple static two-particle irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position. 
Jan Frederik Rentrop and Volker Meden and Severin Georg Jakobs
cond-mat.str-el