Improved treatment of fermion-boson vertices and Bethe-Salpeter equations in nonlocal extensions of dynamical mean field theory

We reconsider the procedure of calculating fermion-boson vertices and the numerical solution of Bethe-Salpeter equations, used in nonlocal extensions of dynamical mean field theory. Because of the frequency dependence of the vertices, a finite-frequency box for matrix inversions is typically used, which often requires some treatment of the asymptotic behavior of vertices. Recently [J. Kunes, Phys. Rev. B 83, 085102 (2011), A. Tagliavini et al., Phys. Rev. B 97, 235140 (2018)] it was proposed to split the considered frequency box into smaller and larger ones; in the smaller-frequency box the numerically exact vertices are used, while beyond this box asymptotics of vertices are applied. Yet, this method requires numerical treatment of vertex asymptotics (including corresponding matrix manipulations) in the larger-frequency box and/or knowledge of fermion-boson vertices, which may not be convenient for numerical calculations. In the present paper we derive the formulas which treat analytically the contribution of vertices beyond the chosen frequency box, such that only numerical operations with vertices in the chosen small-frequency box are required. The method is tested on the Hubbard model and can be used in a broad range of applications of nonlocal extensions of dynamical mean field theory.