Asymmetric Hubbard model in the generating functional method: Spectral functions in the Falicov–Kimball limit

In the framework of the dynamical mean field theory, we investigate the densities of states of the fermionic and bosonic branches of the spectrum of the asymmetric Hubbard model, which is used to describe a strongly correlated two-sort $(A,B)$ system of fermions (electrons). To solve the effective one-site problem, we develop an approximate analytic approach based on the Kadanoff–Baym generating functional method. This technique allows constructing the irreducible part (the mass operator) of the particle Green's function in the form of a formal expansion in powers of the coherent potential. In the first order, the scheme reproduces the so-called generalized approximation Hubbard-III. To improve it, we develop a self-consistent method for calculating both the fermionic and bosonic Green's functions. As $U\to\infty$ in the Falicov–Kimball limit for the asymmetric Hubbard model, when the particles of sort $B$ become localized, we find the spectral densities $\rho_B$ and $\rho_{AB}$ of states of both branches and consider the changes of their forms depending on temperature and particle concentrations. Comparing with the exact thermodynamic dependences $\mu_B(n_B)$, we establish the applicability limits of the self-consistent generalized approximation Hubbard-III.