4 FIG. 2. Graphical representation of Eq. (12). The dashed lines within the boxes indicate charge propagation from the incoming to the corresponding outgoing line. a graphical representation and notation used in Eq. (12). Due to energy conservation in this model the incoming and outgoing frequencies must equal. Notice that the two vertices γ and ϕ never mix up and are completely detached in the solution. The former vertex is relevant for the transport while the latter one for the thermody- namics. It is hence sufficient to take into account only vertex γ for the calculation of the electrical conductivity. We can represent vertex γ via a local Bethe-Salpeter equation with a local irreducible vertex Λ$_{m,n}$. We have Charge conservation leads to a generalized Ward iden- tity that matches a non-equilibrium variation of the self- energy δ Σ$_{m,n}$ with the equilibrium irreducible vertex Λ$_{m,n}$ where we abbreviated The latter representation was derived with the aid of sta- tionarity equations, Eqs. (10). We have derived all the necessary equilibrium quanti- ties needed for the calculation of the electrical conductiv- ity. We must, however, go beyond the local, mean-field limit in order to calculate transport properties. That is, we have to perturb the equilibrium with time-dependent non-local excitations. To do it in a controlled manner we use a perturbation expansion around a mean-field. This can be consistently achieved within an asymptotic expansion in high spatial dimensions. An asymptotic expansion around a mean-field solu- tion in d = ∞ for non-interacting disordered electrons was recently derived by one of us in Ref. 19. The only ingredients of such an expansion are the local mean-field vertex γ and the the non-local one-electron propagator. The relevance and applicability of this expansion goes beyond the model of disordered electrons. It can be ap- plied to any model where energy is conserved in scat- tering events (elastic scatterings) and where only spatial fluctuations matter. The disordered FKM studied here falls into this category and we can hence use the expan- sion concept of Ref. 19 for it. It is a considerable ad- vantage to expand around a mean-field solution, since all the effects of scatterings on impurities are included al- ready in the local vertex γ . The variance of the potential fluctuations is not a small parameter and the expansion terms do not depend on whether the scatterings are due to thermally equilibrated or randomly distributed static impurities, that is, whether the disorder is annealed or quenched. We construct the expansion around a mean field as an asymptotic series on a hyper-cubic lattice in high spatial dimensions. The expansion parameter is the off-diagonal one-electron propagator from the mean-field theory. We define where we denoted ζ = z − Σ( z ) and the local self-energy Σ( z ) is that of the mean-field solution. The off-diagonal two particle bubble is a convolution of the off-diagonal one-electron propagators. We hence define The frequency indices are external parameters and we suppress them when they are not necessary to specify a particular type of the one or two-electron propagators. The asymptotic limit of the full two-particle vertex in high spatial dimensions contains beyond the local mean-field vertex γ also non-local contributions from the electron-hole and electron-electron ladders. 19 It can be represented as follows where we denoted Q = q + k + k ′ the momentum con- served in the electron-electron channel. Notice that the contribution from the electron-hole channel with ¯ χ ( q ) is part of the two-particle vertex from the Coherent Po- tential Approximation (CPA) and can be derived from a Velick´ y-Ward identity. 20 The two-particle vertex from CPA does not carry the full 1 /d correction to the local vertex and moreover it is not electron-hole symmetric on the two-particle level. 19 A consistent extension of the local mean-field two-particle vertex must contain both non-local contributions from the electron-hole and the electron-electron channels as given in Eq. (18). IV. EXPANSION AROUND MEAN FIELD – VERTEX CORRECTIONS TO THE ELECTRICAL CONDUCTIVITY Γ$_{m,n}$ = Λ$_{m,n}$ 1 − Λ$_{m,n}$ G$_{m}$G$_{n}$ . (13) Λ$_{m,n}$ = δ Σ$_{m,n}$ δG$_{m,n}$ = 1 G$_{m}$G$_{n}$ [ 1 − λ − 1 m,n ] (14) λ$_{m,n}$ = 〈 1 − f ( E$_{f}$ + E $_{V}$) [1 + G$_{m}$ (Σ$_{m}$ V )] [1 + G$_{n}$ (Σ$_{n}$ − V ))] + f ( E$_{f}$ + E $_{V}$) [1 + G$_{m}$ (Σ$_{m}$ − V − U )] [1 + G$_{n}$ (Σ$_{n}$ − V − U )] 〉 av . (15) ¯ G ( k , ζ ) = 1 ζ − ϵ ( k ) − ∫ dϵρ ( ϵ ) ζ − ϵ (16) ¯ χ ( ζ, ζ $^{′}$; q ) = 1 N ∑ k ¯ G ( k , ζ ) ¯ G ( k + q , ζ $^{′}$) . (17) Γ$_{kk}$ ′ ( q ) = γ [ 1 + γ ( ¯ χ ( q ) 1 − γ ¯ χ ( q ) + ¯ χ ( Q ) 1 − γ ¯ χ ( Q ) )] (18) GLYPH<A1> Γ n m k l = δ$_{m,l}$δ$_{n,k}$ GLYPH<A2> γ n m n m + δ$_{m,n}$δ$_{k,l}$ GLYPH<A3> ϕ m m k k