9  0  0.001  0.002  0.003  0.004  0.005  0.006  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8 r U FIG. 8. Ratio r = | Δ σ | /σ 0 for filling n = 0 . 7. domly distributed static impurities. We concentrated on quantum coherence effects due to correlated back- scatterings and its impact on the electrical conductiv- ity. We calculated vertex corrections to the mean-field (Drude) conductivity. We used a systematic expansion around the mean-field solution via the asymptotic limit to high spatial dimensions. Our principal finding is that elastic scatterings always lead to diminution of the Drude conductivity. That is, vertex corrections due to elastic scatterings have negative sign. The sign of the vertex correction is determined by its leading high-dimensional term, that is by the numerators in Eq. (22). The sign of the vertex correction is not affected by the type of randomness in the distribution of the scattering impuri- ties. Coulomb interaction in the pure Falicov-Kimaball model has the same effect on the electrical conductivity as variance of fluctuations of the atomic potential in the disordered Anderson model, it hinders diffusion. We explicitly calculated only the leading high- dimensional vertex correction to the Drude zero- temperature conductivity in the paramagnetic phase of FKM. Quantitatively the correction is almost everywhere two orders smaller than the Drude term. It is due to the fact that the Drude conductivity diverges in the limit of the Fermi gas while the vertex correction asymptot- ically approaches a finite value when the interaction is switched off. Only close to the metal-insulator transition in the electron-hole symmetric case the vertex correction is of order of the Drude one, however, they both vanish at the transition point. The leading high-dimensional contribution to the vertex correction of the mean-field conductivity is quantitatively negligible. It is important only for determining the sign of the vertex correction. To obtain a more realistic values of the vertex correc- tions in low-dimensional systems, one has to consider the full representation of the vertex correction from Eq. (22) containing the Cooper pole. Only then we are able to include a sizable impact of spatial dimensionality. The method for evaluating vertex corrections to the mean-field electrical conductivity via an asymptotic ex- pansion in high spatial dimensions is universal and is suitable for any model with elastically scattered elec- trons. We can hence use it to investigate whether the Coulomb interaction in FKM can lead to suppression of diffusion in low-dimensional systems as well as to study an interplay between the Anderson localization and Mott-Hubbard metal-insulator transition in the dis- ordered Falicov-Kimball model. Research on this problem was carried out within project AV0Z10100520 of the Academy of Sciences of the Czech Republic and supported in part by Grant No. 202/07/0644 of the Grant Agency of the Czech Republic. We thank M. ˇ Zonda for illuminating discussions on the numerical solution of the Falicov-Kimball model and J. Kolorenˇ c for discussions on behavior of the Drude con- ductivity in the weak scattering limit. ACKNOWLEDGMENT ∗ janis@fzu.cz 1 L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22 , 997 (1969). 2 P. W. Anderson, Phys. Rev. 109 , 1492 (1958). 3 V. Janiˇs, Phys. Rev. B 40 , 11331 (1989). 4 U. Brandt and C. Mielsch, Z. Physik B 75 , 365 (1989). 5 V. Janiˇs, Z. Physik B 83 , 227 (1990). 6 V. Janiˇs and D. Vollhardt, Phys. Rev. B 46 , 15712 (1992). 7 R. Ramirez, L. M. Falicov, and J. C. Kimball, Phys. Rev. B 2 , 3383 (1970). 8 J. K. Freericks and L. M. Falicov, Phys. Rev. B 45 , 1896 (1992). 9 I. Felner and I. Novik, Phys. Rev B 33 , 617 (1986). 10 C. Ates and K. Ziegler, Phys. Rev. A 71 , 063610 (2005). 11 P. Farkaˇsovsk´ y, Eur. Phys. Lett. 84 , 37010 (2008). 12 A. Gonis, Green Functions for Ordered and Disordered Systems (North Holland, Amsterdam, 1992). 13 B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56 , 1469 (1993). 14 K. Byczuk, Phys. Rev. B 71 , 205105 (2005). 15 J. K. Freericks and V. Zlati´ c, Rev. Mod. Phys. 75 , 1333 (2003). 16 P. de Vries, K. Michielsen, and H. de Raedt, Z. Physik B 92 , 353 (1993), ibid 95 , 475 (1994). 17 V. Janiˇs, J. Kolorenˇ c, and V. ˇ Spiˇ cka, Eur. Phys. J. B 35 , 77 (2003). 18 V. Janiˇs, Phys. Rev. Lett. 83 , 2 781 (1999). 19 V. Janiˇs and J. Kolorenˇ c, Phys. Rev. B 71 , 033103, 245106 (2005). 20 B. Velick´ y, Phys. Rev. 184 , 614 (1969).