The Falicov-Kimball model was introduced in 1969 as a statistical model for metal-insulator transitions; it includes itinerant and localized electrons that mutually interact with a local Coulomb interaction and is the simplest model of electron correlations. It can be solved exactly with dynamical mean-field theory in the limit of large spatial dimensions, which provides an interesting benchmark for the physics of locally correlated systems. In this review, the authors develop the formalism for solving the Falicov-Kimball model from a path-integral perspective and provide a number of expressions for single- and two-particle properties. Many important theoretical results are examined that show the absence of Fermi-liquid features and provide a detailed description of the static and dynamic correlation functions and of transport properties. The parameter space is rich and one finds a variety of many-body features like metal-insulator transitions, classical valence fluctuating transitions, metamagnetic transitions, charge-density-wave order-disorder transitions, and phase separation. At the same time, a number of experimental systems have been discovered that show anomalies related to Falicov-Kimball physics [including ${\mathrm{YbInCu}}_4,$ ${\mathrm{EuNi}}_2({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x{)}_2,$ ${\mathrm{NiI}}_2,$ and ${\mathrm{Ta}}_x\mathrm{N}].$

The temperature dependence of the thermoelectric power (TEP) of metallic systems with cerium and ytterbium ions exhibits characteristic features which we explain by the Coqblin-Schrieffer model (CSM). We specify a given system by the degeneracy and splitting of the crystal-field (CF) levels, the strength of the exchange and potential scattering, and the number of f electrons or f holes; for cerium and ytterbium ions we assume $n_f<~1$ and $n_f^{\mathrm{hole}}<~1,$ respectively. The Kondo temperature $T_K$ is then generated by the “poor man’s scaling”; it separates a local-moment (LM) from a Fermi-liquid (FL) regime. In the LM regime $(T>~T_K)$ we calculate the TEP by a renormalized perturbation expansion, in which the exchange coupling J is also renormalized by the poor man’s scaling. This gives the TEP with a large peak at high temperatures and a sign change at $T_x\simeq \alpha T_K,$ where for parameters used in this paper we have $\alpha$ between roughly 2.5 and 10. For $n_f\simeq 1$ and large CF splitting, we find a broad temperature range $T_K\ll T<~T_x,$ in which the TEP of the CSM is negative. In the FL regime $(T<~T_K)$ we neglect the excited CF states, reduce the CSM to an effective spin-degenerate exchange model, map it on an effective spin-degenerate Anderson model, and calculate the TEP by an expansion in terms of $U\propto 1/J.$ For cerium ions ${(n}_f<~1)$ we obtain in this way the TEP which follows for $T\ll T_K$ a linear FL law, attains at about $T_K/2\mathrm{}$ a positive maximum, and changes sign above $T_K.$ The results pertaining to ytterbium ions ${(n}_f^{\mathrm{hole}}<~1)$ are obtained by reflecting the $n_f<~1$ curves on the temperature axis. The overall results obtained for the CSM in such a way explain the essential features of the temperature, pressure, and doping dependence of the TEP in cerium and ytterbium systems. Unfortunately, neither our high-temperature nor the low-temperature expansion provide the details or locate the minimum of the negative TEP which we find in the transition region between the FL and LM regimes.

``Bad metals'' are known to have a large linear resistivity at high $T$ that is universally seen near the Mott-Hubbard insulating phase. Less well known is that the thermopower $$\alpha${}(T)$ of the Mott-Hubbard systems also exhibits simple universal features: (i) close to the insulating phase, where the resistivity has a low-temperature upturn, $$\alpha${}(T)$ has a pronounced low-$T$ peak that shifts to higher $T$ with doping; (ii) when the resistivity is nearly linear, which occurs at moderate doping, $$\alpha${}(T)$ has a small low-$T$ peak that shifts to lower $T$ with doping and has a high-$T$ sign change; and (iii) at the highest doping, where the resistivity acquires a ${T}^{2}$ term, $$\alpha${}(T)$ is negative and depends monotonically on $T$. The universality $$\alpha${}(T)$ can be understood using the Kelvin formula and the fact that the chemical potential for doped Mott insulators displays similar behavior at high $T$. The universality is illustrated with the exact solution of the simplest model for a doped Mott insulator at high $T$.

Perturbation expansion for the asymmetric single-orbital Anderson Hamiltonian is extended to the case with finite magnetic field, and the self-energy part of the $d$-electron Green function is calculated up to the second order in $u=\frac{U }{\pi \Delta }$ for arbitrary asymmetry $\eta =\frac{1}{2}+\frac{{\epsilon }_d}{U}$ and magnetic field $h$. Zero-temperature density of localized states ${\rho }_{d\sigma }\left(\omega \right)$, magnetic moment $m$ and static spin susceptibility $\chi$, and low-temperature specific heat $C_v=\gamma T$ and magnetoresistance $D$ of a dilute alloy are evaluated within the $u^2$ approximation. Plausible arguments are given (which become exact in the symmetric case) that the higher-order terms do not change the qualitative features of our results. For large enough $u$ the many-body (MB) effects give rise to three different types of behavior, depending on the value of asymmetry: (i) spin-fluctuation behavior for $\eta <\frac{1}{2}$, with large fluctuations of the $d$-level magnetization; (ii) mixed-valence behavior for $\eta \simeq \frac{1}{2}$, with large fluctuations of the $d$-level occupation; (iii) essentially mean-field behavior for $\eta >\frac{1}{2}$, where the MB effects are negligible even for $u\gg 1$. Thus both $\chi$ and $\gamma$ at $h=0\mathrm{}$ are enhanced by the increase of $u$ for $\eta <\frac{1}{2}$, reduced for $\eta >\frac{1}{2}$, and practically unaffected at $\eta \simeq \frac{1}{2}$. As functions of $\eta$ for a fixed $u$, $\chi$ and $\gamma$ have a maximum at $\eta =0\mathrm{}$ and decrease monotonously with increasing $\eta$, and do so more steeply the higher the $u$. For $h\ne 0$ and $\eta \le \frac{1}{2}$, $\chi$ and $\gamma$ decrease monotonously with increasing $h$, the decrease being quicker for higher $u$ and slower for higher $\eta$. For $\eta >\frac{1}{2}$, both