# (p.267) Appendix G Spectral function in the noncrossing approximation (NCA)

# (p.267) Appendix G Spectral function in the noncrossing approximation (NCA)

In this appendix, we present the noncrossing approximation (NCA) results for the spectral function *A*(*ω*), provide the NCA definition of the characteristic scale ${T}_{K}$, study the low-energy spectral features in the vicinity of various fixed points, discuss the changes induced by the crossovers, and explain the behavior of the thermopower $\mathrm{\alpha}(T)$ in terms of the redistribution of spectral weight within the Fermi window. Only the Ce case is considered, in which the ground-state doublet is separated from an excited quadruplet by the crystal filed (CF) splitting Δ and, as argued in the main text, the application of pressure is assumed to increase the hybridization width Γ. The results obtained for $\mathrm{\Gamma}<\mathrm{\Delta}$ are shown in Fig. G.1, where $A(\mathrm{\omega})$ is plotted as a function of frequency for several temperatures. At high temperature, $T\simeq {T}_{\mathrm{\Delta}}$, the spectral function has a broad charge-excitation peak somewhat above ${E}_{f}$ and a narrower resonance of half-width Δ centered below $\mathrm{\mu}$. This low-energy resonance is a many-body effect due to the hybridization of the conduction states with the 4*f*-states and is typical of the exchange scattering on the full multiplet. In this temperature range, the Fermi window contains more spectral weight below than above $\mathrm{\mu}$ (see Fig. G.1 (B)), and $\mathrm{\alpha}(T)<0$. The magnetic susceptibility (Bickers et al., 1987) is Curie–Weiss-like, with a very small Curie–Weiss temperature and a Curie constant that is close to the free Ce-ion value. The maximum of $\mathrm{\alpha}(T)$ at about ${T}_{\mathrm{\alpha}}\simeq {T}_{\mathrm{\Delta}}/2$ is here negative, ${\mathrm{\alpha}}_{\text{max}}<0$, but a slight increase of Γ would make ${\mathrm{\alpha}}_{max}$ positive. All these features are typical of the LM fixed point corresponding to a fully degenerate *f*-state.

At lower temperature, $T<{T}_{\mathrm{\alpha}}$, the CF splits the many-body resonance into two peaks. The larger one grows below $\mathrm{\mu}$ and the smaller one above $\mathrm{\mu}$ (see Fig. G.1 (B)). This asymmetry is enhanced as Γ is reduced, which is typical of the Anderson model with CF splittings (Bickers et al., 1987); the increase of the low-energy spectral weight below $\mathrm{\mu}$ gives rise to a large negative thermopower. A further reduction in temperature leads, for $T\ll {T}_{\mathrm{\Delta}}$, to a rapid growth of an additional peak very close to $\mathrm{\mu}$, such that $A(\mathrm{\omega})$ acquires three pronounced low-energy peaks (see Fig. G.1) (the physical origin of these many-body resonances is explained in detail in Bickers et al. (1987)). The peak centered at ${\mathrm{\omega}}_{0}\ll \mathrm{\Delta}$ is the Kondo resonance and its appearance below $T\le $ 40 K marks the onset of the LM regime associated with the lowest CF level. The two CF peaks centered at about ${\mathrm{\omega}}_{0}\pm \mathrm{\Delta}$ are outside the Fermi window, and do not affect the low-temperature transport and thermodynamics. Once the Kondo peak appears, the Fermi window shows more spectral weight above than below $\mathrm{\mu}$, and $\mathrm{\alpha}(T)$ is positive, (p.268) which is just the opposite of what one finds for $T\ge {T}_{\mathrm{\Delta}}$. The center of the Kondo resonance saturates at low temperatures at the energy ${\mathrm{\omega}}_{0}>0$, which provides the NCA definition of the Kondo scale, ${k}_{B}{T}_{K}={\mathrm{\omega}}_{0}$. In the symmetric Anderson model, ${T}_{K}$ is related to the width of the Kondo resonance, but in the highly asymmetric case we are dealing with here, the current definition is more appropriate. Comparison with numerical renormalization group calculations (Costi et al., 1994) shows that ${\mathrm{\omega}}_{0}$ gives a reliable estimate of the Kondo temperature even for a doubly degenerate Anderson model, and we assume that the NCA definition of ${T}_{K}$ provides the correct Kondo scale of the CF-split single-impurity Anderson model as well. Because the Kondo resonance is asymmetric with respect to the $\mathrm{\omega}=0$ line and has more states above than below $\mathrm{\mu}$, the reduction of temperature enhances $\mathrm{\alpha}(T)$ until it reaches, at ${T}_{K}$, the maximum value ${\mathrm{\alpha}}_{0}$. A further temperature reduction drives the center of the Kondo resonance outside of the Fermi window, and the thermopower drops. However, most Ce and Yb systems with very small ${T}_{K}$ have a phase transition above ${T}_{K}$, and to discuss the normal-state properties of (a)-type systems it is sufficient to consider the NCA solution for $T\ge {T}_{K}$.

An increase of the coupling to $\mathrm{\Gamma}>\mathrm{\Delta}$ has a drastic effect on $A(\mathrm{\omega})$, as illustrated in Fig. G.2, where $A(\mathrm{\omega})$ is plotted for $\mathrm{\Gamma}=0.12$ eV. The charge-excitation peak is transformed into a broad background (see Fig. G.2 (A)), and the only prominent feature at $T\simeq {T}_{\mathrm{\Delta}}$ is the low-energy resonance of half-width Δ centered above $\mathrm{\mu}$. This low-energy resonance is due to the exchange scattering of conduction electrons on the full CF multiplet, which gives rise to the maximum of $\mathrm{\alpha}(T)$ in the LM regime. The Fermi window (see Fig. G.2 (B)) contains more spectral weight above than below $\mathrm{\mu}$, so that $\mathrm{\alpha}(T)>0$. The reduction of temperature below ${T}_{\mathrm{\alpha}}$ removes some spectral weight above $\mathrm{\mu}$ and brings additional spectral weight below $\mathrm{\mu}$, which reduces $\mathrm{\alpha}(T)$ and leads to a minimum. A further reduction in temperature leads to rapid growth of the Kondo (p.269) peak at ${\mathrm{\omega}}_{0}$, and the CF peak at ${\mathrm{\omega}}_{0}+\mathrm{\Delta}$, but the negative CF peak does not develop. That is, an increase in pressure removes the lower CF peak, and shifts the Kondo and the upper CF peak to higher energies, without changing their separation Δ. The Fermi window shows more spectral weight above than below $\mathrm{\mu}$, so that $\mathrm{\alpha}(T)$ is positive and grows as the temperature is lowered. The maximum ${\mathrm{\alpha}}_{0}$ is reached at ${T}_{K}$, when the Kondo resonance is fully developed. The characteristic energy scale is defined again by the position of the Kondo peak, ${k}_{B}{T}_{K}={\mathrm{\omega}}_{0}$, which can be quite large. For $T\le {T}_{K}$, the Fermi window becomes narrower than the Kondo resonance, and $\mathrm{\alpha}(T)$ drops below ${\mathrm{\alpha}}_{0}$. For $T\ll {T}_{K}$, where Fermi liquid behavior is expected (Bickers et al., 1987), the NCA leads to an unphysical peak in the spectral function at $\mathrm{\omega}=\mathrm{\mu}$, which makes ${\mathrm{\rho}}_{\text{mag}}(T)$ and $\mathrm{\alpha}(T)$ much larger than the exact result. However, once ${T}_{K}$ is obtained from the NCA calculations, the low-temperature transport can be inferred from the universal power laws that hold in the Fermi liquid regime, as discussed in the main text.

A further increase in Γ shifts the Kondo and the CF peaks to higher energies, and changes their relative spectral weight, as shown in Fig. G.3, where the low-frequency part of $A(\mathrm{\omega})$ is shown in the (A) for $\mathrm{\Gamma}=2\mathrm{\Delta}$. The Kondo scale is still defined by the center of the Kondo peak, even though the latter is now reduced to a hump on the low-energy side of a large peak centered at ${\mathrm{\omega}}_{0}+\mathrm{\Delta}$. The unphysical NCA spike at $\mathrm{\omega}=0$ can be seen at the lowest temperatures. The thermopower is positive at all temperatures and has only a shoulder below ${T}_{\mathrm{\alpha}}$. A quantitative comparison between ${T}_{K}$, defined by the position of the Kondo resonance, and the position of the Kondo anomaly in $\mathrm{\alpha}(T)$ becomes difficult.

(p.270) Finally, for $\mathrm{\Gamma}>2\mathrm{\Delta}$, we find $A(\mathrm{\omega})$ with a single broad peak centered at $\tilde{{E}_{f}}>0$, as shown in Fig. G.3 (B). The CF excitations are now absent, which is typical for the Anderson model in the vicinity of the valence-fluctuating fixed point. The relevant energy scale at low temperature is defined as ${k}_{B}{T}_{K}=\tilde{{E}_{f}}$, and shows an almost linear dependence on $\mathrm{\Gamma}$. The unphysical spike at $\mathrm{\omega}=0$ appears at higher temperature and is more pronounced than for small Γ. The thermopower is always positive, and grows monotonically from small values at low temperatures toward a high-temperature maximum at ${T}_{\mathrm{\alpha}}$. The initial slope of $\mathrm{\alpha}(T)$ obtained from the NCA results for the transport coefficients is very much overestimated with respect to the Fermi liquid result based on Eq. (14.51). An increase in temperature above ${T}_{\mathrm{\alpha}}$ modifies the excitation spectrum on an energy scale of the order of $\tilde{{E}_{f}}$ and reduces $\mathrm{\alpha}(T)$.

An further increase in Γ (pressure) leaves ${\mathrm{\alpha}}_{max}$ unchanged and enhances ${T}_{\mathrm{\alpha}}$ in a way that appears unrelated to the almost-linear dependence of $\tilde{{E}_{f}}$ on Γ, which suggests that the description of the valence-fluctuating regime requires more than one energy scale.

## Notes:

*Modern Theory of Thermoelectricity*. First Edition. Veljko Zlatić and René Monnier

© Veljko Zlatić and René Monnier 2014. Published in 2014 by Oxford University Press.