Kondo screening and coherence in kagome local-moment metals: Energy scales of heavy fermions in the presence of flat bands

The formation of a heavy Fermi liquid in metals with local moments is characterized by multiple energy and temperature scales, most prominently the Kondo temperature and the coherence temperature, characterizing the onset of Kondo screening and the emergence of Fermi-liquid coherence, respectively. In the standard setting of a wide conduction band, both scales depend exponentially on the Kondo coupling. Here we discuss how the presence of flat, i.e., dispersionless, conduction bands modifies this situation. Focussing on the case of the kagome Kondo lattice model, we utilize a parton mean-field approach to determine the Kondo temperature and the coherence temperature as a function of the conduction-band filling $n_c$, both numerically and analytically. For $n_c$ values corresponding to the flat conduction band located at the Fermi level, we show that the exponential is replaced by a linear dependence for the Kondo temperature and a quadratic dependence for the coherence temperature, while a cubic law emerges for the coherence temperature at $n_c$ corresponding to the band edge between the flat and dispersive bands. We discuss the implications of our results for pertinent experimental data.

Figure 1

Tight-binding band structure on the kagome lattice, Eq. (3), with the flat band located at the bottom [31]. The insets show the lattice and the corresponding Brillouin zone.

Figure 2

Overview of $T=0$ Fermi-level band-structure features of the kagome Kondo lattice as a function of conduction-band filling $n_c$ for the bare conduction band (top) and the hybridized heavy-fermion bands (bottom). Hatched regions correspond to a flat band located at the Fermi level [31].

Figure 3

[(a) and (b)] Kondo temperature and [(c) and (d)] coherence temperature as a function of conduction-band filling $n_c$ and Kondo coupling $J_{\mathrm{K}}$, shown on [(a) and (c)] linear and [(b) and (d)] logarithmic color scales. The vertical lines correspond to the special fillings $n_c=1/3,2/3,1,7/6,4/3$ and $3/2$. Screening is seen to be enhanced in the regime of the flat conduction band, $0<n_c<2/3$.

Figure 4

Kondo temperature $T_{\mathrm{K}}$ vs. Kondo coupling $J_{\mathrm{K}}$ for different fillings $n_c$. The dotted (dashed) line indicates a linear [log-linear, Eq. (23)] dependence for reference.

Figure 5

Coherence temperature $T_{\mathrm{coh}}$ vs Kondo coupling $J_{\mathrm{K}}$ for different fillings $n_c$. The dotted (dashed) lines indicate a quadratic (cubic) dependence, $T_{\mathrm{coh}}\propto J_{\mathrm{K}}^2$ ($T_{\mathrm{coh}}\propto J_{\mathrm{K}}^3$), respectively.

Figure 6

Kondo temperature $T_{\mathrm{K}}$ vs Kondo coupling $J_{\mathrm{K}}$ for the fillings corresponding to the vHs. The dotted line indicates the exponential dependence, $T_{\mathrm{K}}\propto exp(-1/\sqrt{\nu J_{\mathrm{K}}})$, with $\nu =0.024$. The inset illustrates the failure of the standard exponential fitting.

Figure 7

Same as Fig. 6, but now for the coherence temperature $T_{\mathrm{coh}}$, with the fit parameter $\nu =0.011$.

Figure 8

Mean-field results for $J_{\mathrm{K}}=0.97t$ and $n_c=0.03$ where $T_{\mathrm{K}}\approx 0.058t$ and $T_{\mathrm{coh}}\approx 0.001t$. (a) Parton band structure at $T=0$. (b) Entropy per spin, $S\left(T\right)$. (c), (d) Heat capacity, plotted as $C\left(T\right)/T$ up to $T_{\mathrm{K}}$ and $C\left(T\right)$ at very low $T$, respectively.

Figure 9

Numerically determined conduction-electron chemical potential ${\mu }_0\left(T\right)$ for different fillings $n_c$ inside the kagome lattice flat band.

Figure 10

$r$ dependence of the chemical potentials $\mu$ and $-{\lambda }^{\prime }$ as a function, which verifies their leading order behavior with respect to each other and $r$, for a series of fillings $n_c$.

Figure 11

$T_K$ is extracted by fitting the numerical data for $r\left(T\right)$ to $r=A\sqrt{T-T_{\mathrm{K}}}$ near $T_{\mathrm{K}}$. Here we show this fit for $n_c=0.03$ and $J_{\mathrm{K}}=0.97t$; the inset shows $r\left(T\right)$ down to $T=0$.

Figure 12

Numerical data (solid line) for the Kondo temperature $T_{\mathrm{K}}$ with fittings of the analytical result (dashed line) of Eq. (B2) for fillings $0<n_c<2/3$.

Figure 13

Numerical data (solid line) for the coherence temperature $T_{\mathrm{coh}}$ with fittings of the analytical result (dashed line) of Eq. (C4) for fillings $0<n_c<2/3$.

Figure 14

Comparison of the mean-field $T_{\mathrm{coh}}$, calculated either as $r^2(T=0)/D$ (solid line) or as $1/\rho \left(0\right)$ (dashed line), for $n_c=0.2$ ($n_c=0.84$) in the flat (dispersive) band, respectively.