Kondo effect at low electron density and high particle-hole asymmetry in 1D, 2D, and 3D

Using the perturbative scaling equations and the numerical renormalization group, we study the characteristic energy scales in the Kondo impurity problem as a function of the exchange coupling constant $J$ and the conduction-band electron density. We discuss the relation between the energy gain (impurity binding energy) $\mathrm{\Delta }E$ and the Kondo temperature $T_K$. We find that the two are proportional only for large values of $J$, whereas in the weak-coupling limit the energy gain is quadratic in $J$, while the Kondo temperature is exponentially small. The exact relation between the two quantities depends on the detailed form of the density of states of the band. In the limit of low electron density the Kondo screening is affected by the strong particle-hole asymmetry due to the presence of the band-edge van Hove singularities. We consider the cases of one- (1D), two- (2D), and three-dimensional (3D) tight-binding lattices (linear chain, square lattice, cubic lattice) with inverse-square-root, step-function, and square-root onsets of the density of states that are characteristic of the respective dimensionalities. We always find two different regimes depending on whether $T_K$ is higher or lower than $\mu$, the chemical potential measured from the bottom of the band. For 2D and 3D, we find a sigmoidal crossover between the large-$J$ and small-$J$ asymptotics in $\mathrm{\Delta }E$ and a clear separation between $\mathrm{\Delta }E$ and $T_K$ for $T_K<\mu$. For 1D, there is, in addition, a sizable intermediate-$J$ regime where the Kondo temperature is quadratic in $J$ due to the diverging density of states at the band edge. Furthermore, we find that in 1D the particle-hole asymmetry leads to a large decrease of $T_K$ compared to the standard result obtained by approximating the density of states to be constant (flat-band approximation), while in 3D the opposite is the case; this is due to the nontrivial interplay of the exchange and potential scattering renormalization in the presence of particle-hole asymmetry. The 2D square-lattice density of states behaves to a very good approximation as a band with constant density of states.

Figure 1

Overview of the different parameter regimes in the 1D, 2D, and 3D cases. The energy gain $\mathrm{\Delta }E$ is plotted as a function of the Kondo exchange coupling $J$ on a log-log scale. The red and blue lines are the high-$J$ and low-$J$ asymptotics, $\propto J$ and $\propto J^2$, respectively, common for all three dimensions. For the 1D case in (a) we also indicate the intermediate-$J$ regime, which is $\propto J^2$ (green line). The model parameters are indicated in the respective panels. They are chosen so that the band occupancy is comparable in all three cases, $n\approx 0.02$.

Figure 2

(a) and (b) Absolute value of the spin-spin expectation value $|\langle \mathbf{s}·\mathbf{S}\rangle |$, i.e., the correlation between the impurity spin and the conduction-band spin at the location of the impurity, plotted on log-linear and log-log scales. (c) Ratio between the local contribution to the energy gain ${\left(\mathrm{\Delta }E\right)}_{\mathrm{int}}=J\left|\langle \mathbf{s}·\mathbf{S}\rangle \right|\propto J^2$ and the total energy gain $\mathrm{\Delta }E$. The calculations are performed for 2D DOS with $\mu =0.025$.

Figure 3

Relation between the energy gain $\mathrm{\Delta }E$ and the Kondo temperature $T_K$ for three different dimensionalities. The two quantities behave similarly only in the intermediate-coupling regime but have very different weak-coupling asymptotic behavior.

Figure 4

Temperature dependence of the impurity magnetic susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$ and of the spin-spin correlation $\langle \mathbf{s}·\mathbf{S}\rangle$ in the parameter regime where $\mathrm{\Delta }E$ and $T_K$ are already significantly separated.

Figure 5

Asymptotic scaling of the Kondo temperature. We plot the logarithm of $T_K$ versus $1/{\rho }_{0}J$, where ${\rho }_0$ is the density of states in the conduction band at the Fermi level, ${\rho }_0=\rho \left(\mu \right)$. The dashed line indicates the standard result for a flat density of states. The arrows indicate the direction of reduced band filling $n$, i.e., going toward a more asymmetric situation. In (a) we plot $n=0.012$, 0.041, 0.074, 0.12, and 0.5, in (b) we plot $n=0.0011$, 0.0083, 0.085, and 0.39, and in (c) we plot $n=0.0013$, 0.0032, 0.012, 0.035, and 0.5.

Figure 6

Symmetric and antisymmetric combinations of the conduction-band density of states that affects the renormalization flow: ${\rho }_{\pm }\left(\epsilon \right)=\rho \left(\epsilon \right)\pm \rho (-\epsilon )$. Here $\mu =0.1$.

Figure 7

Energy gain $\mathrm{\Delta }E$ in the presence of the magnetic field. Here we use a flat band, and the chemical potential is fixed in the center of the band.

Figure 8

Energy gain $\mathrm{\Delta }E$ in the anisotropic Kondo model with exchange coupling of Ising type, $J_{\parallel }=J$ and $J_{\perp }=0$, and in the isotropic model with exchange coupling of Heisenberg type, $J_{\parallel }=J_{\perp }=J$. The result for the Ising coupling is multiplied by 3 for easier comparison. We use a flat band with $\mu$ fixed in its center. The inset shows the logarithmic derivative of ${\left(\mathrm{\Delta }E\right)}_{\mathrm{Heisenberg}}-3{\left(\mathrm{\Delta }E\right)}_{\mathrm{Ising}}$, thus indicating the power-law exponent of the correction due to spin-flip dynamics.