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) $\mathrm{\ensuremath{\Delta}}E$ 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 $\ensuremath{\mu}$, 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 $\mathrm{\ensuremath{\Delta}}E$ and a clear separation between $\mathrm{\ensuremath{\Delta}}E$ and ${T}_K$ for ${T}_K<\ensuremath{\mu}$. 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.