Scale-free property and edge state of Wilson’s numerical renormalization group

We discuss the scale-free property of Wilson’s numerical renormalization group (NRG) for the Kondo-impurity problem. The single-particle state of the effective Hamiltonian with a cutoff $\Lambda$ is described by the wave-packet basis having the scale-free property. The energy scale of the system can be controlled by the lattice translation of the wave-packet basis with no reference of rescaling of the lattice space. We also analyze the role of the Kondo interaction in the context of wave-packet basis and then discuss the scaling and renormalization of the Kondo coupling. In addition, we clarify the role of the edge state in the lowest energy scale of Wilson NRG.

Figure 1

(Color online) One-particle eigenvalue spectrum of Eq. (2). The horizontal axis $j$ indicates the label of the eigenvalue in increasing order.

Figure 2

(Color online) One-particle wave function $\psi \left(n\right)$ of $j=101$, 130, and 160. The horizontal axis means the cite index $n$. The bulk states $\psi \left(n\right)$ of $j=130$ and 160 can be overlapped with each other by a lattice translation while $\psi \left(n\right)$ of $j=101$ is the edge state.

Figure 3

(Color online) Amplitude of the wave function at the impurity site ${\psi }_j\left(N\right)$ of $N=50$ for $\lambda =0.05$, 0.1, 0.2, and 0.5, where $j$ indicates the label of an eigenstate. We can see the clear exponential dependence on $j$ for the bulk wave function. The small hump around Fermi surface $\left(j\sim 25\right)$ corresponds to the edge states of the lowest energy scale. The hump region of the edge states extends as $\lambda$ becomes small.

Figure 4

(Color online) The linear scale plot of the eigenvalue spectrum around the Fermi surface for $\lambda =0.1$ and $N=200$. Solid symbols denote the $j\text{th}$ eigenvalue. The Fermi surface corresponds to $E=0$, which is indicated by the horizontal dotted line. Solid curves represent exponential dependence of the eigenvalues in the bulk region and the broken line indicates the edge mode Eq. (14).

Figure 5

(Color online) (a) Kondo coupling dependences of $\uparrow$-electron density for $N=12$ with $\lambda =\text{ln}2/2\simeq 0.347\left(\Lambda =\sqrt{2}\right)$, where $j$ denotes the index of the orbitals. The Fermi surface of the free orbital electron corresponds to $j=6.5$. (b) Size dependences of $\uparrow$-electron density for $\lambda \simeq 0.347\left(\Lambda =\sqrt{2}\right)$. The curves indicate the $\uparrow$-electron density at the Fermi surface ${\rho }_{N/2+1\uparrow }^{\left(N\right)}$ for $N=6$, 8, 10, and 12 from down to up. The solid curves of ${\rho }_{N\uparrow }^{\left(N\right)}$ represent the $\uparrow$-electron density at the highest energy-scale orbital.

Figure 6

(Color online) Spin distribution in the real-space representation for $N=14$ and $\lambda =\text{ln}2/2$. The open (solid) circle indicates $J_K=3.0\left(0.5\right)$. $n$ denotes the lattice index, where $n=14$ corresponds to the impurity site. The dotted line means the spin distribution of the free orbital of $j=8$.