2 II. MODEL AND SCALE FREE PROPERTY In Wilson NRG, the essential point is to consider the 1D lattice fermion model mapped from the degenerating free electrons around the Fermi surface. We thus start with the Wilson-type Hamiltonian of the spinless free fermions with the exponentially modulated hopping where c$_{n}$ is a fermion annihilation operator at n th site and N denotes the number of sites. We have also intro- duced Λ ≡ exp( λ ) > 1 for later convenience. Thus n = 1 corresponds to the smallest energy scale and n = N does to the impurity site with the largest energy scale $^{18}$. Al- though, in the original work $^{5}$, the hopping parameter has a supplemental coefficient and n in e λn term takes a half integer, the essential physics is the same as Eq. (1). Let us write the one-particle state as | ψ 〉 = ∑ $_{n}$ψ ( n ) c † n | 0 〉 . Then one-particle Schr¨ odinger equation in the bulk region is Note that Eq. (2) is invariant under the transformation, ψ ( n ) → ( − 1) $^{n}$ψ ( n ) and E → − E , which clearly rep- resents the particle-hole symmetry. Thus, we basically consider the positive energy solution. Since the system has no explicit translational sym- metry, we employ numerical diagonalization of Eq. (2) rather than the usual Fourier analysis, for a finite but sufficiently large system. We assume the free boundary condition and thus what we deal with is the tridiagonal matrix. Figure 1 represents the absolute value of the one-particle spectrum for λ = 0 . 1 and N = 200, where j indicates the label of the eigenvalue in increasing order. The Fermi surface is located between j = 100 and 101, and j ≤ 100 represent negative energy eigenvalues. Thus, the parity in Fig. 1 corresponds to the particle-hole sym- metry. We also show the amplitude of the wavefunctions corresponding to j = 101, 130 and 160th eigenvalues, in Fig. 2. The most important behavior in Fig. 1 is that, as was already mentioned in Ref. $^{5}$, E basically exhibits the exponential dependence E ∝ ± exp( λj ). We call this re- gion of the exponential dependence as “bulk”, since the corresponding wavefunctions are localized in the bulk re- gion of the chain, as can be seen for j = 130 and 160 in Fig.2. On the other hand, we can see that some eigenval- ues near the Fermi surface j ∼ 100 deviate from the bulk lines. We call these states as “edge states”, since they correspond to the edge modes near the Fermi surface, as is the wavefunction of j =101 in Fig. 2. We analyze the bulk part of the spectrum in detail. In connection with the exponential dependence of the eigenvalues, an essential information can be found in the bulk wavefunctions for j = 130 and 160 in Fig. 2. The primary notable point is that these wavefunctions have a very similar wavepacket-like shape, in contrast to the usual plane wave for the uniform chain. The localization of the wavefunction can be qualitatively understood as follows. If a particle carrying a certain energy goes in the larger n region, the particle can not excite the larger energy bonds, into which the wavefunction can not pen- etrate. While the particle goes to the smaller n region, the bond of the smaller coupling can not carry the total energy of the particle and thus the wavefunction decays very rapidly. Another important property of the bulk wavefunctions is that they can be overlapped with each other by the lattice translation; In Fig. 2, the wavefunctions of j =130 and 160 have the very similar shape. Indeed, we can verify that overlap integral of the two wavefunctions after the lattice translation is unity within the computational accuracy. In order to see this property in analytic level, we introduce ψ ( n ) ≡ e − λn/ $^{2}$φ ( n ) . (3) e − $^{λ}$ψ ( n − 1) + ψ ( n + 1) = Ee − $^{λn}$ψ ( n ) . (2) H λ = N − 1 ∑ n =1 e $^{λn}$( c † n $_{+1}$c$_{n}$ + c † $_{n}$c$_{n}$$_{+1}$ ) , (1) FIG. 1: (Color online)One-particle eigenvalue spectrum of Eq. (2). The horizontal axis j indicates the label of the eigenvalue in increasing order. FIG. 2: (Color online)One-particle wavefunction ψ ( n ) of j = 101, 130 and 160. The horizontal axis means the cite index n . The bulk states ψ ( n ) of j = 130 and 160 can be overlapped with each other by a lattice translation, while ψ ( n ) of j = 101 is the edge state. 0 20 40 60 80 100 120 140 160 180 200 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 j | E |  =0.1, N=200 0 100 200 -0.2 0 0.2 0.4 n j=130 j=160 (n) j=101