5 V. EFFECT OF IMPURITY SPIN On the basis of the nature of the wavepacket, let us discuss the effect of the Kondo impurity in Wilson NRG. Assume an impurity spin ⃗ S with the Kondo coupling J$_{K}$ at the right edge of the chain( n = N ). The free electron Hamiltonian H λ is diagonalized in the wavepacket ba- sis, H λ = ∑ jσ E$_{j}$f † $_{jσ}$f$_{jσ}$ , where f$_{jσ}$ is a fermion operator corresponding to the j -th eigenvalue of (1), and the spin index σ is noted explicitly. In the following, the eigen- state described by f$_{j,σ}$ is called as “orbital electron” for convenience. As was discussed in Sec. II, E$_{j}$ ∝ exp(2 λj ) for 1 ≤ j ≤ N/ 2, and E$_{j}$ ∝ exp( λ ( N − 2 j )) for N/ 2 + 1 ≤ j N $^{22}$. On the other hand, the matrix element associated with the impurity is determined by the tail of the wavepacket at the right edge, where ⃗ σ is the S = 1 / 2 spin(not Pauli) matrix associ- ated with the orbital electron, ⃗ S represents the impurity spin of S = 1 / 2, and σ takes σ ∈↑ , ↓ . Here, it should be recalled that, in Wilson NRG, the energy scale of the Kondo coupling itself is scaled up as J$_{K}$e $^{λN}$, in the pro- cess of the iterative increase of the free electron sites. It is instructive to write the matrix elements of H imp in the orbital bases. Using Eq. (11), we have the diagonal parts with respect to the spin as, where Also, the off-diagonal part is given by where ⇑ , ⇓ represent the impurity spin. Here we should remark that the damping factor of the wavepackets near the Fermi-surface( i, j ≃ N/ 2) is canceled out with the scale factor of the Kondo coupling. This implies that the fermionic 2 k$_{F}$ -oscillation of Eq. (9) is correctly recov- ered. Thus, the phase shift of the Fermi-level state due to the impurity scattering can be correctly taken into account, although the wavepackets are resolved with re- spect to the energy scale. The eigenvalues of the impurity term H imp are just given by − 3 e $^{λN}$J$_{K}$ / 4 for the singlet, e $^{λN}$J$_{K}$ / 4 for triplet and the zero energy with the huge degeneracy. By the perturbation of H $_{λ}$, the most of the degenerating zero eigenvalus are lifted to have a “dispersion”, whose maxi- mum energy scale is of order of E$_{N}$ ∼ e $^{λN}$. For J$_{K}$ > 1, we can naively treat H λ as a perturbation against to H $_{imp}$, since “band width” of the free orbital electrons is smaller than the singlet-triplet gap of H $_{imp}$. In the con- text of physics, an orbital electron localizes at the edge and forms a singlet or triplet pairs with the impurity spin. This is the local Fermi liquid fixed point. For the weak coupling case( J$_{K}$ ≪ 1), the dominant part of the Hamiltonian is described by H $_{λ}$, and H imp is a perturbation for small N . As is well known, how- ever, the low-energy physics should flow to the strong coupling limit for sufficiently large N . For simplic- ity, we assume 1 ≤ i, j ≤ N 2 . For 2 M -site lattice translation $^{22}$, then, the scale factor C$_{i,j}$ in (16) and (18) is scaled by C$_{i}$$_{+}$$_{M,j}$$_{+}$$_{M}$ exp(2 λM ), while α$_{i,j}$ in the bulk region is independent of i, j . Thus we can read J$_{K}$e $^{λN}$ψ ∗ $_{i}$( N ) ψ$_{j}$ ( N ) = ˜ J$_{K}$e $^{λN}$ψ ∗ i + $_{M}$( N ) ψ$_{j}$$_{+}$$_{M}$ ( N ) with ˜ J$_{K}$ ≡ J$_{K}$ exp(2 λM ). The role of Kondo coupling of a J$_{K}$ at a certain energy scale e λj can be viewed as that of the scaled Kondo coupling J$_{K}$ exp(2 λM ) in a higher energy scale e λ ( j + M $^{)}$. In the following, we call this fac- tor exp(2 λM ) as “canonical scale factor” of the Kondo coupling. The above scaling argument might suggest that the low-energy physics basically flows to the strong coupling limit by the canonical scale factor e 2 $^{λM}$. However, the actual spectral flow is rather complicated; Since H λ has the same scale factor as H imp as well, the competi- tion of H λ and H imp yields the nontrivial Kondo en- ergy scale. Of course, it is well established that the en- tanglement between the impurity and orbital electrons rapidly develops around the Kondo energy scale given by exp( − 1 / 2 J$_{K}$ ) $_{/}$√$_{2}$$_{J}$$_{K}$$_{.}$ In the present context, this sug- gests that the site dependence of the effective Kondo cou- pling should be deviated from the naive canonical scale factor. In order to see the effect of the Kondo interaction, we perform exact diagonalization of the Wilson’s effective Hamiltonian up to 14 orbitals with the subspace of elec- tron number N and total- S z = 1 / 2(including the impu- rity spin). We use λ = ln 2 / 2 ≃ 0 . 347(Λ = √$_{2) in order}$ to see clearly the role of the impurity within the exact diagonalization. Then, we focus on the nature of the groundstate wavefunction, rather than the spectral flow. Let us write the ↑ -electron density of the groundstate for N -sites system as ρ ( N ) j ↑ ≡ 〈 f † j $_{↑}$f$_{j}$$_{↑}$ 〉 , where j indicates the orbital index. Figure 5(a) shows the J$_{K}$ -dependence of ρ ( N ) j ↑ . Note that, for N = 12, the Fermi surface of the free electron is located between j = 6 and 7. As J$_{K}$ in- creases, the ↑ -electron density of j = 7 approaches 0 . 5, which implies that the impurity spin is screened and then the remaining spin-1/2 chiefly moves to the orbital near the Fermi surface. At the same time, we can see that the density for j = 7 orbitals is also modified from the case of free electrons, even though they are embedded deeply away from the Fermi surface. This implies that the high H imp = J$_{K}$e λN ∑ i,j,σ,σ ′ ψ ∗ $_{i}$( N ) ψ$_{j}$ ( N ) f † $_{jσ}$[ ⃗ σ · ⃗ S ]$_{σ,σ}$ ′ f$_{jσ}$ ′ , (15) 〈 i, σ, S |H imp | j, σ, S 〉 = ± J$_{K}$ 4 α$_{i}$α$_{j}$C$_{i,j}$, (16) 〈 i, ↑ , ⇓ |H imp | j, ↓ , ⇑〉 = 〈 i, ↓ , ⇑ |H imp | j, ↑ , ⇓〉 = J$_{K}$ 2 α$_{i}$α$_{j}$C$_{i,j}$, (18) C$_{i,j}$ ≡      e λ ( N − i − j $^{)}$, 1 ≤ i, j ≤ N 2 e λ ( − i + j − $^{1)}$, 1 i ≤ N 2 , N 2 + 1 ≤ j ≤ N e λ ( i − j − $^{1)}$, N 2 + 1 i ≤ N, 1 j N 2 e λ ( − N + i + j − $^{2)}$, N 2 + 1 ≤ i, j ≤ N . (17)