content_image/419524.jpg content_image/419522.jpg content_image/419523.jpg content_image/419527.jpg content_image/419525.jpg
(3), the tail of the single fermion wavepacket is described by the damping factor e − λn/ $^{2}$, when the packet is suffi- ciently away from the impurity. Thus, we can write the amplitude of the wavefunction at the impurity site as 

<img src='content_image/419524.jpg'>

where α$_{j}$ is a coefficient of order of unity for j -th wavepacket. Note that α$_{j}$ is a constant in the bulk region. However, we should also remark that, as λ becomes small, the small hump region extends around Fermi surface( j ∼ 25); The states in the hump region correspond to the edge states of the lowest energy scale, which will be discussed in the next section. 

In this section, we would like to discuss the bound- ary effects in Wilson NRG, which can be seen around the Fermi surface( E = 0) corresponding to the left edge of the chain. Figure 4 shows the magnification of the eigenvalue spectrum around E = 0( j ∼ 100), where the scale of the vertical axis is linear. As was mentioned, the energy eigenvalue distribution in the bulk part has the exponential dependence, which is represented as broken curves. However, we can see that the spectrum in the vicinity the Fermi surface deviates from the exponential dependence and becomes rather linear with respect to index j . These states have the lowest energy scale near the Fermi surface, which implies that the property of the edge state is important. Indeed we can see that the wave- function of j = 101 in Fig. 2 localizes at the edge region. Although the shape of this wavefunction itself is similar to the right side of the bulk wavepacket, the eigenvalue distribution exhibits different structure from the bulk re- gion. 

In order to analyze the edge states, we consider another continuum limit, which may be similar to the usual ap- proach for the fermion with the linear dispersion. We rewrite φ ( n ) = e ± iπn/ $^{2}$η ( n ), where the phase factor ba- sically corresponds to the right or left going Fermi wave numbers of the uniform case. Then, equation (4) becomes 

Writing x = λn again and maintaining the leading term of O ( λ ), we have (2 iλ d dx ∓ ¯ Ee − $^{x}$) η ( x ) = 0 . The solution of this equation can be easily obtained as 

where C is a certain constant. For x ≫ 1, η → const and thus ψ ∼ e − x/ $^{2}$e ± iπx/ 2 $^{λ}$. For x ≪ − 1, however, η be- comes highly oscillating, and if its period becomes smaller than the lattice space, this continuum limit breaks down. Moreover, the wavefunction | ψ ( x ) | 2 ∼ e − x → ∞ in 

<img src='content_image/419522.jpg'>

<img src='content_image/419523.jpg'>

## IV. EDGE STATE 

x → −∞ . In this sense, the wavefunction (13) is pro- hibited in the bulk limit. If the system has the left edge, however, such a breakdown can be avoided. When the boundary condition is ψ (0) = ψ ( N + 1) = 0, the energy eigenvalue is determined as 

where q = ± 1 / 2 , ± 3 / 2 · · · 21 is a quantum number mea- sured from the Fermi surface. In fact, we have E = 0 . 2988 for λ = 0 . 1 and q = 1 / 2, which is in good agree- ment with the numerical result 0 . 2989 in Fig.4. As ¯ E increases, the oscillation of Eq. (13) becomes significant and then the wavefunction crosses over to the bulk wave- function (8), away from the edge. Since the phase dif- ference between n = 0 and 1 in Eq. (13) is about ¯ E/ 2, the criterion of the crossover is estimated as E ≃ ¯ E ∼ 2, which is also consistent with Fig. 4. 

Although the lowest energy behavior is described by the boundary excitation (14), the role of the edge state attracts less interest so far. One reason for this is that the practical value of the cutoff is Λ 2 ≃ 1 . 5 ∼ 2 . 5, for which the level space becomes too sparse to detect the linear region in the spectrum. Here, it should be noted that the generalized Wilson NRG scheme with much smaller Λ(= 1 . 02) exactly extracts the low-energy spectrum of the critical XXZ chain. 17 Another reason is that, as in Fig. 4, the edge wavefunction smoothly varies its shape into the bulk wavepacket and the tail of the edge wave- function shows the same dumping | ψ | 2 ∼ e − x as the bulk wavefunction. Thus, one can increase number of sites seamlessly in NRG iterations, with no explicit reference of the edge state. Finally, we make a comment on the right edge corresponding to the highest energy scale; In contrast to the left edge above, the universal eigenvalue structure can not be seen for the right-edge states. 

<img src='content_image/419527.jpg'>

FIG. 4: (Color online)The linear scale plot of the eigenvalue spectrum around the Fermi surface for λ = 0 . 1 and N = 200. Solid symbols denote the j 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 indi- cates the edge mode (14). 

<img src='content_image/419525.jpg'>

