Numerical Renormalization Group Calculations for Impurity ...

4.3. Structure of the quantum critical points 39
ε
3
...
p=3
0
ε
ε
1
−1
p=1
p=−1
ε −3
...
p=−3
Figure 4.5: Single-particle spectrum of the free conduction electron chain
Eq. (A.12). The ground state is given by all the levels with p < 0 filled.
and
∑
≡
p
p=N
∑
p=−N,p odd
. (4.21)
Note that an equally spaced spectrum of single-particle levels is only recovered
in the limit Λ → 1; see Fig. 6 in (Bulla, Hewson and Zhang 1997) for the case
r = 0.
The RG analysis of Section 4.2 tells us that the critical fixed point is perturbatively
accessible from the LM one using a Kondo-type coupling as perturbation.
We thus focus on the operator
H ′ N = α(r)f(N) ⃗ S imp · ⃗s 0 , (4.22)
with the goal to calculate the many-body spectrum of the critical fixed point via
perturbation theory in H N ′ for small r. The function α(r) contains the fixedpoint
value of the Kondo-type coupling, and f(N) will be chosen such that H N
′
is exactly marginal, i.e., the effect of H N ′ governs the scaling of the many-particle
spectrum itself. The scaling analysis of Section 4.2, Eq. (4.8), Eq. (4.10), suggests
a parametrization of the coupling as
α(r) = µ−r
ρ 0
αr, (4.23)
where ρ 0 , is the prefactor in the density of states, and µ is a scale of order of the
bandwidth-such a factor is required here to make α a dimensionless parameter.
Thus, the strength of perturbation increases linearly with r at small r (where
µ −r /ρ 0 = D + O(r) for a featureless |ω| r density of states).
The qualitative influence of the operator ⃗ S imp ·⃗s 0 on the many-particle states
has been discussed in general in (Gonzalez-Buxton and Ingersent 1998) for finite