Numerical Renormalization Group Calculations for Impurity ...

3.2. Summary of the Basic Techniques 17
by U and the two subsystems are coupled via an energy-dependent hybridization
V (ε k ).
The Hamiltonian in Eq. (3.3) can be written into a form which is more convenient
for the derivation of the NRG equations:
∑
H = ε f f −1σf † −1σ + Uf † −1↑ f −1↑f † −1↓ f −1↓
σ
+ ∑ σ
∫ 1
−1
dε g(ε)a † εσ a εσ + ∑ σ
∫ 1
−1
dε h(ε) ( f † −1σ a εσ + a † εσ f −1σ)
, (3.4)
where we introduced a one-dimensional energy representation for the conduction
band with band cut-offs at ±1, dispersion g(ε) and hybridization h(ε). The
band operators fulfill the standard fermionic commutation rules [ a † εσ, a ε ′ σ ′ ]
=
δ(ε − ε ′ )δ σσ ′.
The Hamiltonian in Eq. (3.3) is equivalent to the Hamiltonian in Eq. (3.4)
when we select g(ε) and h(ε) to satisfy the following condition:
where ε(x) is the inverse of g(ε), i.e.
∂ε(x)
∂x h(ε(x))2 = V (ε(x)) 2 ρ(ε(x)) = 1 ∆(x), (3.5)
π
ε(g(x)) = x, (3.6)
and ρ(ε) is the density of states for the free conduction electrons (Bulla, Pruschke
and Hewson 1997).
As a first step of discretization, we divide a conduction band into N-intervals
{I n },
∫ 1
dε → ∑ ∫
dε, (3.7)
−1 n I n
with I n = [ε n , ε n+1 ] , (n = 0, 1, ..., N − 1, ε 0 = −1 and ε N = 1) and replace the
operators of conduction electrons with the Fourier components in each interval
I n ,
a (†)
pσ,n = √ 1 dε a
dn
∫I (†)
ε,σe −i2pπ|ε|/dn , (3.8)
n
with d n = |ε n+1 −ε n | and p = 0, 1, 2, 3, · · ·. At the end, we drop all the non-zero p-
terms and write the Hamiltonian only with the zero-th Fourier components, a (†)
0σ,n,
in each interval. This, so called p = 0 approximation, is the first approximation
in NRG.
Let us look into the hybridization and the kinetic term of the Hamiltonian to
check the validity of the approximation. The hybridization term is given as
H hyb = ∑ ∫
dε h n f −1σa † εσ + h.c.. (3.9)
n I n