Numerical Renormalization Group Calculations for Impurity ...

3.2. Summary of the Basic Techniques 21
NRG deals with those many electronic degrees of freedom in a certain sequence
(iteratively). How do we distribute the huge number of electrons into a
sequence of diagonalization steps? How can one describe the correlations among
the electrons in different steps? Section 3.2.2 is devoted to answer the questions.
3.2.2 Iterative diagonalization of a semi-infinite chain
Let us start from the Hamiltonian with p = 0 Fourier components only.
∑
H = ε f f † −1σ f −1σ + Uf † −1↑ f −1↑f † −1↓ f −1↓ + ∑ ∑
ξ n a † nσ a nσ
σ
σ n
+ √ ∑
η 0 (f −1σf † 0σ + f 0σf † −1σ ), (3.26)
with
σ
f 0σ =
η 0
= ∑ n
1 ∑
√ ¯h n a nσ ,
η0
n
¯h 2 n. (3.27)
Now we drop the index p (= 0) from the conduction operators (a (†)
0σ,n → a nσ).
(†)
The well-known Lanczos algorithm for converting matrices to a tridiagonal form
maps the Hamiltonian in Eq. (3.26) into a semi-infinite chain.
∑
H = ε f f −1σf † −1σ + Uf † −1↑ f −1↑f † −1↓ f −1↓ + √ ∑
η 0 (f −1σf † 0σ + f 0σf † −1σ )
+ ∑ σ
σ
∑
n
[
]
ε n f nσ † f nσ + t n (f nσ † f n+1σ + f † n+1σ f nσ)
σ
(3.28)
where operators f nσ (†) (n = 1, 2, ...) are represented as a linear combination of
conduction operators a (†)
mσ by a real orthogonal transformation U (U T U = UU T =
1, U ∗ = U):
f nσ = ∑ U nm a mσ . (3.29)
m
The parameters of the semi-infinite chain are calculated recursively with the
relations (Bulla, Lee, Tong and Vojta 2005),
ε m
= ∑ n
ξ n U 2 mn ,
t 2 m
= ∑ n
[(ξ n − ε m )U mn − t m−1 U m−1n ] 2 , (3.30)
U m+1n = 1
t m
[(ξ n − ε m )U mn − t m−1 U m−1n ] ,