Numerical Renormalization Group Calculations for Impurity ...

3.2. Summary of the Basic Techniques 23
The semi-infinite chain is solved iteratively by starting from H 0 and successively
adding the next site.
H 0 = H imp + √ ∑
ξ 0 (f † −1σ f 0σ + f † 0σ f ∑
−1σ) + ε 0 f † 0σ f 0σ
H 1
H 2
H 3
...
H N+1
σ
∑
∑
= H 0 + t 0 (f 0σf † 1σ + f 1σf † 0σ ) + ε 1
σ
σ
σ
f † 1σf 1σ
∑
= H 1 + t 1 (f † 1σ f 2σ + f † 2σ f ∑
1σ) + ε 2 f † 2σ f 2σ (3.35)
σ
∑
∑
= H 2 + t 2 (f 2σf † 3σ + f 3σf † 2σ ) + ε 3
σ
σ
σ
f † 3σf 3σ
∑
= H N + t N (f † Nσ f N+1σ + f † N+1σ f ∑
Nσ) + ε N+1 f † N+1σ f N+1σ
σ
To prevent the rapid growth of the Hilbert space, it is indispensable to discard
some of eigenstates before including an additional conduction site to the Hamiltonian.
Let us assume that the Hamiltonian of the N −1-th iterative step, H N−1 ,
yields M of eigenstates,
H N−1 |φ (N−1)
n
〉 = E n
(N−1) |φ n (N−1) 〉 (3.36)
with n = 1, 2, ....M.
The matrix representation of a new Hamiltonian H N is based on the product
states
|ψ nm (N) 〉 = |φ(N−1) n 〉 ⊗ |m〉, (n = 1, 2, ..., M, m = 1, 2, ..., l), (3.37)
where {|m〉|m = 1, 2, ..., l} corresponds to a basis for a new site. In fermionic
cases
|Ω〉 = |0〉,
| ↑〉 = f † N↑ |0〉,
| ↓〉 = f † N↓ |0〉,
The matrix elements of the new Hamiltonian H N is
| ↑↓〉 = f † N↑ f † N↓
|0〉. (3.38)
σ
〈ψ (N)
n ′ m ′ |H N |ψ (N)
nm 〉
= 〈m ′ |m〉〈φ (N−1)
n ′
+ t N−1 〈φ (N−1)
n ′
|H N−1 |φ (N−1)
n
〉 + ε N 〈φ (N−1)
n ′
|f † N−1σ |φ(N−1) nσ 〉〈m ′ |f Nσ |m〉
|φ n
(N−1) 〉〈m ′ |f † Nσ f Nσ|m〉
+ t N−1 〈φ (N−1)
n
|f ′ N−1σ |φ nσ (N−1) 〉〈m ′ |f † Nσ
|m〉. (3.39)
with |ψ (N−1)
nm
〉 = |φ n
(N−1) 〉 ⊗ |m〉 and |m〉 ∈ {|Ω〉, | ↑〉, | ↓〉, | ↑↓〉}.
In NRG, we truncate the matrix of Hamiltonian by keeping the first N s × N s
elements out of the M × M ones and discarding the remnants.