Numerical Renormalization Group Calculations for Impurity ...

3.3. Flow diagrams and Fixed points 27
basis {|ψ (N)
n 〉}.
H 0 =
H 1 =
.
.
.
H N =
H N+1 =
∑N s
n=1
∑N s
n=1
∑N s
n=1
∑N s
n=1
E (0)
n |ψ(0) n 〉〈ψ(0) n |
E (1)
n |ψ (1)
n 〉〈ψ (1)
n |
E (N)
n
E (N+1)
n
|ψ n
(N) 〉〈ψ n (N) |
|ψ n
(N+1) 〉〈ψ n (N+1) | (3.60)
where H m |ψ (m)
n
〉 = E (m)
n
|ψ (m)
n 〉, (m = 0, 1, ..., N + 1 and n = 1, ..., N s ).
The iterative Hamiltonian H N approaches to the fixed point H ∗ as the eigenstates
{|ψ (N)
n 〉} converges to constant states {|ψ ∗ n 〉}:
lim
N→∞ |ψ(N) n 〉 = |ψ∗ n 〉, (3.61)
for n = 1, ..., N s .
Once the iterative Hamiltonian is very close to a fixed point, the mapping R
hardly affects the structure of Hamiltonian but changes the overall energy-scale
as α. 9
A sequence of transformations gives
H N+1 = R α (H N ) = α H N + O(1/N)
H N+2
H N+3
= R α (H N+1 ) = α 2 H N + O(1/N)
= R α (H N+2 ) = α 3 H N + O(1/N)
... (3.62)
If we define an renormalized Hamiltonian ¯H N where overall energy scale is divided
by α N , ( ¯H N = H N × 1
α N )
¯H N+1 = ¯H N + O(1/N)
¯H N+2 = ¯H N + O(1/N)
¯H N+3 = ¯H N + O(1/N)
9 In fermionic (bosonic) NRG, α = 1/ √ Λ (1/Λ).
... (3.63)