PhD Thesis - Universität Augsburg

32 2. Density-Matrix Renormalization Group
Here we only used the orthonormality of the site local basis {|s j 〉}.
Equations (2.25) and (2.26) can be used to find a matrix representation of any local
operator, that acts on some site j < i in the basis B(L, i).
It remains to find a matrix representation of the product of two local operators ˆQ k and
Ô j , acting on the different sites k and j (k < j), respectively, e in the basis B(L, l). For this
we start with the L(k)-representation Q L(k) of ˆQ k (2.25) and using (2.26) iteratively obtain
its L(j − 1)-representation. Then using (2.8) we write the product of Q L(j−1) and matrix
representation of Ôj (O j ) in the basis B(L, j) and hence obtain the L(j)-representation of
ˆQ k Ô j ( ˆQ k Ô j ) L(j)
= 〈α
α j|( ′ ˆQ k Ô j )|α ′
j ,α j 〉
j
⎛
⎞ ⎛
⎞
∑
= ⎝〈α j−1|〈s ′ ′ j| A ∗ j(s ′ j) α ′ ⎠
j−1 ,α ′ ( ˆQ k Ô j ) ⎝ ∑
A
j
j (s j ) αj−1 ,α j
|α j−1 〉|s j 〉 ⎠
α ′ j−1 ,α′ j
α j−1 ,α j
⎛
⎞
⎛
⎞
= ⎝ ∑
A ∗ j (s′ j ) ⎠
α ′ Q L(j−1)
j−1 ,α′ j α ′
α ′ j−1 j−1O j ⎝ ∑
A ,α s ′ j ,s j (s j ) αj−1
⎠
,α
j
j
j−1 ,α′ j
α j−1 ,α j
∑
= A ∗ j (s′ j ) α ′ A j−1 ,α′ j j (s j ) α j−1 ,α j
Q L(j−1)
α ′ j−1 j−1O j , (2.27a)
,α s ′ j ,s j
α ′ j−1 ,α j−1,s ′ j ,s j
or in matrix notation
( ˆQ k Ô j ) L(j) = ∑ s ′ j ,s j
A † j (s′ j )QL(j−1) A j (s j ) O j s ′ j ,s j. (2.27b)
Finally, once more using (2.26) iteratively, from (2.27) we obtain its L(l)-representation.
This can be easily generalized for the product of more than two operators each acting on
a different site, e.g.,
Ô ν = Ô1 ν Ô2 ν · · ·Ôl ν . (2.28)
O L(l) (2.24) is then constructed by summing up L(l)-representations Oν
L(l) of Ô ν . Note
that we can take the sum but not the product of operators in the same L(l)-representation;
since B(L, l) is the basis of the block subspace and not the complete one,
( ˆQÔ)L(l) α ′ l ,α l
= 〈α ′ l |( ˆQÔ)|α l 〉 ≠ ∑ α ′′
l
〈α l ′ ′′
| ˆQ|α l 〉〈α′′ l |Ô|α l 〉 = [OL(l) Q L(l) ] α ′
l ,α l
. (2.29)
As mentioned at the beginning, typically most of Ôν j in Ôν are just identity operators Î,
corresponding to identity matrices in the site-local basis. In case of fermionic operators, due
to the anticommutation relations, one should take care about “-” signs where it is necessary.
e For k = j we take the product in the local basis of the site k and obtain a new local operator.