PhD Thesis - Universität Augsburg

28 2. Density-Matrix Renormalization Group
The states of the considered structure had appeared in the literature in many different
contexts and under different names before the invention of DMRG. The simplest case of
(2.14), corresponding to a homogeneous state with the same matrices for all sites, was first
considered in the eighties [60, 61] and occured as a ground state of certain spin chains with
competing interactions [141]. The best-known example is the spin-one chain with bilinear
and biquadratic interactions and a certain ratio of the couplings, where the valence-bond
ground state [2, 3] can be written in this form using (2 × 2) matrices.
In this thesis, we focus only on the case of open boundary condition (MPS (2.11)).
The systems with periodic boundaries can be studied within these states too, however the
effective dimensions a j tend to the square of that required for open boundary conditions
[150].
2.2.1 MPS, Blocks, and a Superblock
There is another way of expressing c(s) in MPS. Instead of starting from the site 1 and
growing the system by subsequent adding from the right the sites 2, 3, and so on (like
in the previous subsection), one can start from the site L and add from the left the sites
L − 1, L − 2, . . . , 1.
Similar to the procedure outlined in the previous subsection we start with some orthonormal
basis {|β L 〉} in H L and express it in terms of |s L 〉
where
|β L 〉 =
〈β ′ L|β L 〉 = δ β ′
L ,β L
⇐⇒
d L
∑
s L =1
B L (s L ) βL |s L 〉 , (2.15)
d L
∑
s L =1
B ∗ L(s L ) β ′
L
B L (s L ) βL
= δ β ′
L ,β L
. (2.16)
After adding from the left first the (L − 1)-th site, then the (L − 2)-th, and so on,
at step L − k = j we can write a basis of H R(j) ⊂ H j ⊗ H R(j+1) ⊂ ⊗ L
i=j H i in terms of
|s j 〉 ⊗ |β j+1 〉 as
with
b j+1
∑
d j
∑
β j+1 =1 s j =1
|β j 〉 =
d j
∑
b j+1
∑
s j =1 β j+1 =1
B j (s j ) βj ,β j+1
|s j 〉 ⊗ |β j+1 〉 , (2.17)
B ∗ j (s L−1) βj ,β j+1
B L−1 (s L−1 ) βj ,β j+1
= δ β ′
j ,β j
= 〈β ′ j |β j 〉 . (2.18)
Substituting |β j 〉 from the previous steps in (2.17)
∑
|β j 〉 = [B j (s j )B j+1 (s j+1 ) · · ·B L (s L )] βj |s j 〉 ⊗ |s j+1 〉 ⊗ · · · ⊗ |s L 〉 , (2.19)
s j ,s j+1 ,...,s L