PhD Thesis - Universität Augsburg

34 2. Density-Matrix Renormalization Group
Figure 2.1: Sketch showing superblock consisting of the system and the environment
blocks, and the system block constructed with a block L(l) and a site l.
and the corresponding B i (s i ) matrices. Forthcoming sections will describe the construction
and the optimization of all A l (s l ) and B i (s i ) for the entire system (l, i = 1, . . .,L).
In the above outlined “growing” procedure the block size is increased iteratively, adding
one site at every step, until the desired length L is reached. f Consequently, the block at
some step becomes a part of another, bigger one at the following steps. To avoid strong
boundary effects during this procedure, White proposed [255, 256] to embed the block,
which we might call the system block, into another one called the environment block, and
search for the reduced representation of it in the environment given by the superblock
(system + environment block) state. g Later in the subsection we will see that embedding
into the environment for avoiding strong boundary effects has a deeper meaning, but at
this stage let us follow a historical development of this concept.
Let us look into this matter more precisely. Assume that at some point we reached
a block of length l with a state space of dimension a l spanned by a set of orthonormal
vectors {|α l 〉}. Now one site is added to it resulting in a system block (S) which has a
n S = a l · d l+1 dimensional Hilbert space H S with a basis {|α l 〉 ⊗ |s l+1 〉 ≡ |α l s l+1 〉} =: {|i〉}.
To avoid strong boundary effects we embed the system block into an “environment” (E)
consisting with remaining sites of the lattice. This is schematically depicted in Fig. 2.1.
At this point and for the following discussion it is not important whether the environment
contains all remaining sites or just a part of them. Let n E be the size and {|j〉}
the complete orthonormal basis of the Hilbert space H E of the environment block. Now
we wish to determine a procedure which finds a set of orthonormal system-block states
{|α l+1 〉} that spans an m := a l+1 n S dimensional subspace H L(l+1) of H S and gives the
f It is not necessary to have a fixed L from the beginning. It can be identified as the size of the system
for which the convergence in some quantity was reached.
g Sometimes the name system/environment block is missleading, because at the end we are interested in
features of the entire system (superblock). It is less confusing just to use the names left and right blocks,
but it still will be a good concept to give a name system to the block whose states are going to be optimized
and the rest name as an environment.