PhD Thesis - Universität Augsburg

36 2. Density-Matrix Renormalization Group
∑
ˆρ E = Tr S |ψ〉〈ψ| = nE ∑
[ψ † ψ] jj ′|j〉〈j ′ | = nE [V D 2 V † ] jj ′|j〉〈j ′ |
jj ′ jj ′
∑
= nE λ 2 µ|v µ 〉〈v µ | = n ∑Sch
λ 2 µ|v µ 〉〈v µ | .
µ=1
µ=1
(2.38b)
So even if system and environment are different, both reduced density matrices have the
same eigenvalue spectrum and hence the same number of nonzero eigenvalues bounded
by the smaller of the dimensions of system and environment. Their eigenvalues are given
by the squared singular values of the state coefficient matrix and the eigenvectors — |u µ 〉
for ˆρ S and |v µ 〉 for ˆρ E — corresponding to nonzero eigenvalues λ 2 µ are the orthonormal
vectors of the Schmidt decomposition (2.37). Therefore the analysis of the reduced density
matrices or the Schmidt decomposition yields exactly the same information. Note that the
columns of U (V ) are the vector representations of the eigenstates of the system-block
(environment-block) reduced density matrix in the basis {|i〉} ({|j〉}).
Now we come back to the original problem of finding a procedure for the optimal reduction
of the system-block state space from the n S -dimensional H S to the m-dimensional
H L(l+1) . If m n Sch , then from (2.37) and (2.38b) it follows that the projection to the subspace
spanned by the system-block reduced density-matrix eigenstates |u µ 〉 with nonzero
eigenvalues λ 2 µ is enough to reduce the Hilbert-space size and to leave the superblock state
|ψ〉 unchanged. What remains is the case when m < n Sch and hence it is not possible to
find a projection which does not alter |ψ〉. In what follows I will provide three different
criteria for determining the optimal reduction procedures, which, as we will see, lead to
the same space-reduction schemes.
(i) Optimization of expectation values [257]: If the superblock is in a pure state |ψ〉
(2.34), the physical state of the system block is fully described through a reduced
density matrix ˆρ S (2.38a).
Consider some bounded operator Ô acting on the system block (‖Ô‖ =
max φ |〈φ|Ô|φ〉/〈φ|φ〉| ≡ c Ô
< ∞). The expectation value of Ô is
〈ψ|Ô|ψ〉
〈Ô〉 = = Tr Sˆρ S Ô (2.39)
〈ψ|ψ〉
that can be expressed in the reduced density-matrix eigenbasis as
n S
〈Ô〉 = ∑
λ 2 µ 〈u µ |Ô|u µ 〉 . (2.40)
µ=1
Now, since we wish to reduce the system-block space size to the m, it is reasonable
to consider a projection onto the subspace spanned by the m dominant eigenvectors
(those with the largest eigenvalues) of the reduced density matrix in order to obtain