PhD Thesis - Universität Augsburg

40 2. Density-Matrix Renormalization Group
decomposition, i.e., the µ-th largest Schmidt coefficients correspond to the same local
basis element (see Eqs. (2.37) and (2.51)). j With Von Neumann’s trace inequality
(2.54), the maximum possible fidelity between |ψ〉 and | ˜ψ〉 is
max F(|ψ〉〈ψ|, | ˜ψ〉〈 ˜ψ|) =
{|ũ µ〉},{|ṽ µ〉}
(
∑ m
) 2
λ µ˜λµ (2.55)
where the sum runs only up to m since ˜λ µ>m = 0. Associating the singular values
λ µ and ˜λ µ with m-dimensional vectors λ := (λ 1 , λ 2 , . . .,λ m ) (|λ| =
√ ∑m
µ=1 λ2 µ ) and
˜λ := (˜λ 1 , ˜λ 2 , . . ., ˜λ m ) (|˜λ| = 1), respectively,
max
|˜λ|=1
( m
∑
µ=1
) 2
λ µ˜λµ = max(λ · ˜λ) 2 =
|˜λ|=1
µ
m∑
λ 2 µ = 1 − ǫ ⇐⇒ λ ‖ ˜λ . (2.56)
Therefore, | ˜ψ〉 that approximates |ψ〉 in a sense of maximal fidelity is given by
µ=1
| ˜ψ〉 =
1
√
m∑
λ 2 µ
µ=1
m∑
λ µ |u µ 〉|v µ 〉 . (2.57)
µ=1
Once more, the projection of the system-block state to the subspace spanned by the m
eigenvectors of the reduced density matrix with the largest eigenvalues, is the optimal
reduction procedure for the system-block state space.
As we have seen, all the considered optimization criteria lead to the same procedure for the
optimal reduction of the system-block state space. The new state space is spanned by a set
of m dominant eigenvectors {|u µ 〉 = ∑ i U iµ|i〉; µ = 1, . . .,m} of the reduced density matrix
ˆρ S and since the elements of this set are orthonormal it can be considered as an effective
basis {|α l+1 〉} = {|u µ 〉} of the block L(l). The matrix elements of the basis transformation
matrices then read
A l+1 (s l+1 ) αl ,α l+1
= 〈α l s l+1 |α l+1 〉 = U αl s l+1 ,α l+1
. (2.58)
The outlined procedure of the block state-space optimal reduction is successful and a
substantial reduction can be achieved if the density-matrix eigenvalues fall down quickly.
Typically this is the case for one-dimensional quantum lattice systems with a finite energy
gap. That is why DMRG algorithms, which we describe in the upcoming sections
(Sections 2.4 and 2.5) have been quite successful in studying such systems [99, 100, 196].
j
Consider SVD of ψ, ψ = UDV † , where D is a diagonal matrix with singular values of ψ ordered in
non-increasing sequence. TrU † DV ˜ψ = TrDV ˜ψU † and if V ˜ψU † = ˜D, where ˜D is a diagonal matrix with
singular values of ˜ψ ordered in non-increasing sequence, then Trψ † ˜ψ = ∑ n
k=1 s k(ψ)s k ( ˜ψ).