PhD Thesis - Universität Augsburg

42 2. Density-Matrix Renormalization Group
In order to represent the global (e.g. superblock) states accurately, it is necessary to
take into account entanglement between a portion of the lattice sites (system block) and the
rest of the lattice (which we “mimic” with an environment block). Therefore, the optimal
system-block space-reduction procedure should preserve system-environment entanglement
[73, 74, 75, 191]. Neglecting it completely and considering the states of the isolated system
block, causes the removal of the key parts necessary for reproducing the entanglement in the
final state. This is also equivalent to the application of extra strong boundary conditions
to the subsystem and is the reason why Wilson’s numerical RG [260], with those isolated
block states, fails to find the ground state of the regular lattice systems, which in general
is entangled [191, 255, 256]. Note however, that for an accurate representation of the true
global state, the appropriate choice of the environment is also very important [191]. A good
environment has to contain as much as possible degrees of freedom, that are maximally
entangled with the system-block states. In general, it is not possible to pick the best
environment from the beginning, due to the same reason of an exponentially large Hilbert
space of the subsystem with all remaining sites. Therefore, it can be only constructed
and improved iteratively. In the following sections we will see that the first part, namely
the construction of the environment blocks just like the system blocks is performed by the
so-called infinite-system DMRG algorithm and then the finite-system DMRG algorithm
improves them iteratively.
According to quantum information theory the amount of entanglement between two
parts can be measured with the entanglement entropy [184], namely the von Neumann
entropy of the reduced density matrix ˆρ S ,
∑n Sch
S vN (ˆρ S ) ≡ −Tr(ˆρ S log 2 (ˆρ S )) = − λ 2 µ log 2(λ 2 µ ) . (2.61)
µ=1
This measure vanishes for a product state and it is maximal for the flat probability distribution
λ 2 µ = 1/n Sch, where S vN = − ∑ µ 1/n Sch log 2 (1/n Sch ) = log 2 (n Sch ). Therefore we
always have n Sch 2 S vN
. This quantity imposes a useful bound on the minimal number
m of kept states during the reduction process. Therefore, the true global state can be approximated
accurately with the MPS, if the bipartition entanglements of the subsystems
constituting the entire system are bounded or maximally grow logarithmically with large
subsystem sizes (effecting in polynomial growth for the matrix dimension). In one spatial
dimension, the entanglement entropy of a block of l contiguous sites typically increases
with l until l becomes of the order of the correlation length ξ in the system, at this point
it saturates to some value S max , whereas it diverges logarithmically at a quantum critical
point [191, 246]. The question of representing a quantum state in terms of matrix products
has recently been investigated in more details [240, 241].