are shifted completely to a second paper.

The idea of DMRG {#dmrgintro}
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The very standard real-space RG approach is best explained for a spin Hamiltonian $H$ on a one dimensional lattice as visualized in figure \[1Dchain\].

The dots represent the individual spins which are grouped together by breaking up the chain into blocks as visualized in figure \[blocks\] for a particular block composition of two sites.

We like to establish a notation in which small letters refer to the single site spins and capital letters denote the blocks. The block Hamiltonian for the block with the index $I$ is then denoted as $H_I$. The idea of real-space RG is then to replace each block of the single spins by one effective [*block-spin*]{}, which leads to a renormalized block-spin Hamiltonian $H_{I'}$. The calculation of the block-spins from the blocks composed of single site spins is carried out by a [*RG transformation*]{} $R$, which can be defined in various ways [@bl82], for example by projecting the block on the low lying spectrum [@wh93]. In summary a RG approach is designed to split of the whole system into subsystems called blocks for which it is possible to reduce the degrees of freedom. 