  We present a completely unbiased and controlled numerical method to solve
quantum impurity problems in d-dimensional lattices. This approach is based on
a canonical transformation, of the Lanczos form, where the complete lattice
Hamiltonian is exactly mapped onto an equivalent one dimensional system, in the
same spirit as Wilson's numerical renormalization. The method is particularly
suited to study systems that are inhomogeneous, and/or have a boundary. As a
proof of concept, we use the density matrix renormalization group to solve the
equivalent one-dimensional problem. The resulting dimensional reduction
translates into a reduction of the scaling of the entanglement entropy by a
factor $L^{d-1}$, where L is the linear dimension of the original d-dimensional
lattice. This allows one to calculate the ground state of a magnetic impurity
attached to an LxL square lattice and an LxLxL cubic lattice with L up to 140
sites. We also study the localized edge states in graphene nanoribbons by
attaching a magnetic impurity to the edge or the center of the system. For
armchair metallic nanoribbons we find a slow decay of the spin correlations as
a consequence of the delocalized metallic states. In the case of zigzag
ribbons, the decay of the spin correlations depends on the position of the
impurity. If the impurity is situated in the bulk of the ribbon, the decay is
slow as in the metallic case. On the other hand, if the adatom is attached to
the edge, the decay is fast, within few sites of the impurity, as a consequence
of the localized edge states, and the short correlation length. The mapping can
be combined with ab-initio band structure calculations to model the system, and
to understand correlation effects in quantum impurity problems from first
principles.
