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Lanczos transformation for quantum impurity problems in d-dimensional lattices: application to graphene nanoribbons

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.