Lanczos transformation for quantum impurity problems ind-dimensional lattices: Application to graphene nanoribbons

Abstract

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, and Haydock's recursion method. We introduce many-body interactions in the form of a Kondo or Anderson impurity and we solve the low-dimensional problem using the density matrix renormalization group. The technique is particularly suited to study systems that are inhomogeneous, and/or have a boundary. 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 L × L square lattice and an L × L × L 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 starting from first principles.

Spatial structure of correlations around a quantum impurity at the edge of a two-dimensional topological insulator

“…We have applied the Lanczos transformation method combined with the DMRG [35][36][37] , to study the many-body ground state of a quantum (S = 1/2) impurity (modeled as an Anderson impurity) coupled to the edge of a zigzag nanoribbon of stanene, a slightly buckled (non-planar) honeycomb lattice of Sn atoms, which hosts a topologically protected metallic edge state. The main motivation was to study the detailed spatial structure of the spin correlations between the quantum impurity and the electrons in the host, which characterize the Kondo ground state.…”

“…By generalizing the ideas introduced in Ref. 35 for single impurity problems, we reduce a complex lattice geometry to a single chain, or a multi-leg ladder in the case of multiple impurities.…”

“…II we present the model and a very brief description of the numerical method used (for details the reader is referred to previous papers by the authors [35][36][37] ). In Sec.…”

Nonperturbative effects and indirect exchange interaction between quantum impurities on metallic (111) surfaces

“…In such cases, reliable non-perturbative techniques are required for an unbiased analysis. Such methods, based on the density matrix renormalization group (DMRG) group, have recently been developed 31,32 . They allow one to study the full many-body problem with a realistic description of the band structure obtained from atomistic first principles calculations.…”

Dilute antiferromagnetism in magnetically doped phosphorene

“…[20,22]. The full many-body calculation can in turn be carried out using the density matrix renormalization group (DMRG) algorithm [23][24][25] with high accuracy and without approximations.…”