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arXiv:1307.8132 · Physical Review B

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

C. A. Büsser, George Balster Martins, Adrian E. Feiguin

Published Dec 11, 2013• 35 citations
arXivHTMLPDF

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 ${L}^{d\ensuremath{-}1}$, 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 $L\ifmmode\times\else\texttimes\fi{}L$ square lattice and an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ 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.

DOI
10.1103/physrevb.88.245113

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We present a completely unbiased and controlled numerical method to solve quantum impurity problems in ddd-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 Ld\ensuremath−1{L}^{d\ensuremath{-}1}Ld\ensuremath−1, where LLL is the linear dimension of the original ddd-dimensional lattice. This allows one to calculate the ground state of a magnetic impurity attached to an L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL square lattice and an L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiL cubic lattice with LLL 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.
We introduce a block Lanczos (BL) recursive technique to construct quasi-one-dimensional models, suitable for density-matrix renormalization group (DMRG) calculations, from single- as well as multiple-impurity Anderson models in any spatial dimensions. This new scheme, named BL-DMRG method, allows us to calculate not only local but also spatially dependent static and dynamical quantities of the ground state for general Anderson impurity models without losing elaborate geometrical information of the lattice. We show that the BL-DMRG method can be easily extended to treat a multiorbital Anderson impurity model where not only inter- and intraorbital Coulomb interactions but also Hund's coupling and pair hopping interactions are included. We also show that the symmetry adapted BL bases can be utilized, when it is appropriate, to reduce the computational cost. As a demonstration, we apply the BL-DMRG method to three different models for graphene with a structural defect and with a single hydrogen or fluorine absorbed, where a single Anderson impurity is coupled to conduction electrons in the honeycomb lattice. These models include (i) a single adatom on the honeycomb lattice, (ii) a substitutional impurity in the honeycomb lattice, and (iii) an effective model for a single carbon vacancy in graphene. Our analysis of the local dynamical magnetic susceptibility and the local density of states at the impurity site reveals that, for the particle-hole symmetric case at half-filling of electron density, the ground state of model (i) behaves as an isolated magnetic impurity with no Kondo screening, while the ground state of the other two models forms a spin-singlet state where the impurity moment is screened by the conduction electrons. We also calculate the real-space dependence of the spin-spin correlation functions between the impurity site and the conduction sites for these three models. Our results clearly show that, reflecting the presence or absence of unscreened magnetic moment at the impurity site, the spin-spin correlation functions decay as $\ensuremath{\propto}$ ${r}^{\ensuremath{-}3}$, differently from the noninteracting limit ($\ensuremath{\propto}$ ${r}^{\ensuremath{-}2}$), for model (i) and as $\ensuremath{\propto}$ ${r}^{\ensuremath{-}4}$, exactly the same as the noninteracting limit, for models (ii) and (iii) in the asymptotic rrr, where rrr is the distance between the impurity site and the conduction site. Finally, based on our results, we shed light on recent experiments on graphene where the formation of local magnetic moments as well as the Kondo-like behavior have been observed.
We review a completely unbiased and controlled numerical approach to quantum impurity problems consisting of exactly mapping the complete lattice Hamiltonian onto an equivalent one dimensional system through a unitary transformation. The resulting dimensional and entanglement reduction allows one to study single particle problems on arbitrary ddd-dimensional lattices using the density matrix renormalization group (DMRG) method. The real-space resolution allows one to position the impurities at the boundary or bulk of the sample and to study screening effects due to edge or surface modes. We describe how to generalize this approach to multi-impurity problems, discuss applications and possible extensions.
We present an extensive study of the two-impurity Kondo problem for spin-1 adatoms on a square lattice using an exact canonical transformation to map the problem onto an effective one-dimensional system that can be numerically solved using the density matrix renormalization group method. We provide a simple intuitive picture and identify the different regimes, depending on the distance between the two impurities, Kondo coupling JK{J}_{K}JK​, longitudinal anisotropy DDD, and transverse anisotropy EEE. In the isotropic case, two impurities on opposite (the same) sublattices have a singlet (triplet) ground state. However, the energy difference between the triplet ground state and the singlet excited state is very small and we expect an effectively fourfold-degenerate ground state, i.e., two decoupled impurities. For large enough JK{J}_{K}JK​ the impurities are practically uncorrelated forming two independent underscreened states with the conduction electrons, a clear nonperturbative effect. When the impurities are entangled in an RKKY-like state, Kondo correlations persist and the two effects coexist: the impurities are underscreened, and the dangling spin-1/21/21/2 degrees of freedom are responsible for the interimpurity entanglement. We analyze the effects of magnetic anisotropy in the development of quasiclassical correlations.
The focus of this thesis is set on the calculation of spectral functions for low-dimensional quantum systems. At first the quantum many-body problem is introduced and the related spectral functions of interest are defined. In a next step the applied numerical algorithms are presented: Exact diagonalization & Lanczos algorithm, the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) in the formulation of matrix product states (MPS). The main focus is set on the calculation of spectral functions with these algorithms. In particular for the DMRG a new algorithm is presented which combines the Lanczos algorithm with matrix product states. The presented algorithms are applied to different problems. The first system denotes a copper crystal with magnetic (cobalt/iron) impurities. These materials are known Kondo-systems, i.e. they show an untypical increase in the resistivity with decreasing temperature. At low temperatures local moments will develop, which will be screened from the surrounding conduction band electrons. The ground state of the system is a long-range strongly correlated many-body state between conduction band electrons and local moments. In the singleparticle impurity spectral function a so-called Kondo resonance at the Fermi energy is formed below the Kondo temperature. In scanning tunneling microscope measurements the spectral function at the surface - the so-called local density of states - of the copper crystals with buried impurities in varying depth was measured. The goal of this study is to verify the existence of long-range Kondo signatures via a comparison of the spectral functions for impurities with different depth below the surface. Therefore in this work numerical simulations are used to calculate the local density of states at the surface. The single impurity Anderson model (SIAM) is applied which shows the Kondo resonance in the impurity spectral function. In order to calculate the local density of states at the surface of the copper crystal the equations of motion of the SIAM and the Dyson equation are used and then solved by a combination of numerical renormalization group and band structure calculations of the copper crystal (linear combination of atomic orbitals). The simulation for impurities in different depth below the surface and the experimental data are in very good agreement and hence support the measurement of a long-range Kondo signature. Furthermore the simulations allow to determine the Kondo temperature of a single impurity. The focus of the second part is set on the one-dimensional Heisenberg model. The spectral function of interest is here the dynamic spin structure factor. The newly developed MPS-Lanczos algorithm within the DMRG is used to calculate the dynamic spin structure factor. The results are compared to analytic results from the Bethe ansatz. In the end the new algorithm is discussed in respect to existing methods within the DMRG to calculate spectral functions.
We study the ground state properties of a single magnetic impurity in the honeycomb lattice. To understand the non-perturbative properties, we use the density matrix renormalization group method where the honeycomb lattice model is mapped onto an effective one-dimensional model by means of the Lanczos recursion technique. We apply this method to three different models: (i) a magnetic adatom on the honeycomb lattice, (ii) a substitutional magnetic impurity in the honeycomb lattice, and (iii) a defect in the honeycomb lattice. Here, we focus only on the particle-hole symmetric cases and find that the ground state of model (i) is qualitatively different from the others: the magnetic impurity spin behaves like an isolated spin in model (i) while it is screened in models (ii) and (iii) by conduction electrons to form a Kondo singlet like state.
Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.