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Frontiers in Physics

A Numerically Exact Approach to Quantum Impurity Problems in Realistic Lattice Geometries

Andrew Allerdt, Adrian E. Feiguin

Published Jun 12, 2019• 16 citations
Publisher

Abstract

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 $d$-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.

DOI
10.3389/fphy.2019.00067

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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 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 ${L}^{d\ensuremath{-}1}$, 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 $L\ifmmode\times\else\texttimes\fi{}L$ square lattice and an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ cubic lattice with 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.
The density matrix renormalization group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of one-dimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In this article, we review these recent developments. Following a description of the real-space DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to quantum chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm. We close with a description of the outlook for future applications of the method.
A new approach to large-scale nuclear structure calculations, based on the density matrix renormalization group (DMRG), is described. The DMRG method, which was developed originally in the framework of low-dimensional quantum lattices, has recently been extended to finite Fermi systems. Following a general description of this extended DMRG methodology, we briefly review its first application to a pairing hamiltonian. We then discuss its application to a schematic nuclear model involving many identical nucleons constrained to move in a single large-j shell. In those cases in which exact diagonalization of the hamiltonian is possible, the method is able to reproduce the exact results with extreme precision. Results are also presented for a model problem in which exact solution is not feasible.
Historically, the RKKY or indirect exchange, interaction has been accepted as being able to be described by second order perturbation theory. A typical universal expression is usually given in this context. This approach, however, fails to incorporate many body ef- fects, quantum fluctuations, and other important details. In Chapter 2, a novel numerical approach is developed to tackle these problems in a quasi-exact, non-perturbative manner. Behind the method lies the main concept of being able to exactly map an n-dimensional lattice problem onto a 1-dimensional chain. The density matrix renormalization group algorithm is then employed to solve the newly cast Hamiltonian.
In these lecture notes, we present a pedagogical review of a number of related numerically exact approaches to quantum many‐body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson’s Numerical Renormalization Group (NRG) and White’s Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low‐dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.
The density matrix renormalization group (DMRG) is one of the most powerful numerical methods for studying two-dimensional quantum lattice systems, despite a perception that it is only suitable for one dimension. Reviewing past applications of DMRG in 2D demonstrates its success in treating a wide variety of problems, although it remains underutilized in this context. We present techniques for performing cutting-edge 2D DMRG studies including methods for ensuring convergence, extrapolating finite-size data, and extracting gaps and excited states. Finally, we consider a selection of recently developed 2D tensor network methods and compare the performance of one of these to 2D DMRG.