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Density Functional Theory for the Hubbard-Holstein model

Pernilla Helmer

Published Jan 1, 2018• 2 citations
OpenAlex

Abstract

The physics of interacting many-body systems is a very complex and challenging subject. Therefore one in general needs to use simplified models and approximate methods for studying these systems theoretically. In this work we study the exchange-correlation (xc) potential for the Hubbard-Holstein (HH) model for the application of Density Functional Theory (DFT) to the same. The Hohenberg-Kohn theorem and the two sets of Kohn-Sham equations are derived for the HH-model extending a procedure outlined in previous work. The xc-potentials for the fermions and bosons respectively are then derived analytically as functions of the density $n$ and phononic displacement $x$ for a single HH-site in contact with a heat bath, generalizing the procedure for the pure Hubbard model. The potential for the HH-site is found to have a smaller gap at half filling than for the Hubbard model, due to a rescaling of the effective electron-electron interaction. It also has an additional term linear in $(n-1)$. The xc-potentials are used to study two simple systems, an Andersson impurity dimer and a four-site chain of HH-sites. The results are compared to those of an exact solution and to the Hartree-Fock (HF) solution. For the dimer the DFT-method performs consistently better than the HF-method, while the results for the chain are more ambiguous. However, the central result of this thesis is to produce xc-potentials for an infinitely coordinated Bethe lattice, using Dynamical Mean Field Theory (DMFT). The potentials for a small but representative set of interaction parameters are presented and compared to the analytical potential. The potentials are similar in that they do not exhibit much structure, and they behave very similarly far from half filling. Close to half filling they show more differences, as the analytic potential is always discontinuous around half filling while the potential attained via DMFT is only discontinuous for large electron-electron interactions.

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The physics of interacting many-body systems is a very complex and challenging subject. Therefore one in general needs to use simplified models and approximate methods for studying these systems theoretically. In this work we study the exchange-correlation (xc) potential for the Hubbard-Holstein (HH) model for the application of Density Functional Theory (DFT) to the same. The Hohenberg-Kohn theorem and the two sets of Kohn-Sham equations are derived for the HH-model extending a procedure outlined in previous work. The xc-potentials for the fermions and bosons respectively are then derived analytically as functions of the density nnn and phononic displacement xxx for a single HH-site in contact with a heat bath, generalizing the procedure for the pure Hubbard model. The potential for the HH-site is found to have a smaller gap at half filling than for the Hubbard model, due to a rescaling of the effective electron-electron interaction. It also has an additional term linear in (n−1)(n-1)(n−1). The xc-potentials are used to study two simple systems, an Andersson impurity dimer and a four-site chain of HH-sites. The results are compared to those of an exact solution and to the Hartree-Fock (HF) solution. For the dimer the DFT-method performs consistently better than the HF-method, while the results for the chain are more ambiguous. However, the central result of this thesis is to produce xc-potentials for an infinitely coordinated Bethe lattice, using Dynamical Mean Field Theory (DMFT). The potentials for a small but representative set of interaction parameters are presented and compared to the analytical potential. The potentials are similar in that they do not exhibit much structure, and they behave very similarly far from half filling. Close to half filling they show more differences, as the analytic potential is always discontinuous around half filling while the potential attained via DMFT is only discontinuous for large electron-electron interactions.
The use of Density Functional Theory (DFT) has become a standard procedure in the recent years for the simulation of nanostructures. However it is well known that DFT suffers from severe limitations when covering the description of strongly correlated systems for which those approximations are not good.\n\nStrongly correlated systems can be described by means of tight-binding models. Such models are much simpler than the more realistic real-space models as they depend on few parameters and in some cases can be solved algebraically.\n\nThe first three chapters of this thesis are devoted to the use of the DFT machinery to the description of such lattice models in order to assess the true limitations when DFT is applied to strongly correlated systems.\n\nThe fourth and fifth chapters are oriented to the direct application of DFT to the study of two different and interesting problems: the search of molecular magnets with large magnetic anisotropy energy (MAE) and the study of molecular junctions using graphene leads.\n\nIn the first chapter we derive the exact functional and the exchange-correlation energies that let us building the Kohn-Sham (KS) potentials of the single and double site Anderson-Hubbard models in order to find out if the exact DFT gives the correct description of such systems. We find that this description is in general incomplete.\n\nThe second chapter presents an analysis of the double-site asymmetric Hubbard model with 2 electrons. The properties of its ground and excited states are discussed. We then use the Mean-Field (MF) and the Bethe-ansatz LDA (BALDA) approximations to assess their performance as a function of the external potential-interactions ratio. We then apply the concepts of DFT to this model, establishing its KS description and the corresponding adiabatic connection. A parametrized functional is also devised using a method that allows improving the accuracy of any approximated functional whenever the relation between the kinetic and the potential correlation terms is known.\n\nThe third chapter covers the DFT estimation of the fundamental gap in strongly correlated systems. We study the asymmetric Hubbard dimer. We find the exact quasi-particle excitation spectrum, including the ionization potential and electron affinity, which let us derive the real gap. We set a method to calculate the KS potentials for non-integer particle numbers as a necessary step to find the KS Green's function. We prove that the Koopmans' and the Janak's theorems are obeyed. We find that the KS spectrum is in general incorrect, except for the HOMO by definition, when strong correlations are dominant. The KS gap is always smaller than the true gap with interactions and we find the role of the derivative discontinuity on this.\n\nIn the fourth chapter we focus on the practical use of DFT and we test several organometallic molecules based with the objective of finding stable molecular magnets at room temperatures. The current technology uses thin-sheets for manufacturing magnetic bits for data storage. Different alternatives such as the use of single molecular magnets or clusters of transition metals to reduce the physical size of a bit. Heavy metal dimers may show huge magnetic anisotropies provided by a strong spin-orbit coupling (SOC). We analyze the magnetic properties of several organometallic molecules based on transition metals to determine if such huge anisotropies survive when the metal is encapsulated into the organic molecule.\n\nThe fifth chapter analyzes the use of graphene in the development of nanoelectronic circuits based on molecular junctions. The graphene has unique properties excellently matching the requirements for ideal electrodes. However the use of graphene nanoelectronics is difficulty because the shape of the graphene edges can change arbitrarily the coupling and switching functionality of the molecules attached. The aim of this chapter is to investigate the electronics and transport properties of different edge morphologies of graphene-wedge single-molecule junctions. It is shown that some graphene-wedge BDT and BPD junctions show negative differential resistance (NDR), spin-filtering (SP) behavior and Fano resonances.
This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy-Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the 'failure' of the Kohn-Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree-Fock and Bethe ansatz local density approximation) to show their successes and limitations. We also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for density functional energy components, several limits of the exact functional (weak- and strong-coupling, symmetric and asymmetric), various adiabatic connection results, proofs of exact conditions for this model, and the origin of the Hubbard model from a minimal basis model for stretched H2.
The derivative discontinuity of the exchange-correlation (xc) energy at an integer particle number is a property of the exact, unknown xc functional of density functional theory (DFT) which is absent in many popular local and semilocal approximations. In lattice DFT, approximations exist which exhibit a discontinuity in the xc potential at half-filling. However, due to convergence problems of the Kohn-Sham (KS) self-consistency cycle, the use of these functionals is mostly restricted to situations where the local density is away from half-filling. Here a numerical scheme for the self-consistent solution of the lattice KS Hamiltonian with a local xc potential with rapid (or quasidiscontinuous) density dependence is suggested. The problem is formulated in terms of finite-temperature DFT where the discontinuity in the xc potential emerges naturally in the limit of zero temperature. A simple parametrization is suggested for the xc potential of the uniform one-dimensional (1D) Hubbard model at finite temperature which is obtained from the solution of the thermodynamic Bethe ansatz. The feasibility of the numerical scheme is demonstrated by application to a model of fermionic atoms in a harmonic trap. The corresponding density profile exhibits a plateau of integer occupation at low temperatures which melts away for higher temperatures.
This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of Density Functional Theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy-Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green’s function, demonstrating the ‘failure ’ of the Kohn-Sham method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree-Fock and Bethe Ansatz Local Density Approximation) to show their successes and limitations. We also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for Density
In this work, nonadiabatic exchange-correlation (XC) potentials for time-dependent density-functional theory (TDDFT) for strongly correlated materials are derived in the limits of strong and weak correlations. After summarizing some essentials of the available dynamical mean-field theory (DMFT) XC potentials valid for these systems, we present details of the Sham–Schluter equation approach that we use to obtain, in principle, an exact XC potential from a many-body theory solution for the nonequilibrium electron self-energy. We derive the XC potentials for the one-band Hubbard model in the limits of weak and strong on-site Coulomb repulsion. To test the accuracy of the obtained potentials, we compare the TDDFT results obtained with these potentials with the corresponding nonequilibrium DMFT solution for the one-band Hubbard model and find that the agreement between the solutions is rather good. We also discuss possible directions to obtain a universal XC potential that would be appropriate for the case of intermediate interaction strengths, i.e., a nonadiabatic potential that can be used to perform TDDFT analysis of nonequilibrium phenomena, such as transport and other ultrafast properties of materials with any strength of electron correlation at any value in the applied perturbing field.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.