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arXiv:0708.3265

Ground and excited-state fermions in a 1D double-well, exact and time-dependent density-functional solutions

Rudolph J Magyar

Published Aug 24, 2007• 1 citation
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Abstract

Two of the most popular quantum mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a 1D double-well potential, a model of atoms in a quasi-1D optical lattice or electrons of a Hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly-correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density functional theory (DFT) functional and 2-site Hubbard models. A 1D adiabatic LDA kernel is constructed for use in time-dependent density functional theory (TDDFT), and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant (LC) density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably with the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with non-adiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.

DOI
10.48550/arxiv.0708.3265

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Two of the most popular quantum mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a 1D double-well potential, a model of atoms in a quasi-1D optical lattice or electrons of a Hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly-correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density functional theory (DFT) functional and 2-site Hubbard models. A 1D adiabatic LDA kernel is constructed for use in time-dependent density functional theory (TDDFT), and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant (LC) density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably with the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with non-adiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.
Two of the most popular quantum-mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a one-dimensional (1D) double-well potential, a model of atoms in a quasi-1D optical lattice, or electrons of a hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density-functional theory (DFT) functional and two-site Hubbard models. A 1D adiabatic local-density approximation kernel is constructed for use in time-dependent density-functional theory (TDDFT) and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably to the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with nonadiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.
We illustrate the scope of Time Dependent Density Functional Theory (TDDFT) for strongly correlated (lattice) models out of equilibrium. Using the exact many body time evolution, we reverse engineer the exact exchange correlation (xc) potential vxcv_{xc}vxc​ for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation (ALDA) to vxcv_{xc}vxc​ for the 1D Hubbard model and compare it to exact results, to gain insight about approximate xc potentials. Finally, we provide some remarks on the v-representability for the 1D Hubbard model.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.
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.
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.