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arXiv:1009.3416 · Physical Review B

Density functional theory for a model quantum dot: Beyond the local-density approximation

Stefan Schenk, P. Schwab, Michael Dzierzawa, Ulrich Eckern

Published Mar 16, 2011• 15 citations
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Abstract

We study both static and transport properties of model quantum dots, employing density functional theory as well as (numerically) exact methods. For the lattice model under consideration the accuracy of the local-density approximation generally is poor. For weak interaction, however, accurate results are achieved within the optimized effective potential method, while for intermediate interaction strengths a method combining the exact diagonalization of small clusters with density functional theory is very successful. Results obtained from the latter approach yield very good agreement with density matrix renormalization group studies, where the full Hamiltonian consisting of the dot and the attached leads has to be diagonalized. Furthermore we address the question of whether static density functional theory is able to predict the exact linear conductance through the dot correctly---with, in general, a negative answer.

DOI
10.1103/physrevb.83.115128

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We study both static and transport properties of model quantum dots, employing density functional theory as well as (numerically) exact methods. For the lattice model under consideration the accuracy of the local-density approximation generally is poor. For weak interaction, however, accurate results are achieved within the optimized effective potential method, while for intermediate interaction strengths a method combining the exact diagonalization of small clusters with density functional theory is very successful. Results obtained from the latter approach yield very good agreement with density matrix renormalization group studies, where the full Hamiltonian consisting of the dot and the attached leads has to be diagonalized. Furthermore we address the question of whether static density functional theory is able to predict the exact linear conductance through the dot correctly---with, in general, a negative answer.
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.
We present a method which uses density functional theory (DFT) to treat transport through a single molecule connected to two conducting leads for the weak and intermediate coupling. This case is not accessible to standard nonequilibrium Green's function calculations. Our method is based on a mapping of the Hamiltonian on the molecule to a limited set of many-body eigenstates. This generates a many-body Hamiltonian with parameters obtained from ground-state local (spin) density approximation-DFT calculations. We then calculate the transport using many-body Green's function theory. We compare our results with existing density matrix renormalization group calculations for spinless and for spin-1/2 fermion chains and find good agreement.
Providing sufficient accuracy at low-computational cost, not surprisingly, density-functional theory ( DFT ) is a standard tool in theoretical material science. Similarly, quantum-electrodynamical density-functional theory ( QEDFT ) provides a promising route to describe experiments at the interface of quantum optics and quantum chemistry. Formally, in the Kohn-Sham ( KS ) formulation of DFT and QEDFT the exact many-particle Schrödinger equation is replaced by a set of coupled non-linear one-particle differential equations. The electron-electron, the electron-ion and the electron-photon interactions of the interacting system are mimicked by their corresponding effective KS potential with an, in principle unknown exchange-correlation ( xc ) term. In practice, DFT and QEDFT require approximations for this xc potential. The quality of which fully determines how much the DFT and QEDFT observables deviate from the exact ones. Understanding the shortcomings and improving the quality of approximate functionals makes the study of the exact functional of model systems indispensable. Here, we employ exact diagonalization for the Hamiltonian of different types of real-space lattice models in one spatial dimension to access the ground-state wave function, and thus to explicitly construct the exact functionals of DFT and QEDFT . To study the functional behavior of purely electronic systems, we consider two singlet-state electrons moving in an external potential with two connected atomic wells of variable depth and distance. To study coupled electron-photon systems, we consider the Rabi Hamiltonian with variable electron-photon coupling strength. By varying the model parameters, we systematically construct the functionals for systems from weak to strong coupling between the building blocks of the global system. Using the concept of the effective density-to- potential map, we discuss the intrasystem steepening which is an exact condition on the xc functional that captures the static correlation present in the system and carve out how it relates to the well-known intersystem derivative discontinuity. We further construct arbitrary observables as functionals of the internal variables which are directly affected by such features of the effective density-to-potential map. By constructing the reduced dipole map in the four-particle case of coupled electron-photon system, we show that the density-to-potential map helps to identify and understand recently discovered physical effects in the field of strongly coupled electron-photon systems. This work provides a contribution to the development of approximate function- als required in practical density-functional calculations of arbitrary experimentally accessible observables. Both the intrasystem steepening and the intersystem derivative discontinuity play a crucial role for systems with highly-localized electrons. Including both features in approximate functionals will improve the accu- racy of DFT evaluation of observables for such systems. Those results provide novel insight into the properties of the exact xc potentials in QEDFT . This work opens a new route to construct new functionals using coarse-grained or embed- ding approaches within the general and exact frameworks of DFT and QEDFT.
We calculate steady-state properties of a strongly correlated quantum dot under voltage bias by means of nonequilibrium cluster perturbation theory and the nonequilibrium variational cluster approach, respectively. Results for the steady-state current are benchmarked against data from accurate matrix product state based time evolution. We show that for low to medium interaction strength, nonequilibrium cluster perturbation theory already yields good results, while for higher interaction strength the self-consistent feedback of the nonequilibrium variational cluster approach significantly enhances the accuracy. We report the current-voltage characteristics for different interaction strengths. Furthermore we investigate the nonequilibrium local density of states of the quantum dot and illustrate that within the variational approach a linear splitting and broadening of the Kondo resonance is predicted which depends on interaction strength. Calculations with applied gate voltage, away from particle-hole symmetry, reveal that the maximum current is reached at the crossover from the Kondo regime to the doubly occupied or empty quantum dot. Obtained stability diagrams compare very well to recent experimental data [A. V. Kretinin et al., Phys. Rev. B 84, 245316 (2011)].
Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.
Scattering states fulfill the correct boundary conditions of a current carrying open quantum system. Discretizing the energy continuum of these states allows for employing Wilson's numerical renormalization group approach without violating the boundary conditions by using a finite size system. We evolve the analytically known steady-state density operator for a non-interacting quantum-system at finite bias to the full interacting problem by the time-dependent numerical renormalization group after switching on the local charging energy. Using a newly developed algorithm for steady-state nonequilibrium Green functions, we can calculate the current I as function of bias voltage V for arbitrary temperature and magnetic field. A comparison with second-order and GW Kadanoff-Baym-Keldysh results shows excellent agreement for weak interaction strength U .