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Physical review. B, Condensed matter

Local-density approach and quasiparticle levels for generalized Hubbard Hamiltonians

Pablo Jauralde Pou, Rúben Pérez, Fernando Flores, A. Levy Yeyati, A. Martín‐Rodero, Jose Manuel Blanco, Francisco José García-Vidal, José Ortega

Published Aug 15, 2000• 59 citations
Publisher

Abstract

This paper presents a general method to describe and analyze electron correlation effects in local-orbital electronic structure calculations using a generalized Hubbard Hamiltonian. In our approach, we first introduce a local density formalism where the total energy of the system is obtained as a function of the orbital occupancies ${{n}_{i}}$ associated with each local orbital; in particular, exchange and correlation local potentials are presented for a multilevel case. In parallel, using the dynamical mean field approximation, a many-body solution is obtained by means of a local self-energy that appropriately interpolates between the low and high correlation limits. We also show that the local density and the many-body solutions are linked through charge consistency conditions. These two solutions are applied to a multilevel Anderson impurity and to a multiband Hubbard lattice, our results showing the high accuracy of the approach presented in this paper. Further on, we discuss how to apply our previous analysis to the case of crystals and molecules and analyze several examples: bulk Si, and HF and ${\mathrm{H}}_{2}\mathrm{O}$ molecules. The good results obtained for these cases show that our approach for the description of correlation effects offers an interesting alternative to the well-established density functional methods based on the calculation of the electron density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r}).$

DOI
10.1103/physrevb.62.4309

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Development and applications of density matrix functional theory of generalized Hubbard Models

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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.

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Quasiparticle Many-Body Dynamics of Highly Correlated Electronic Systems

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Density Functional Theory for the Hubbard-Holstein model

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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 $n$ 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.

•Jan 1, 2018•2 citations
This paper presents a general method to describe and analyze electron correlation effects in local-orbital electronic structure calculations using a generalized Hubbard Hamiltonian. In our approach, we first introduce a local density formalism where the total energy of the system is obtained as a function of the orbital occupancies ni{{n}_{i}}ni​ associated with each local orbital; in particular, exchange and correlation local potentials are presented for a multilevel case. In parallel, using the dynamical mean field approximation, a many-body solution is obtained by means of a local self-energy that appropriately interpolates between the low and high correlation limits. We also show that the local density and the many-body solutions are linked through charge consistency conditions. These two solutions are applied to a multilevel Anderson impurity and to a multiband Hubbard lattice, our results showing the high accuracy of the approach presented in this paper. Further on, we discuss how to apply our previous analysis to the case of crystals and molecules and analyze several examples: bulk Si, and HF and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecules. The good results obtained for these cases show that our approach for the description of correlation effects offers an interesting alternative to the well-established density functional methods based on the calculation of the electron density \ensuremathρ(r\ensuremath→).\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r}).\ensuremathρ(r\ensuremath→).
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
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.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.
The self-consistent theory of the correlation effects in Highly Correlated Systems(HCS) is presented. The novel Irreducible Green's Functions(IGF) method is discused in detail for the Hubbard model and random Hubbard model. The interpolative solution for the quasiparticle spectrum, which is valid for both the atomic and band limit is obtained. The (IGF) method permits to calculate the quasiparticle spectra of many-particle systems with the complicated spectra and strong interaction in a very natural and compact way. The inelastic scattering corrections leads to the damping of the quasiparticles and are the main topic of the present consideration. The calculation of the damping has been done in a self-consistent way for both limits. For the random Hubbard model the weak coupling case has been considered and the self-energy operator has been calculated using the combination of the IGF method and Coherent Potential Approximation (CPA). The other applications of the method to s-f model, Anderson model, Heisenberg antiferromagnet, electron-phonon interaction models are discussed briefly.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength Uc{U}_{c}Uc​ and exhibits strong indications of a lower critical occupation number nc{n}_{c}nc​. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.
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.