arXiv:cond-mat/9910264 · Physical review. B, Condensed matter

Density-matrix functional theory of the Hubbard model: An exact numerical study

Román López-Sandoval, G. M. Pastor

Published Jan 15, 2000• 45 citations

Abstract

A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites ${N}_{a},$ band filling ${N}_{e},$ and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on ${N}_{a},$ ${N}_{e},$ and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive $(U>0)$ and attractive $(U<0)$ interactions are contrasted.

DOI: 10.1103/physrevb.61.1764

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Electronic properties of the dimerized one-dimensional Hubbard model using lattice density-functional theory

Román López-Sandoval, G. M. Pastor

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Interaction-energy functional for lattice density functional theory: Applications to one-, two-, and three-dimensional Hubbard models

Román López-Sandoval, G. M. Pastor

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Scaling and transferability of the interaction-energy functional of the inhomogeneous Hubbard model

Matthieu Saubanère, G. M. Pastor

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

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Jan 25, 20120 citations

arXiv:cond-mat/9910264 · Physical review. B, Condensed matter

Density-matrix functional theory of the Hubbard model: An exact numerical study

Román López-Sandoval, G. M. Pastor

Published Jan 15, 2000• 45 citations

arXiv HTML PDF

Abstract

A density-functional theory for many-body lattice models is considered in which the single-particle density matrix

{\ensuremath{\gamma}}_{\mathrm{ij}}

is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional

W[{\ensuremath{\gamma}}_{\mathrm{ij}}].

W[{\ensuremath{\gamma}}_{\mathrm{ij}}]

is expressed as the sum of Hartree-Fock energy

{E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]

and the correlation energy

{E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].

Exact results are obtained for

{E}_{C}({\ensuremath{\gamma}}_{12})

of the Hubbard model on various periodic lattices, where

{\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}

for all nearest neighbors i and j. The functional dependence of

{E}_{C}({\ensuremath{\gamma}}_{12})

is analyzed by varying the number of sites

{N}_{a},

band filling

{N}_{e},

and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of

{E}_{C}({\ensuremath{\gamma}}_{12})

are discussed in the limits of weak

({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})

and strong

({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})

electronic correlations, and in the crossover region

({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).

Using an appropriate scaling we observe that

{\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}

has a pseudo-universal behavior as a function of

{g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).

The fact that

{\ensuremath{\varepsilon}}_{C}{(g}_{12})

depends weakly on

{N}_{a},

{N}_{e},

and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of

{E}_{C}({\ensuremath{\gamma}}_{12})

for repulsive

(U>0)

and attractive

(U<0)

interactions are contrasted.

DOI: 10.1103/physrevb.61.1764

Related papers

Electronic properties of the dimerized one-dimensional Hubbard model using lattice density-functional theory

Román López-Sandoval, G. M. Pastor

The dimerized one-dimensional Hubbard model is studied 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. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength $U/t$ and of the alternation $\ensuremath{\delta}t$ of the hopping integrals ${t}_{\mathrm{ij}}$ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.The dimerized one-dimensional Hubbard model is studied 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. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength $U/t$ and of the alternation $\ensuremath{\delta}t$ of the hopping integrals ${t}_{\mathrm{ij}}$ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.

arXiv:cond-mat/0210658Physical review. B, Condensed matterJan 24, 200324 citations

Interaction-energy functional for lattice density functional theory: Applications to one-, two-, and three-dimensional Hubbard models

Román López-Sandoval, G. M. Pastor

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 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.

arXiv:cond-mat/0311470Physical Review BFeb 13, 200429 citations

Scaling and transferability of the interaction-energy functional of the inhomogeneous Hubbard model

Matthieu Saubanère, G. M. Pastor

The inhomogeneous Hubbard model is investigated in the framework of lattice density-functional theory (LDFT) by considering the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites as the basic variable of the many-body problem. The domain of representability of ${\ensuremath{\gamma}}_{ij}$ is determined for charge-density wave states on finite bipartite lattices. Levy's constrained search of the interaction-energy functional $W[{\ensuremath{\gamma}}_{ij}]$ is numerically solved by applying the Lanczos method to an effective Hubbard-type model. The exact functional dependence of $W[{\ensuremath{\gamma}}_{ij}]$ is analyzed by varying systematically the charge transfer $\ensuremath{\Delta}n={\ensuremath{\gamma}}_{22}\ensuremath{-}{\ensuremath{\gamma}}_{11}$, the degree of electron delocalization ${g}_{12}$ between the sublattices, the number of sites ${N}_{a}$ , and the band filling $n=({\ensuremath{\gamma}}_{11}+{\ensuremath{\gamma}}_{22})/2={N}_{e}/{N}_{a}$. For each $\ensuremath{\Delta}n$ the properties of $W$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, as well as in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}\ensuremath{\le}{\ensuremath{\gamma}}_{12}\ensuremath{\le}{\ensuremath{\gamma}}_{12}^{0})$. It is shown that $W$ follows quite closely a simple scaling behavior as a function of $\ensuremath{\Delta}n$ and ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$. The very good transferability of $W(\ensuremath{\Delta}n,{g}_{12})$ for different ${N}_{a}$ , $n$ and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.The inhomogeneous Hubbard model is investigated in the framework of lattice density-functional theory (LDFT) by considering the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites as the basic variable of the many-body problem. The domain of representability of ${\ensuremath{\gamma}}_{ij}$ is determined for charge-density wave states on finite bipartite lattices. Levy's constrained search of the interaction-energy functional $W[{\ensuremath{\gamma}}_{ij}]$ is numerically solved by applying the Lanczos method to an effective Hubbard-type model. The exact functional dependence of $W[{\ensuremath{\gamma}}_{ij}]$ is analyzed by varying systematically the charge transfer $\ensuremath{\Delta}n={\ensuremath{\gamma}}_{22}\ensuremath{-}{\ensuremath{\gamma}}_{11}$, the degree of electron delocalization ${g}_{12}$ between the sublattices, the number of sites ${N}_{a}$, and the band filling $n=({\ensuremath{\gamma}}_{11}+{\ensuremath{\gamma}}_{22})/2={N}_{e}/{N}_{a}$. For each $\ensuremath{\Delta}n$ the properties of $W$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, as well as in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}\ensuremath{\le}{\ensuremath{\gamma}}_{12}\ensuremath{\le}{\ensuremath{\gamma}}_{12}^{0})$. It is shown that $W$ follows quite closely a simple scaling behavior as a function of $\ensuremath{\Delta}n$ and ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$. The very good transferability of $W(\ensuremath{\Delta}n,{g}_{12})$ for different ${N}_{a}$, $n$ and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.

Physical Review BJun 1, 200917 citations

Research of the density-matrix functionals of the periodic Hubbard model

Zheng Yong-lin

The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). It is considered that the interaction energy functional w[γij,S] of the Hubbard model is investigated as a function of the single-particle density matrix γij with respect to the lattice sites, and total spin S. Exact numerical results are obtained for w[γij,S] on rings, where γij=γ12 for all nearest neighbors. The properties of w[γij,S] are discussed in the limits of weak (γ012) and strong (γ∞ij) electronic correlations, and in the crossover region γ∞12γ12γ012. Scaling of w[γij,S] in units of the uncorrelated energy w0[γ0ij,S] shows a pseudo-universal behavior as a function of g12=(γ12-γ∞12)/(γ012-γ∞12). In addition, The functional dependence of w[γij,S] is analyzed for different S.The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). It is considered that the interaction energy functional w[γij,S] of the Hubbard model is investigated as a function of the single-particle density matrix γij with respect to the lattice sites, and total spin S. Exact numerical results are obtained for w[γij,S] on rings, where γij=γ12 for all nearest neighbors. The properties of w[γij,S] are discussed in the limits of weak (γ012) and strong (γ∞ij) electronic correlations, and in the crossover region γ∞12γ12γ012. Scaling of w[γij,S] in units of the uncorrelated energy w0[γ0ij,S] shows a pseudo-universal behavior as a function of g12=(γ12-γ∞12)/(γ012-γ∞12). In addition, The functional dependence of w[γij,S] is analyzed for different S.

Journal of Atomic and Molecular PhysicsJan 1, 20070 citations

Lattice density-functional theory of the attractive Hubbard model

Matthieu Saubanère, G. M. Pastor

The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy $E=T+W$ is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength $|U|/t$ , onsite potential $\ensuremath{\varepsilon}/t$, and band filling $n={N}_{e}/{N}_{a}$ . Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy $E=T+W$ is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength $|U|/t$, onsite potential $\ensuremath{\varepsilon}/t$, and band filling $n={N}_{e}/{N}_{a}$. Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.

Physical Review BSep 17, 201412 citations

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

Matthieu Saubanère

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.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.

Jan 25, 20120 citations

A density-functional theory for many-body lattice models is considered in which the single-particle density matrix

{\ensuremath{\gamma}}_{\mathrm{ij}}

is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional

W[{\ensuremath{\gamma}}_{\mathrm{ij}}].

W[{\ensuremath{\gamma}}_{\mathrm{ij}}]

is expressed as the sum of Hartree-Fock energy

{E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]

and the correlation energy

{E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].

Exact results are obtained for

{E}_{C}({\ensuremath{\gamma}}_{12})

of the Hubbard model on various periodic lattices, where

{\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}

for all nearest neighbors i and j. The functional dependence of

{E}_{C}({\ensuremath{\gamma}}_{12})

is analyzed by varying the number of sites

{N}_{a},

band filling

{N}_{e},

and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of

{E}_{C}({\ensuremath{\gamma}}_{12})

are discussed in the limits of weak

({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})

and strong

({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})

electronic correlations, and in the crossover region

({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).

Using an appropriate scaling we observe that

{\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}

has a pseudo-universal behavior as a function of

{g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).

The fact that

{\ensuremath{\varepsilon}}_{C}{(g}_{12})

depends weakly on

{N}_{a},

{N}_{e},

and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of

{E}_{C}({\ensuremath{\gamma}}_{12})

for repulsive

(U>0)

and attractive

(U<0)

interactions are contrasted.

U/t

and of the alternation $\ensuremath{\delta}t$ of the hopping integrals

{t}_{\mathrm{ij}}

${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.

{g}_{12}

between the sublattices, the number of sites

{N}_{a}

, and the band filling $n=({\ensuremath{\gamma}}_{11}+{\ensuremath{\gamma}}_{22})/2={N}_{e}/{N}_{a}$. For each $\ensuremath{\Delta}n$ the properties of

W

are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, as well as in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}\ensuremath{\le}{\ensuremath{\gamma}}_{12}\ensuremath{\le}{\ensuremath{\gamma}}_{12}^{0})$. It is shown that

W

follows quite closely a simple scaling behavior as a function of $\ensuremath{\Delta}n$ and ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$. The very good transferability of $W(\ensuremath{\Delta}n,{g}_{12})$ for different

{N}_{a}

n

and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.

The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy

E=T+W

is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength

|U|/t

, onsite potential $\ensuremath{\varepsilon}/t$, and band filling

n={N}_{e}/{N}_{a}

. Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.