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Journal of Atomic and Molecular Physics

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

Zheng Yong-lin

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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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 interaction energy of the Hubbard model W[γ ] is investigated as a functional of the single-particle density matrix γ. W[γ ] plays a central role in the lattice density-functional theory of strongly correlated electrons. It is determined exactly in the framework of Levy's constrained-search formulation by using sparse-matrix diagonalization or renormalization-group numerical methods. Results for representative examples of one-dimensional chains, small clusters, spin-polarized systems, and attractive interactions are presented. The functional dependence of W[γ ] is analysed. Introducing an appropriate scaling it is shown that W has pseudo-universal behaviour as a function of , where γ ij refers to the bond order between nearest neighbours and () refers to the corresponding limit of weak (strong) correlations. Simple and accurate approximations to W can be inferred for general applications.
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 Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, 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 Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, 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)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
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 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/tU/tU/t and of the alternation $\ensuremath{\delta}t$ of the hopping integrals tij{t}_{\mathrm{ij}}tij​ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ 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 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 g12{g}_{12}g12​ between the sublattices, the number of sites Na{N}_{a}Na​, and the band filling $n=({\ensuremath{\gamma}}_{11}+{\ensuremath{\gamma}}_{22})/2={N}_{e}/{N}_{a}$. For each $\ensuremath{\Delta}n$ the properties of WWW 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 WWW 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 Na{N}_{a}Na​, nnn and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.
The inhomogeneous Hubbard model is investigated in one and two dimensions by using lattice density-functional theory (LDFT). The ground-state energy E=T+WE=T+WE=T+W is regarded as a functional of the single-particle density matrix $\ensuremath{\gamma}$, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energy, and $W[\ensuremath{\gamma}]$ the interaction energy. Besides the known functional $T[\ensuremath{\gamma}]$ we propose a simple scaling approximation to the interaction energy $W[\ensuremath{\gamma}]$, which is based on exact results for the Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications we consider the Hubbard model on one- and two-dimensional bipartite lattices. Several ground-state properties are determined including the kinetic and Coulomb energy, density distribution, nearest-neighbor bond order, and charge gap. Comparison with exact Lanczos diagonalizations shows that LDFT with the scaled dimer approximation to $W[\ensuremath{\gamma}]$ yields a quite accurate description of the interplay between correlations and charge redistributions, from weak to strong coupling regimes, and for all band fillings. Goals and limitations of the present approach are discussed.