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Physical Review B

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

Matthieu Saubanère, G. M. Pastor

Published Jun 1, 2009• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.79.235101

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The inhomogeneous Hubbard model is investigated in the framework of lattice density-functional theory (LDFT) by considering the single-particle density matrix \ensuremathγij{\ensuremath{\gamma}}_{ij}\ensuremathγij​ with respect to the lattice sites as the basic variable of the many-body problem. The domain of representability of \ensuremathγij{\ensuremath{\gamma}}_{ij}\ensuremathγij​ is determined for charge-density wave states on finite bipartite lattices. Levy's constrained search of the interaction-energy functional W[\ensuremathγij]W[{\ensuremath{\gamma}}_{ij}]W[\ensuremathγij​] is numerically solved by applying the Lanczos method to an effective Hubbard-type model. The exact functional dependence of W[\ensuremathγij]W[{\ensuremath{\gamma}}_{ij}]W[\ensuremathγij​] is analyzed by varying systematically the charge transfer \ensuremathΔn=\ensuremathγ22\ensuremath−\ensuremathγ11\ensuremath{\Delta}n={\ensuremath{\gamma}}_{22}\ensuremath{-}{\ensuremath{\gamma}}_{11}\ensuremathΔn=\ensuremathγ22​\ensuremath−\ensuremathγ11​, 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=(\ensuremathγ11+\ensuremathγ22)/2=Ne/Nan=({\ensuremath{\gamma}}_{11}+{\ensuremath{\gamma}}_{22})/2={N}_{e}/{N}_{a}n=(\ensuremathγ11​+\ensuremathγ22​)/2=Ne​/Na​. For each \ensuremathΔn\ensuremath{\Delta}n\ensuremathΔn the properties of WWW are discussed in the limits of weak (\ensuremathγ12\ensuremath≃\ensuremathγ120)({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})(\ensuremathγ12​\ensuremath≃\ensuremathγ120​) and strong (\ensuremathγ12\ensuremath≃\ensuremathγ12\ensuremath∞)({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})(\ensuremathγ12​\ensuremath≃\ensuremathγ12\ensuremath∞​) electronic correlations, as well as in the crossover region (\ensuremathγ12\ensuremath∞\ensuremath≤\ensuremathγ12\ensuremath≤\ensuremathγ120)({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}\ensuremath{\le}{\ensuremath{\gamma}}_{12}\ensuremath{\le}{\ensuremath{\gamma}}_{12}^{0})(\ensuremathγ12\ensuremath∞​\ensuremath≤\ensuremathγ12​\ensuremath≤\ensuremathγ120​). It is shown that WWW follows quite closely a simple scaling behavior as a function of \ensuremathΔn\ensuremath{\Delta}n\ensuremathΔn and g12=(\ensuremathγ12\ensuremath−\ensuremathγ12\ensuremath∞)/(\ensuremathγ120\ensuremath−\ensuremathγ12\ensuremath∞){g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})g12​=(\ensuremathγ12​\ensuremath−\ensuremathγ12\ensuremath∞​)/(\ensuremathγ120​\ensuremath−\ensuremathγ12\ensuremath∞​). The very good transferability of W(\ensuremathΔn,g12)W(\ensuremath{\Delta}n,{g}_{12})W(\ensuremathΔn,g12​) for different Na{N}_{a}Na​, nnn and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.
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 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.
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.
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 attractive Hubbard model is investigated in the framework of 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 ${\ensuremath{\gamma}}_{ij}$ 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|U|/t∣U∣/t, onsite potential $\ensuremath{\varepsilon}/t$, and band filling n=Ne/Nan={N}_{e}/{N}_{a}n=Ne​/Na​. 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.
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