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Physical review. B, Condensed matter

Tight-binding electrons on open chains: Density distribution and correlations

U. Busch, K. A. Penson

Published Dec 15, 1987• 17 citations
Publisher

Abstract

The ground state of the tight-binding electrons on finite open chains is studied for arbitrary band filling. Analytic expressions for the distribution of electron density and the density-density correlation functions within the system are found and discussed as a function of chain size. Particular attention is paid to the study of boundary effects.

DOI
10.1103/physrevb.36.9271

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The ground state of the tight-binding electrons on finite open chains is studied for arbitrary band filling. Analytic expressions for the distribution of electron density and the density-density correlation functions within the system are found and discussed as a function of chain size. Particular attention is paid to the study of boundary effects.
In this paper, we investigate the open tight-binding model with NNN sites coupled to two reservoirs on its edges with the nonequilibrium Green function method to understand effects of open boundaries. As a result, we obtain an analytical expression of the electron density in the steady state and in the transient regime for N→∞N \rightarrow \inftyN→∞ and at absolute zero temperature. From the expression of the electron density in the steady state, we show that the open boundaries do not affect the qualitative behavior of the electron density and the phase transition which has been observed for the isolated tight-binding model also exists in our open case. Using the expression of the time-dependent electron density, we show that a two-step decay appears in the relaxation process purely caused by open boundaries. In addition, we show that the dependence of speed of convergence on boundary coupling strength qualitatively changes at a certain value of the boundary parameters. This is a result of special eigenvalues of a non-hermitian matrix whose imaginary parts do not vanish even in the limit N→∞N \rightarrow \inftyN→∞.
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.
It is shown that the ground-state optical absorption spectrum of a dimerised strongly correlated half-filled Hubbard chain can be expressed in terms of local densities of states of a corresponding cleaved one-electron tight-binding chain with appropriate alternating hopping matrix elements. A general continued-fraction expression is presented which gives, in principle, the local density of states for any arbitrary distribution of hopping elements in a nearest-neighbour one-electron tight-binding chain, and from which, as an application, the absorption spectrum of the dimerised Hubbard chain can be calculated.