Density-functional study of the Mott gap in the Hubbard model

Abstract

We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA). Unlike conventional functionals the BA-LDA has an explicit derivative discontinuity. We demonstrate that as a consequence of this discontinuity the BA-LDA yields the correct Mott gap, independently of the strength of the correlations. A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived. We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations. 71.15.Mb, 71.10.Pm, 71.10.Fd The intricacies of the correlation-induced metalinsulator transition ('Mott transition') have fascinated and challenged physicists for many years, and a multitude of methods has been brought to bear on the problem [1]. Here we study the Mott insulator from the point of view of density-functional theory (DFT), a method whose full potential for exploring the Mott phenomenon has not been widely recognized. It may be particularly timely to study the prospects for a DFT treatment of the Mott insulating phase because this is one of the two main phases of the one-dimensional Hubbard model (1DHM), the paradigmatic model for low-dimensional quantum systems. The recent interest in nanotechnology has brought quasi one-dimensional systems, such as quantum wires and carbon nanotubes, into the focus of current research in many-body physics and materials science. DFT, on the other hand, is the de facto standard method for the calculation of the electronic structure of materials, but its applicability to the very peculiar phases typical of low-dimensional quantum systems has not been systematically investigated. Here we report on the application of a very recently developed density functional [2] to the Mott insulating phase of the 1DHM.DFT is often applied in a single-particle mode, in which the eigenvalues of the Kohn-Sham (KS) equation are interpreted as a mean-field approximation to the quasiparticle energies. It is well known that this approximation breaks down for systems with strong correlations, of which the Mott insulator is an example. On the other hand, as long as one restricts oneself to working with the particle density, the total ground-state energy, and quantities derivable from these, DFT is a rigorous manybody theory, whose predictive power is only limited by the quality of available approximations for the exchangecorrelation (xc) functional. It is in this many-body mode that we employ DFT here.The choice of the appropriate approximation for the xc functional depends on the system under study. Here we investigate the 1DHM, since this model represents a particularly clear case of a Mott transition, free of unessential complications: for nonzero interaction strength (U > 0) the 1DHM is a Mott insulator at half filling, whereas it is a Luttinger liquid (i.e., a one-dimens...

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