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arXiv:1403.5080 · Physical Review B

Solving lattice density functionals close to the Mott regime

Zu‐Jian Ying, Valentina Brosco, J. Lorenzana

Published May 27, 2014• 14 citations
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Abstract

We study a lattice version of the local density approximation (LDA) based on Bethe ansatz (BALDA). Contrary to what happens in density functional theory in the continuum and despite its name, BALDA displays some very nonlocal features and it has a discontinuous functional derivative. The same features prevent the convergence of the self-consistent Kohn-Sham cycle thus hindering the study of BALDA solutions close to a Mott phase or in the Coulomb blockade regime. Here, we propose a numerical approach which, differently from previous works, does not introduce ad hoc parameters to smear out the singularity. Our results are relevant for all lattice models where BALDA is applied ranging from Kondo systems to harmonically trapped Hubbard fermions. As an example, we apply the method to the study of a one-dimensional lattice model with Hubbard interaction and a staggered potential which can be driven from an ionic to a Mott-insulating state. In the Mott regime, the presence of a ``vacuum'' allows us to calculate the different contribution to the gap and to highlight an ultranonlocality of BALDA.

DOI
10.1103/physrevb.89.205130

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We study a lattice version of the local density approximation (LDA) based on Bethe ansatz (BALDA). Contrary to what happens in density functional theory in the continuum and despite its name, BALDA displays some very nonlocal features and it has a discontinuous functional derivative. The same features prevent the convergence of the self-consistent Kohn-Sham cycle thus hindering the study of BALDA solutions close to a Mott phase or in the Coulomb blockade regime. Here, we propose a numerical approach which, differently from previous works, does not introduce ad hoc parameters to smear out the singularity. Our results are relevant for all lattice models where BALDA is applied ranging from Kondo systems to harmonically trapped Hubbard fermions. As an example, we apply the method to the study of a one-dimensional lattice model with Hubbard interaction and a staggered potential which can be driven from an ionic to a Mott-insulating state. In the Mott regime, the presence of a ``vacuum'' allows us to calculate the different contribution to the gap and to highlight an ultranonlocality of BALDA.
The derivative discontinuity of the exchange-correlation (xc) energy at an integer particle number is a property of the exact, unknown xc functional of density functional theory (DFT) which is absent in many popular local and semilocal approximations. In lattice DFT, approximations exist which exhibit a discontinuity in the xc potential at half-filling. However, due to convergence problems of the Kohn-Sham (KS) self-consistency cycle, the use of these functionals is mostly restricted to situations where the local density is away from half-filling. Here a numerical scheme for the self-consistent solution of the lattice KS Hamiltonian with a local xc potential with rapid (or quasidiscontinuous) density dependence is suggested. The problem is formulated in terms of finite-temperature DFT where the discontinuity in the xc potential emerges naturally in the limit of zero temperature. A simple parametrization is suggested for the xc potential of the uniform one-dimensional (1D) Hubbard model at finite temperature which is obtained from the solution of the thermodynamic Bethe ansatz. The feasibility of the numerical scheme is demonstrated by application to a model of fermionic atoms in a harmonic trap. The corresponding density profile exhibits a plateau of integer occupation at low temperatures which melts away for higher temperatures.
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.
We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.
We present an extensive numerical study of the ground-state properties of confined repulsively interacting fermions in one-dimensional optical lattices. Detailed predictions for the atom-density profiles are obtained from parallel Kohn-Sham density-functional calculations and quantum Monte Carlo simulations. The density-functional calculations employ a Bethe ansatz based local-density approximation for the correlation energy that accounts for Luttinger-liquid and Mott-insulator physics. Semianalytical and fully numerical formulations of this approximation are compared with each other and with a cruder Thomas-Fermi-type local-density approximation for the total energy. Precise quantum Monte Carlo simulations are used to assess the reliability of the various local-density approximations, and in conjunction with these provide a detailed microscopic picture of the consequences of the interplay between particle-particle interactions and confinement in one-dimensional systems of strongly correlated fermions.
We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.
We present a theoretical study on the response properties to an external electric field of strongly correlated one-dimensional metals. Our investigation is based on the recently developed Bethe ansatz local-density approximation (BALDA) to the density functional theory formulation of the Hubbard model. This is capable of describing both Luttinger liquid and Mott-insulator correlations. The BALDA calculated values for the static linear polarizability are compared with those obtained by numerically accurate methods, such as exact (Lanczos) diagonalization and the density-matrix renormalization group, over a broad range of parameters. In general BALDA linear polarizabilities are in good agreement with the exact results. The response of the exact exchange and correlation potential is found to point in the same direction of the perturbing potential. This is well reproduced by the BALDA approach, although the fine details depend on the specific parameterization for the local approximation. Finally we provide a numerical proof for the nonlocality of the exact exchange and correlation functional.