Lattice density-functional theory on graphene

Abstract

A density-functional approach on the hexagonal graphene lattice is developed using an exact numerical solution to the Hubbard model as the reference system. Both nearest-neighbour and up to third nearest-neighbour hoppings are considered and exchange-correlation potentials within the local density approximation are parameterized for both variants. The method is used to calculate the ground-state energy and density of graphene flakes and infinite graphene sheet. The results are found to agree with exact diagonalization for small systems, also if local impurities are present. In addition, correct ground-state spin is found in the case of large triangular and bowtie flakes out of the scope of exact diagonalization methods.

Interacting fermions in one-dimensional disordered lattices: Exploring localization and transport properties with lattice density-functional theories

“…In 2D, the Hubbard model has been investigated via DFT on the graphene lattice. 64 To date, the simple cubic lattice is the only 3D case considered in the literature, 62 with the ground-state energy of the uniform system computed within dynamical mean field theory (DMFT). 65,66 Differently from the 1D case, here a discontinuity in v xc appears only for U > U Mott c , a DFT description of the onset of the Mott-Hubbard metal-insulator regime at a finite U .…”

Density-functional theory on graphs

“…This interaction has a similar form as the familiar Coulomb interaction in continuum systems, which is also diagonal in position basis. The form (12) without an external potential v will later be denoted as the internal part H 0 that typically contains the kinetic energy and the interaction W (usually of the form of a two-body interaction like above). An external potential v : X → R (equivalent to v ∈ R M ) alone acting on the many-particle wave function becomes…”

“…Lemma 4. Let H be like in (12) including W diagonal in the {e I } I basis and G(h) connected, then G(H) is connected as well.…”

“…In the theoretical development of density-functional theory (DFT) following the pioneering paper of Hohenberg and Kohn [1], the works of Levy [2] and Lieb [3] marked cornerstones on which practically all the following investigations built. DFT has since been extended to many different settings, including lattice systems [4][5][6][7][8][9] that in practice mostly appear in the form of the Hubbard model [10][11][12][13]. The solidity of the theoretical investigations primarily established for continuum systems has led many to believe that the same statements automatically hold for fermionic lattice systems.…”

Does Hohenberg-Kohn Theorem apply to a general quantum system?

“…As in the original HK theorem, the key is to establish a one-to-one correspondence between O and g. If so, by varying Ĥint and { Ôi }, the territory of DFT is safely expanded not only to general quantum systems [5][6][7][8][9][10][11][12][13], but also to general quantum properties in additon to the GS energy [14].…”