Electronic structures and topological properties in nickelates $Ln_{n+1}$Ni$_n$O$_{2n+2}$
After the significant discovery of the hole-doped nickelate compound Nd$_{0.8}$Sr$_{0.2}$NiO$_2$, an analysis of the electronic structure, orbital components, Fermi surfaces and band topology could be helpful to understand the mechanism of its superconductivity. Based on the first-principles calculations, we find that Ni $3d_{x^2-y^2}$ states contribute the largest Fermi surface. $Ln~5d_{3z^2-r^2}$ states form an electron pocket at $Γ$, while $5d_{xy}$ states form a relatively bigger electron pocket at A. These Fermi surfaces and symmetry characteristics can be reproduced by our two-band model, which consists of two elementary band representations: $B_{1g}@1a~\oplus~A_{1g}@1b$. We find that there is a band inversion near A, giving rise to a pair of Dirac points along A--M below the Fermi level once including spin-orbit coupling. Furthermore, we have performed the LDA+Gutzwiller calculations to treat the strong correlation effect of Ni 3d orbitals. In particular, the bandwidth of $3d_{x^2-y^2}$ has been renormalized largely. After the renormalization of the correlated bands, the Ni $3d_{xy}$ states and the Dirac points become very close to the Fermi level. Thus, a hole pocket at A could be introduced by hole doping, which may be related to the observed sign change of Hall coefficient. By introducing an additional Ni $3d_{xy}$ orbital, the hole-pocket band and the band inversion can be captured in our modified model. Besides, the nontrivial band topology in the ferromagnetic two-layer compound La$_3$Ni$_2$O$_6$ is discussed and the band inversion is associated with Ni $3d_{x^2-y^2}$ and La $5d_{xy}$ orbitals.