Existence of nodal line semi-metal in a generalized three dimensional Haldane model

TL;DR

This paper introduces a three-dimensional generalization of the Haldane model on a diamond lattice, revealing a nodal line semi-metal phase with unique topological properties and surface states.

Contribution

It constructs a novel 3D Haldane-like model with complex hopping and inversion symmetry breaking, analyzing its spectrum and topological phases.

Findings

Existence of six pairs of anisotropic gapless points with linear dispersion.

Gapless phase exhibits non-zero Chern numbers on certain planes.

Gapped phase is a trivial insulator with zero Chern number and Hopf index.

Abstract

We construct and study a time reversal broken tight binding model on diamond lattice with complex next-nearest-neighbour hopping which can be thought of as a generalisation of two dimensional Haldane model in three dimension. The model also breaks inversion symmetry owing to sub-lattice dependent chemical potential. We calculate the spectrum of the model and find the existence of six pairs of anisotropic gapless points with linear dependence on momentum. The coordinates of the gapless points are ($2 \pi, \pi \pm k_0,0),~ (2 \pi, \pi \pm k_0,2 \pi)$ and their possible permutations . The condition for gapless spectrum is very similar to the two dimensional case. Each gapless points are having well defined chirality and in the gapless phase specific set of planes have non-zero Chern number. The gapped phase is a trivial bulk insulator which has vanishing Chern number as well as Hopf index. The model belongs to the symmetry class AIII according to the ten-fold way of classification. Surprisingly the gapless phase does contain a gapped surface state where as the gapped state has a gapless surface states as found in (1,1,1) direction.