Diagnosis of topological nodal lines with nontrivial monopole charge in the presence of rotation symmetries

Abstract

We discuss a general diagnosis scheme for topological nodal lines protected by time-reversal symmetry and space-inversion symmetry, in the absence of spin-orbit coupling. It is shown that when a shallow band inversion (with a small inverted gap) appears at the $\Gamma$-point in the Brillouin zone, nodal lines are generated along with the inversion. Much information on the types and numbers of these nodal lines can be related to the rotation eigenvalues and inversion eigenvalues of the inverted bands. In addition, we establish a $\mathbb{Z}$-type monopole charge for nodal lines with fourfold or sixfold rotation axis. We point out that the new monopole charge is only well-defined in a fragile sense, that is, when the orbitals near the Fermi energy are restricted to those having the same sign under a twofold rotation. The diagnosis scheme goes beyond the one discussed in Phys. Rev. X 8, 031069 (2018), particularly when there is a band crossing along some high-symmetry line, or when the nodal line has non-unity monopole charge.

arXiv comment17 pages and 3 figures

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• Unremovable linked nodal structures protected by crystalline symmetries in stacked bilayer graphene with Kekulé texture
• Movable but unavoidable nodal lines through high-symmetry points in two-dimensional materials