Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges

Abstract

We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying Z_{2} monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion P and time reversal T when spin-orbit coupling is negligible. In contrast to the well-known PT-symmetric nodal lines protected only by the π Berry phase, in which a single nodal line can exist, the nodal lines with Z_{2} monopole charges should always exist in pairs. We show that a pair of nodal lines with Z_{2} monopole charges is created by a double band inversion process and that the resulting nodal lines are always linked by another nodal line formed between the two topmost occupied bands. It is shown that both the linking structure and the Z_{2} monopole charge are the manifestation of the nontrivial band topology characterized by the second Stiefel-Whitney class, which can be read off from the Wilson loop spectrum. We show that the second Stiefel-Whitney class can serve as a well-defined topological invariant of a PT-invariant two-dimensional insulator in the absence of Berry phase. Based on this, we propose that pair creation and annihilation of nodal lines with Z_{2} monopole charges can mediate a topological phase transition between a normal insulator and a three-dimensional weak Stiefel-Whitney insulator. Moreover, using first-principles calculations, we predict ABC-stacked graphdiyne as a nodal line semimetal (NLSM) with Z_{2} monopole charges having the linking structure. Finally, we develop a formula for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion-invariant momenta, which is used to prove the quantized bulk magnetoelectric response of NLSMs with Z_{2} monopole charges under a T-breaking perturbation.

Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric

“…1(a) carries a Z 2 monopole charge [12]; this can be derived from the second Stiefel-Whitney invariant defined (a) on a unitary sphere that wraps the ring [28]. Besides, a direct identification of this topological invariant can be obtained by calculating a Wilson loop (associated with the touching bands) around the ring [28]; we note that Wilson loops can be measured through interferometry in cold-atom setups [60]. The absence of Berry curvature indicates that such topological defects cannot be signaled through standard topological responses [8].…”

Discovering Topological Surface States of Dirac Points

“…The gapless spectra indicate the nontrivial monopole charge of Z 2 = 1. Since this Z 2 charge can be protected by either one of the three G i T [27], we can break the other two or one G i T to get the Z 2 nodal ring [35][36][37][38][39] and Weyl dipoles [27,31]. These symmetry-breaking cases are illustrated in Fig.…”

Higher-order topological superconductivity of spin-polarized fermions

“…Interestingly, the two nodal loops at k y = 0 are linked with the nodal loop of the normal state, which appears as the crossing between the occupied bands of the BdG Hamiltonian. This linking structure indicates that the zero-energy nodes carry nontrivial monopole charges [29]. To see the higher-order topology of this phase, we consider the open boundary condition with 20 × 20 unit cells along x and y and the periodic boundary condition along the z direction.…”

“…In the case of the monopole nodal line superconductor, the nodes are fully gapped when T symmetry is broken due to spin-orbit coupling. The resulting gapped superconductor is a second-order TSC hosting chiral hinge states [8,29,39]. In fact, in the normal state, NLSM transforms to a Weyl semimetal by spin-orbit coupling as long as the parity configuration does not change.…”