Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges (1803.11416v2)

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 $\pi$ 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 {\it double band inversion} (DBI) process, and that the resulting nodal lines are always {\it 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 {\it 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 (2D) 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 (3D weak SWI). 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.

Band Topology and Linking Structure of Nodal Line Semimetals with $Z_2$ Monopole Charges

The paper "Band Topology and Linking Structure of Nodal Line Semimetals with $Z_2$ Monopole Charges" presents a rigorous exploration of the band topology inherent to topological semimetals with nodal lines that carry $Z_2$ monopole charges. The paper is particularly focused on three-dimensional systems where inversion ($P$) and time-reversal ($T$) symmetries are present, especially when spin-orbit coupling is negligible.

The authors aim to differentiate the $PT$-symmetric nodal lines safeguarded by the $\pi$ Berry phase from nodal lines endowed with $Z_2$ monopole charges. The paper posits that nodal lines carrying $Z_2$ monopole charges must obligatorily appear in pairs. These pairs are generated through a double band inversion (DBI) process, which is coupled with a linking structure of nodal lines. This linking involves an additional nodal line formed between the two highest occupied bands.

The investigation reveals that both the linking structure and the incorporation of the $Z_2$ monopole charge are manifestations of the non-trivial band topology. This can be characterized by the second Stiefel-Whitney class, a topological invariant that can be deduced from the spectral progression of the Wilson loop. In the absence of Berry phase, the second Stiefel-Whitney class serves as a topological invariant of a $PT$-invariant two-dimensional insulator. This characteristic plays a central role in mediating topological phase transitions between various forms of insulators, specifically from normal insulators to a three-dimensional weak Stiefel-Whitney insulator (3D weak SWI).

The paper also includes findings from first-principles calculations predicting ABC-stacked graphdiyne as a nodal line semimetal (NLSM) with $Z_2$ monopole charges that exhibit a linking structure. This prediction, founded on the numerical solutions to the band topology, showcases the practicality of these theoretical constructs in real materials.

A notable contribution of this research is the formula developed for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion-invariant momenta. This formula proves the quantized bulk magnetoelectric response of the $Z_2$-charged nodal line semimetals under a perturbation that breaks time-reversal symmetry.

Practical and Theoretical Implications

Practically, the research opens pathways for new material discoveries, particularly in identifying $Z_2$-nontrivial nodal lines. The implications of utilizing materials such as graphdiyne in the field of electronic devices lie in their unique topological properties, which could enhance device efficiency and functionality.

Theoretically, the identification of $Z_2$ monopole charges contributes significantly to expanding the landscape of topological phases in condensed matter physics. The linking structure and the derived formula for computing topological invariants from parity eigenvalues offer a new dimension in understanding the switch between various topological states.

Future Directions

The paper implies potential avenues for exploring additional materials hosting complex nodal structures. Furthermore, as the prominence of these topological insights grows, the development of experimental techniques capable of verifying theoretical predictions, such as those involving graphdiyne, will be invaluable.

Conclusively, this paper is an essential reference for researchers working on topological mechanics in condensed matter, offering robust mathematical frameworks to dissect and anticipate unique phases in semimetallic systems. As advancements in simulation and material synthesis continue, leveraging the findings herein could catalyze transformative progress in electronic materials science.