Recently, the experimental realization of Dirac fermions in ${\beta }_{12}$- and ${\chi }_3$-boron sheets has attracted tremendous attention and simulated a broad exploration for the novel topologically nontrivial states in two-dimensional materials. Herein, by combining first-principles and tight-binding calculations, we discover a coexistence of open nodal arc and closed nodal loop produced by three-band touching in the recently synthesized $\beta$-boron sheet [Q. Zhong et al., Phys. Rev. Mater. 1, 021001 (2017)]. The symmetry analysis reveals that the nodal lines are protected by both time-reversal symmetry (TRS) and mirror-reflection symmetry (MRS). Intriguingly, when TRS is broken, e.g., via introducing magnetic proximity effect, a topological phase transition occurs, namely, the original spin-degenerate nodal line (loop) decays into fully spin-polarized nodal line (loop). Furthermore, when Rashba spin-orbit coupling (SOC) is turned on to break the MRS, spin-up and -down states are shifted toward opposite momentum directions, leading to the emergence of two new nodal loops around the $\mathrm{\Gamma }$ point. In addition, inclusion of both Rashba SOC and exchange field leads to band-gap opening of the Dirac points and nodal lines. Our findings not only reveal a form of nodal line in the $\beta$-boron sheet, but also offer an alternative approach to realize spin-polarized nodal line semimetals with promising applications in spintronic devices.

• Received 10 July 2019
• Revised 29 August 2019

DOI:https://doi.org/10.1103/PhysRevB.100.115423