A second-order topological insulator (SOTI) in $d$ spatial dimensions features topologically protected gapless states at its ($d-2$)-dimensional boundary at the intersection of two crystal faces, but is gapped otherwise. As a novel topological state, it has been attracting great interest, but it remains a challenge to identify a realistic SOTI material in two dimensions (2D). Here, based on combined first-principles calculations and theoretical analysis, we reveal the already experimentally synthesized 2D material graphdiyne as the first realistic example of a 2D SOTI, with topologically protected 0D corner states. The role of crystalline symmetry, the robustness against symmetry breaking, and the possible experimental characterization are discussed. Our results uncover a hidden topological character of graphdiyne and promote it as a concrete material platform for exploring the intriguing physics of higher-order topological phases.

We theoretically study the three-dimensional topological semimetals with nodal surfaces protected by crystalline symmetries. Different from the well-known nodal-point and nodal-line semimetals, in these materials, the conduction and valence bands cross on closed nodal surfaces in the Brillouin zone. We propose different classes of nodal surfaces, both in the absence and in the presence of spin-orbit coupling (SOC). In the absence of SOC, a class of nodal surfaces can be protected by space-time inversion symmetry and sublattice symmetry and characterized by a ${\mathbb{Z}}_2$ index, while another class of nodal surfaces are guaranteed by a combination of nonsymmorphic twofold screw-rotational symmetry and time-reversal symmetry. We show that the inclusion of SOC will destroy the former class of nodal surfaces but may preserve the latter provided that the inversion symmetry is broken. We further generalize the result to magnetically ordered systems and show that protected nodal surfaces can also exist in magnetic materials without and with SOC, given that certain magnetic group symmetry requirements are satisfied. Several concrete nodal-surface material examples are predicted via the first-principles calculations. The possibility of multi-nodal-surface materials is discussed.

Response properties that are purely intrinsic to physical systems are of paramount importance in physics research, as they probe fundamental properties of band structures and allow quantitative calculation and comparison with experiment. For anomalous Hall transport in magnets, an intrinsic effect can appear at the second order to the applied electric field. We show that this intrinsic second-order anomalous Hall effect is associated with an intrinsic band geometric property—the dipole moment of Berry-connection polarizability (BCP) in momentum space. The effect has scaling relation and symmetry constraints that are distinct from the previously studied extrinsic contributions. Particularly, in antiferromagnets with $\mathcal{PT}$ symmetry, the intrinsic effect dominates. Combined with first-principles calculations, we demonstrate the first quantitative evaluation of the effect in the antiferromagnet ${\mathrm{Mn}}_2\mathrm{Au}$. We show that the BCP dipole and the resulting intrinsic second-order conductivity are pronounced around band near degeneracies. Importantly, the intrinsic response exhibits sensitive dependence on the Néel vector orientation with a $2\pi$ periodicity, which offers a new route for electric detection of the magnetic order in $\mathcal{PT}$-invariant antiferromagnets.

We propose a universal practical approach to realize magnetic second-order topological insulator (SOTI) materials, based on properly breaking the time reversal symmetry in conventional (first-order) topological insulators. The approach works for both three dimensions (3D) and two dimensions (2D), and is particularly suitable for 2D, where it can be achieved by coupling a quantum spin Hall insulator with a magnetic substrate. Using first-principles calculations, we predict bismuthene on EuO(111) surface as the first realistic system for a two-dimensional magnetic SOTI. We explicitly demonstrate the existence of the protected corner states. Benefitting from the large spin-orbit coupling and sizable magnetic proximity effect, these corner states are located in a boundary gap $\sim 83\mathrm{meV}$, and hence can be readily probed in experiment. By controlling the magnetic phase transition, a topological phase transition between a first-order TI and a SOTI can be simultaneously achieved in the system. The effect of symmetry breaking, the connection with filling anomaly, and the experimental detection are discussed.

Topological states of matter and two-dimensional (2D) magnetism are two fascinating topics attracting tremendous interest in current research. In this work, we explore their interplay in a single 2D material system by proposing a different topological quantum state of matter—the 2D Weyl half-semimetal (WHS), which features 2D Weyl points at the Fermi level belonging to a single spin channel, such that the low-energy electrons are described by fully spin polarized 2D Weyl fermions. We provide the condition to realize this state, which requires an in-plane magnetization and a preserved vertical mirror symmetry. Remarkably, we prove that the WHS state is a critical state sitting at the topological phase transition between two quantum anomalous Hall (QAH) insulator phases with opposite Chern numbers, such that a switching of the QAH states as well as the direction of chiral edge channels can be readily achieved by rotating the magnetization direction. Furthermore, we predict a concrete 2D material, monolayer ${\mathrm{PtCl}}_3$, as a candidate for realizing the 2D WHS state and the above intriguing effects. Our findings open up a new direction of research at the confluent point of topology and magnetism in two dimensions, and the revealed route towards switchable QAH phases will enable new designs of topological nanoelectronic devices.

Nonsymmorphic symmetries, which involve fractional lattice translations, can generate exotic types of fermionic excitations in crystalline materials. Here we propose a topological phase arising from nonsymmorphic symmetries—the hourglass Dirac chain metal, and predict its realization in the rhenium dioxide. We show that ReO₂ features hourglass-type dispersion in the bulk electronic structure dictated by its nonsymmorphic space group. Due to time reversal and inversion symmetries, each band has an additional two-fold degeneracy, making the neck crossing-point of the hourglass four-fold degenerate. Remarkably, close to the Fermi level, the neck crossing-point traces out a Dirac chain—a chain of connected four-fold-degenerate Dirac loops—in the momentum space. The symmetry protection, the transformation under symmetry-breaking, and the associated topological surface states of the Dirac chain are revealed. Our results open the door to an unknown class of topological matters, and provide a platform to explore their intriguing physics.