Two-dimensional higher-order topology in monolayer graphdiyne

Bulk-boundary correspondence is a fundamental property of topological phases, where gapped bulk states in *d*-dimensions support metallic states in (*d*-1)-dimensional surfaces. Higher-order topological insulators (HOTIs) violate this, exhibiting gapless excitations localized in a subspace with dimension lower than (*d*-1), such as corners or hinges, when crystalline symmetry is preserved. Examples of 3D HOTIs include rhombohedral bismuth, SnTe (with strain), MoTe₂ and WTe₂, and Bi₂SmSe₂. 2D second-order topological insulators (SOTIs) hosting corner states are less common. Phosphorene, while proposed as a candidate, isn't a genuine 2D HOTI due to corner states originating from charge polarization. Twisted bilayer graphene is another potential candidate, but requires precise control of twist angle and chemical potential. In 2D, *d*th-order TIs are obstructed atomic insulators; stable topology requires 1<*k*<*d*. Even without chiral symmetry, a 2D system can inherit nontrivial band topology manifested as a filling anomaly and corner charge accumulation (a 2D TOTI). This study proposes monolayer graphdiyne (MGD) as a realistic 2D HOTI candidate.

The introduction thoroughly reviews existing literature on higher-order topological insulators, both in 3D and 2D. It discusses known examples of 3D HOTIs and highlights the scarcity of experimentally realizable 2D SOTIs. The limitations of previously proposed 2D candidates, such as phosphorene and twisted bilayer graphene, are critically analyzed. The theoretical framework of 2D TOTIs, characterized by filling anomalies and corner charge accumulation in the absence of chiral symmetry, is also established, laying the groundwork for the study's central hypothesis.

The study employs a combination of first-principles density functional theory (DFT) calculations using the Vienna ab initio simulation package (VASP) and tight-binding model analysis. DFT calculations, utilizing the projector augmented-wave (PAW) method and generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) scheme, determine the electronic band structure of monolayer graphdiyne. A 12x12x1 k-point mesh and a 500 eV cutoff energy were used for the bulk calculations, while only the Γ point was used for finite-size structures. The tight-binding model, focusing on the pz orbitals and considering nearest-neighbor hopping parameters, is developed to describe the low-energy band structure. The topological invariants w1α and w2 are calculated using parity eigenvalues at time-reversal invariant momenta (TRIMs) and Wilson loop calculations. The Wilson loop calculations are performed to verify the orbital dependence of w2 and to identify the origin of the higher-order topology. For the ABC-stacked graphdiyne, a tight-binding model using pz orbitals is employed to investigate hinge modes. The Wannier function analysis is used to understand the higher order topology from the chemical bonding perspective. Finite-size structures of MGD with and without hydrogen passivation are modeled to analyze corner charge accumulation.

Monolayer graphdiyne (MGD) is identified as a 2D higher-order topological insulator with a nontrivial topological invariant w2 = 1, protected by inversion symmetry. While the low-energy band structure can be approximated using only the pz orbitals, the higher-order topology is only correctly captured when the core levels (s and pxy orbitals) are included. This is demonstrated through parity eigenvalue analysis and Wilson loop calculations. The w2 invariant is shown to be zero when considering only pz orbitals but becomes 1 when core levels are included. A filling anomaly is observed in finite-size MGD structures, where the half-filling condition cannot be satisfied due to the presence of corner states. To resolve this anomaly, extra electrons must be added, resulting in a half-integral charge accumulation at the My-invariant corners. This corner charge accumulation is directly observed in DFT calculations on a finite-size MGD structure. The addition of hydrogen atoms at corners, except at My-invariant corners, does not alter the topological invariant w2. The nontrivial w2 of MGD is shown to affect the band structure of ABC-stacked graphdiyne, leading to monopole nodal lines and hinge states. Tight-binding calculations, initially using only pz orbitals, revealed hinge states consistent with the w2 = 1 topology in a specific k-space region. The Wannier function analysis shows that the higher-order topology is related to the chemical bonding of s, px, and py orbitals across the unit cell boundary.

The findings demonstrate that MGD is a novel example of a 2D higher-order topological insulator, providing a realistic material realization of this topological phase. The work highlights the importance of considering core level contributions to accurately capture the higher-order band topology and emphasizes the utility of filling anomaly as a signature of this topology in systems lacking chiral symmetry. The connection between the 2D topology of MGD and the 3D topology of ABC-stacked graphdiyne suggests a pathway for designing 3D topological materials. The study significantly extends the scope of HOTI materials and provides a general approach for identifying and characterizing higher-order topological phases.

This study successfully demonstrates monolayer graphdiyne (MGD) as a realistic 2D higher-order topological insulator, characterized by a filling anomaly and corner charge accumulation. The nontrivial topology is shown to be dependent on the inclusion of core electronic states and is further confirmed through Wilson loop calculations. The connection to the 3D ABC-stacked graphdiyne provides a route for exploring 3D topological materials. Future research could explore other 2D materials with similar properties and investigate the potential applications of MGD in topological devices.

The study primarily relies on theoretical calculations. Experimental verification of the predicted higher-order topology and corner charges in MGD is needed. The tight-binding model used for ABC-stacked graphdiyne simplifies the system by focusing on pz orbitals; including other orbitals in the tight-binding calculation would enhance the accuracy. The study focuses on a specific stacking order of graphdiyne; exploring other stacking arrangements would provide a comprehensive understanding of the topological properties. Furthermore, the effects of disorder and interactions, which are not considered in this work, could influence the topological properties.