Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe2_2 (X=Mo,W)

In recent years, transition metal dichalcogenides (TMDs) have garnered great interest as topological materials -- monolayers of centrosymmetric $\beta$-phase TMDs have been identified as 2D topological insulators (TIs), and bulk crystals of noncentrosymmetric $\gamma$-phase MoTe$_2$ and WTe$_2$ have been identified as type-II Weyl semimetals. However, ARPES and STM probes of these TMDs have revealed huge, "arc-like" surface states that overwhelm, and are sometimes mistaken for, the much smaller topological surface Fermi arcs of bulk type-II Weyl points. In this letter, we use first-principles calculations and (nested) Wilson loops to analyze the bulk and surface electronic structure of both $\beta$- and $\gamma$-MoTe$_2$, finding that $\beta$-MoTe$_2$ ($\gamma$-MoTe$_2$ gapped with symmetry-preserving distortion) is an inversion-symmetry-indicated $\mathbb{Z}_{4}$-nontrivial ($noncentrosymmetric, non$-$symmetry$-$indicated$) higher-order TI (HOTI) driven by double band inversion. Both structural phases of MoTe$_2$ exhibit the same surface features as WTe$_2$, revealing that the large Fermi arcs are in fact not topologically trivial, but are rather the characteristic split and gapped fourfold surface states of a HOTI. We also show that, when the effects of SOC are neglected, $\beta$-MoTe$_2$ is a nodal-line semimetal with $\mathbb{Z}_{2}$-nontrivial monopole nodal lines (MNLSM). This finding confirms that MNLSMs driven by double band inversion are the weak-SOC limit of HOTIs, implying that MNLSMs are higher-order topological $semimetals$ with flat-band-like hinge states, which we find to originate from the corner modes of 2D "fragile" TIs.Comment: Final version, 5 pg main text + 18 pg supplement, 4 + 6 figures, abstract abridged for arXiv posting - see paper for full abstrac

Strong and fragile topological Dirac semimetals with higher-order Fermi arcs

Dirac and Weyl semimetals both exhibit arc-like surface states. However, whereas the surface Fermi arcs in Weyl semimetals are topological consequences of the Weyl points themselves, the surface Fermi arcs in Dirac semimetals are not directly related to the bulk Dirac points, raising the question of whether there exists a topological bulk-boundary correspondence for Dirac semimetals. In this work, we discover that strong and fragile topological Dirac semimetals exhibit one-dimensional (1D) higher-order hinge Fermi arcs (HOFAs) as universal, direct consequences of their bulk 3D Dirac points. To predict HOFAs coexisting with topological surface states in solid-state Dirac semimetals, we introduce and layer a spinful model of an s–d-hybridized quadrupole insulator (QI). We develop a rigorous nested Jackiw–Rebbi formulation of QIs and HOFA states. Employing ab initio calculations, we demonstrate HOFAs in both the room- (α) and intermediate-temperature (α″) phases of Cd3As2, KMgBi, and rutile-structure (β′-) PtO2

Topological zero-dimensional defect and flux states in three-dimensional insulators

In insulating crystals, it was previously shown that defects with two fewer dimensions than the bulk can bind topological electronic states. We here further extend the classification of topological defect states by demonstrating that the corners of crystalline defects with integer Burgers vectors can bind 0D higher-order end (HEND) states with anomalous charge and spin. We demonstrate that HEND states are intrinsic topological consequences of the bulk electronic structure and introduce new bulk topological invariants that are predictive of HEND dislocation states in solid-state materials. We demonstrate the presence of first-order 0D defect states in PbTe monolayers and HEND states in 3D SnTe crystals. We relate our analysis to magnetic flux insertion in insulating crystals. We find that π-flux tubes in inversion- and time-reversal-symmetric (helical) higher-order topological insulators bind Kramers pairs of spin-charge-separated HEND states, which represent observable signatures of anomalous surface half quantum spin Hall states

Wallpaper Fermions and the Nonsymmorphic Dirac Insulator

Recent developments in the relationship between bulk topology and surface crystal symmetry have led to the discovery of materials whose gapless surface states are protected by crystal symmetries. In fact, there exists only a very limited set of possible surface crystal symmetries, captured by the 17 "wallpaper groups." We show that a consideration of symmetry-allowed band degeneracies in the wallpaper groups can be used to understand previous topological crystalline insulators, as well as to predict new examples. In particular, the two wallpaper groups with multiple glide lines, $pgg$ and $p4g$, allow for a new topological insulating phase, whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion. Like the surface state of a conventional topological insulator, the surface Dirac fermion in this "nonsymmorphic Dirac insulator" provides a theoretical exception to a fermion doubling theorem. Unlike the surface state of a conventional topological insulator, it can be gapped into topologically distinct surface regions while keeping time-reversal symmetry, allowing for networks of topological surface quantum spin Hall domain walls. We report the theoretical discovery of new topological crystalline phases in the A$_2$B$_3$ family of materials in SG 127, finding that Sr$_2$Pb$_3$ hosts this new topological surface Dirac fermion. Furthermore, (100)-strained Au$_2$Y$_3$ and Hg$_2$Sr$_3$ host related topological surface hourglass fermions. We also report the presence of this new topological hourglass phase in Ba$_5$In$_2$Sb$_6$ in SG 55. For orthorhombic space groups with two glides, we catalog all possible bulk topological phases by a consideration of the allowed non-abelian Wilson loop connectivities, and we develop topological invariants for these systems. Finally, we show how in a particular limit, these crystalline phases reduce to copies of the SSH model.Comment: Final version, 6 pg main text + 29 pg supplement, 6 + 13 figure

Magnetic Topological Quantum Chemistry

For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids - Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.Comment: Final version, 10 pg main text + 184 pg appendix, 5 + 25 figure