Engineering Fragile Topology in Photonic Crystals: Topological Quantum Chemistry of Light

In recent years, there have been rapid advances in the parallel fields of electronic and photonic topological crystals. Topological photonic crystals in particular show promise for coherent transport of light and quantum information at macroscopic scales. In this work, we apply for the first time the recently developed theory of "Topological quantum chemistry" to the study of band structures in photonic crystals. This method allows us to design and diagnose topological photonic band structures using only group theory and linear algebra. As an example, we focus on a family of crystals formed by elliptical rods in a triangular lattice. We show that the symmetry of Bloch states in the Brillouin zone can determine the position of the localized photonic wave packets describing groups of bands. By modifying the crystal structure and inverting bands, we show how the centers of these wave packets can be moved between different positions in the unit cell. Finally, we show that for shapes of dielectric rods, there exist isolated topological bands which do not admit a well-localized description, representing the first physical instance of "fragile" topology in a truly noninteracting system. Our work demonstrates how photonic crystals are the natural platform for the future experimental investigation of fragile topological bands.Comment: v1. 4 pages + references main text, 5+epsilon page supplementary material v2. Published version, 4pgs + references. Supplemental material available at https://doi.org/10.1103/PhysRevResearch.1.03200

Topological phonon analysis of the 2D buckled honeycomb lattice: an application to real materials

By means of group theory, topological quantum chemistry, first-principles and Monte Carlo calculations, we analyze the topology of the 2D buckled honeycomb lattice phonon spectra. Taking the pure crystal structure as an input, we show that eleven distinct phases are possible, five of which necessarily have non-trivial topology according to topological quantum chemistry. Another four of them are also identified as topological using Wilson loops in an analytical model that includes all the symmetry allowed force constants up to third nearest neighbors, making a total of nine topological phases. We then compute the ab initio phonon spectra for the two-dimensional crystals of Si, Ge, P, As and Sb in this structure and construct its phase diagram. Despite the large proportion of topological phases found in the analytical model, all of the crystals lie in a trivial phase. By analyzing the force constants space using Monte Carlo calculations, we elucidate why topological phonon phases are physically difficult to realize in real materials with this crystal structure

Higher-Order Topological Insulators

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk, but conducting surface states which are topologically protected by time-reversal (or spatial) symmetries. Here, we extend the notion of three-dimensional topological insulators to systems that host no gapless surface states, but exhibit topologically protected gapless hinge states. Their topological character is protected by spatio-temporal symmetries, of which we present two cases: (1) Chiral higher-order topological insulators protected by the combination of time-reversal and a four-fold rotation symmetry. Their hinge states are chiral modes and the bulk topology is $\mathbb{Z}_2$-classified. (2) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs and the bulk topology is $\mathbb{Z}$-classified. We provide the topological invariants for both cases. Furthermore we show that SnTe as well as surface-modified Bi$_2$TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.Comment: 8 pages (4 figures) and 16 pages supplemental material (7 figures

Electronic properties, correlated topology and Green's function zeros

There is extensive current interest about electronic topology in correlated settings. In strongly correlated systems, contours of Green's function zeros may develop in frequency-momentum space, and their role in correlated topology has increasingly been recognized. However, whether and how the zeros contribute to electronic properties is a matter of uncertainty. Here we address the issue in an exactly solvable model for Mott insulator. We show that the Green's function zeros contribute to several physically measurable correlation functions, in a way that does not run into inconsistencies. In particular, the physical properties remain robust to chemical potential variations up to the Mott gap as it should be based on general considerations. Our work sets the stage for further understandings on the rich interplay among topology, symmetry and strong correlations.Comment: 15 pages, 3 figure

Monopole-like orbital-momentum locking and the induced orbital transport in topological chiral semimetals

The interplay between chirality and topology nurtures many exotic electronic properties. For instance, topological chiral semimetals display multifold chiral fermions that manifest nontrivial topological charge and spin texture. They are an ideal playground for exploring chirality-driven exotic physical phenomena. In this work, we reveal a monopole-like orbital-momentum locking texture on the three-dimensional Fermi surfaces of topological chiral semimetals with B20 structures (e.g., RhSi and PdGa). This orbital texture enables a large orbital Hall effect (OHE) and a giant orbital magnetoelectric (OME) effect in the presence of current flow. Different enantiomers exhibit the same OHE which can be converted to the spin Hall effect by spin-orbit coupling in materials. In contrast, the OME effect is chirality-dependent and much larger than its spin counterpart. Our work reveals the crucial role of orbital texture for understanding OHE and OME effects in topological chiral semimetals and paves the path for applications in orbitronics, spintronics, and enantiomer recognition.Comment: 23 pages, 5 figure

Large anomalous Hall, Nernst effect and topological phases in the 3d-4d/5d based oxide double perovskites

Magnetism and spin-orbit coupling are two fundamental and interconnected properties of oxide materials, that can give rise to various topological transport phenomena, including anomalous Hall and anomalous Nernst effects. These transport responses can be significantly enhanced by designing an electronic structure with a large Berry curvature. In this context, rocksalt-ordered double perovskites (DP), denoted as A2BB’O6, with two distinct transition metal sites are very powerful platforms for exploration and research. In this work, we present a comprehensive study based on the intrinsic anomalous transport in cubic and tetragonal stable DP compounds with 3d-4d/5d elements. Our findings reveal that certain DP compounds show a large anomalous Hall effect, displaying topological band crossings in the proximity of the Fermi energy

Spin-momentum locking from topological quantum chemistry: applications to multifold fermions

In spin-orbit coupled crystals, symmetries can protect multifold degeneracies with large Chern numbers and Brillouin zone spanning topological surface states. In this work, we explore the extent to which the nontrivial topology of chiral multifold fermions impacts the spin texture of bulk states. To do so, we formulate a definition of spin-momentum locking in terms of reduced density matrices. Using tools from the theory of topological quantum chemistry, we show how the reduced density matrix can be determined from the knowledge of the basis orbitals and band representation forming the multifold fermion. We show how on-site spin orbit coupling, crystal field splitting, and Wyckoff position multiplicity compete to determine the spin texture of states near chiral fermions. We compute the spin texture of multifold fermions in several representative examples from space groups $P432$ (207) and $P2_13$ (198). We show that the winding number of the spin around the Fermi surface can take many different integer values, from zero all the way to $\pm 7$. Finally, we conclude by showing how to apply our theory to real materials using the example of PtGa in space group $P2_13$.Comment: 28 pages, 6 figure