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Register a new userHigher-order topological insulators and superconductors protected by inversion symmetry
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Eslam Khalaf
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We study surface states of topological crystalline insulators and superconductors protected by inversion symmetry. These fall into the category of higher-order topological insulators and superconductors which possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. We show that the surface states of higher-order topological insulators and superconductors can be thought of as globally irremovable topological defects and provide a complete classification of these inversion-protected phases in any spatial dimension for the ten symmetry classes by means of a layer construction. Furthermore, we discuss possible physical realizations of such states starting with a time-reversal invariant topological insulator (class AII) in three dimensions or a time-reversal invariant topological superconductor (class DIII) in two or three dimensions. The former can be used to build a three-dimensional second-order topological insulator which exhibits one-dimensional chiral or helical modes propagating along opposite edges, whereas the latter enables the construction of three-dimensional third-order or two-dimensional second-order topological superconductors hosting Majorana zero modes localized to two opposite corners. Being protected by inversion, such states are not pinned to a specific pair of edges or corners thus offering the possibility of controlling their location by applying inversion-symmetric perturbations such as magnetic field.
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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.
Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a three-dimensional crystal. A second-order topological phase can be induced by the presence of a bulk crystalline symmetry. Building on Shiozaki and Satos complete classification of bulk crystalline phases with an order-two crystalline symmetry [Phys. Rev. B {bf 90}, 165114 (2014)], such as mirror reflection, twofold rotation, or inversion symmetry, we classify all corresponding second-order topological insulators and superconductors. The classification also includes antiunitary symmetries and antisymmetries.
Pursuing topological phase and matter in a variety of systems is one central issue in current physical sciences and engineering. Motivated by the recent experimental observation of corner states in acoustic and photonic structures, we theoretically study the dipolar-coupled gyration motion of magnetic solitons on the two-dimensional breathing kagome lattice. We calculate the phase diagram and predict both the Tamm-Shockley edge modes and the second-order corner states when the ratio between alternate lattice constants is greater than a critical value. We show that the emerging corner states are topologically robust against both structure defects and moderate disorders. Micromagnetic simulations are implemented to verify the theoretical predictions with an excellent agreement. Our results pave the way for investigating higher-order topological insulators based on magnetic solitons.
We study disorder effects in a two-dimensional system with chiral symmetry and find that disorder can induce a quadrupole topological insulating phase (a higher-order topological phase with quadrupole moments) from a topologically trivial phase. Their topological properties manifest in a topological invariant defined based on effective boundary Hamiltonians, the quadrupole moment, and zero-energy corner modes. We find gapped and gapless topological phases and a Griffiths regime. In the gapless topological phase, all the states are localized, while in the Griffiths regime, the states at zero energy become multifractal. We further apply the self-consistent Born approximation to show that the induced topological phase arises from disorder renormalized masses. We finally introduce a practical experimental scheme with topoelectrical circuits where the predicted topological phenomena can be observed by impedance measurements. Our work opens the door to studying higher-order topological Anderson insulators and their localization properties.
We identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy Fermi arcs at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as mirror, twofold rotation, or inversion.
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