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Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of $e^2/h$.
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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.
We demonstrate, both theoretically and experimentally, the concept of non-linear second-order topological insulators, a class of bulk insulators with quantized Wannier centers and a bulk polarization directly controlled by the level of non-linearity. We show that one-dimensional edge states and zero-dimensional corner states can be induced in a trivial crystal insulator made of evanescently coupled resonators with linear and nonlinear coupling coefficients, simply by tuning the excitation intensity. This allows global external control over topological phase transitions and switching to a phase with non-zero bulk polarization, without requiring any structural or geometrical changes. We further show how these non-linear effects enable dynamic tuning of the spectral properties and localization of the topological edge and corner states. Such self-induced second-order topological insulators, which can be found and implemented in a wide variety of physical platforms ranging from electronics to microwaves, acoustics, and optics, hold exciting promises for reconfigurable topological energy confinement, power harvesting, data storage, and spatial management of high-intensity fields.
We show that a two-dimensional semiconductor with Rashba spin-orbit coupling could be driven into the second-order topological superconducting phase when a mixed-pairing state is introduced. The superconducting order we consider involves only even-parity components and meanwhile breaks time-reversal symmetry. As a result, each corner of a square-shaped Rashba semiconductor would host one single Majorana zero mode in the second-order nontrivial phase. Starting from edge physics, we are able to determine the phase boundaries accurately. A simple criterion for the second-order phase is further established, which concerns the relative position between Fermi surfaces and nodal points of the superconducting order parameter. In the end, we propose two setups that may bring this mixed-pairing state into the Rashba semiconductor, followed by a brief discussion on the experimental feasibility of the two platforms.
Two-dimensional second-order topological superconductors host zero-dimensional Majorana bound states at their boundaries. In this work, focusing on rotation-invariant crystalline topological superconductors, we establish a bulk-boundary correspondence linking the presence of such Majorana bound states to bulk topological invariants introduced by Benalcazar et al. We thus establish when a topological crystalline superconductor protected by rotational symmetry displays second-order topological superconductivity. Our approach is based on stacked Dirac Hamiltonians, using which we relate transitions between topological phases to the transformation properties between adjacent gapped boundaries. We find that in addition to the bulk rotational invariants, the presence of Majorana boundary bound states in a given geometry depends on the interplay between weak topological invariants and the location of the rotation center relative to the lattice. We provide numerical examples for our predictions and discuss possible extensions of our approach.
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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