Projected branes as platforms for crystalline, superconducting, and higher-order topological phases

Abstract

Projected branes are constituted by a small subset of sites of a higher-dimensional crystal, otherwise placed on a hyperplane oriented at an irrational or a rational slope therein, for which the effective Hamiltonian is constructed by systematically integrating out the sites of the parent lattice that fall outside such branes [Commun. Phys. 5, 230 (2022)]. Specifically, when such a brane is constructed from a square lattice, it gives rise to an aperiodic Fibonacci quasicrystal or its rational approximant (displaying emergent periodicity) in one dimension. In this work, starting from square lattice-based models for topological crystalline insulators, protected by the discrete fourfold rotational (C4) symmetry, we show that the resulting one-dimensional projected topological branes encode all the salient signatures of such phases in terms of robust endpoint zero-energy modes, quantized local topological markers, and mid-gap modes bound to dislocation lattice defects, despite such linear branes lacking the C4 symmetry of the original lattice. Furthermore, we show that such branes can also feature all the hallmarks of two-dimensional strong and weak topological superconductors through Majorana zero-energy bound states residing near their endpoints and at the core of dislocation lattice defects, besides possessing suitable quantized local topological markers. Finally, we showcase a successful realization of a square lattice-based second-order topological insulator with the characteristic corner-localized zero modes (protected by composites of nonspatial and crystalline symmetries) in its geometric descendant one-dimensional quasicrystalline or crystalline branes that generically feature a quantized localizer index, but support the endpoint zero-energy mode only at one of its endpoints when it passes through a corner of the parent crystal. Otherwise, topological endpoint modes at slightly higher energies (still separated from bulk states) appear at both ends of the brane, when they are slightly far from the square lattice corners. Possible engineered quantum and metamaterial-based platforms to experimentally harness our theoretically proposed topological crystalline and higher-order insulating as well as superconducting branes are discussed.

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