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

Projected branes are constituted by only 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 quasi-crystal or its rational approximant in one dimension. In this work, starting from square lattice-based models for topological crystalline insulators, protected by the discrete four-fold rotational ($C_4$) 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 being devoid of the $C_4$ 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 incarnation of a square lattice-based second-order topological insulator with the characteristic corner-localized zero modes in its geometric descendant one-dimensional quasi-crystalline or crystalline branes that feature a quantized localizer index and endpoint zero-energy modes only when one of its end points passes through a corner of the parent crystal. Possible designer quantum and meta material-based platforms to experimentally harness our theoretically proposed topological branes are discussed.