Projected topological branes

Nature harbors crystals of dimensionality (d) only up to three. Here we introduce the notion of projected topological branes (PTBs): Lower-dimensional branes embedded in higher-dimensional parent topological crystals, constructed via a geometric cut-and-project procedure on the Hilbert space of the parent lattice Hamiltonian. When such a brane is inclined at a rational or an irrational slope, either a new lattice periodicity or a quasicrystal emerges. The latter gives birth to topoquasicrystals within the landscape of PTBs. As such PTBs are shown to inherit the hallmarks, such as the bulk-boundary and bulk-dislocation correspondences, and topological invariant, of the parent topological crystals. We exemplify these outcomes by focusing on two-dimensional parent Chern insulators, leaving its signatures on projected one-dimensional (1D) topological branes in terms of localized endpoint modes, dislocation modes and the local Chern number. Finally, by stacking 1D projected Chern insulators, we showcase the imprints of three-dimensional Weyl semimetals in d = 2, namely the Fermi arc surface states and bulk chiral zeroth Landau level, responsible for the chiral anomaly. Altogether, the proposed PTBs open a realistic avenue to harness higher-dimensional (d > 3) topological phases in laboratory.