Projected Topological Branes

TL;DR

This paper introduces projected topological branes (PTBs), lower-dimensional structures embedded in higher-dimensional topological crystals, revealing new topological phenomena and signatures akin to higher-dimensional phases through geometric cut-and-project methods.

Contribution

It proposes a novel framework for constructing and analyzing lower-dimensional topological structures within higher-dimensional crystals using a geometric cut-and-project approach.

Findings

PTBs inherit bulk-boundary and bulk-dislocation correspondences.

Signatures of higher-dimensional topological phases appear in projected lower-dimensional branes.

Stacked 1D PTBs mimic 3D Weyl semimetal features such as Fermi arcs and chiral zeroth Landau levels.

Abstract

Nature harbors crystals of dimensionality ($d$) only up to three. Here we introduce the notion of \emph{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, 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, 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.