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Abstract

We report finite-size topology in the quintessential time-reversal (TR) invariant systems, the quantum spin Hall insulator (QSHI) and the three-dimensional, strong topological insulator (STI) - previously-identified helical or Dirac cone boundary states of these phases hybridize in wire or slab geometries with one open boundary condition for finite system size, and additional, topologically protected, lower-dimensional boundary modes appear for open boundary conditions in two or more directions and coexist with the response signatures of the higher-dimensional topological bulk. We explicitly demonstrate this coexistence for both the QSHI in a ribbon geometry and the STI in a slab geometry. For the quasi-one-dimensional (q(2-1)D) QSHI, we find topologically protected, quasi-zero-dimensional (q(2-2)D) boundary states within the hybridization gap of the helical edge states, determined from q(2-1)D bulk topology characterized by topologically nontrivial Wilson loop spectra. We show this finite-size topology furthermore occurs in 1T"-WTe2 in ribbon geometries with sawtooth edges, based on analysis of a tight-binding model derived from density-functional theory calculations, motivating experimental investigation of our results. In addition, we find quasi-two-dimensional (q(3-1)D) finite-size topological phases occur for the STI, yielding helical boundary modes distinguished from those of the QSHI by a nontrivial magneto-electric polarizability linked to the original 3D bulk STI. Finite-size topological phases therefore exhibit signatures associated with the nontrivial topological invariant of a higher-dimensional bulk, clearly distinguishing them from previously-known topological phases. Finally, we find the q(3-2)D STI also exhibits finite-size topological phases, finding the first signs of topologically protected boundary modes of codimension greater than one due to finite-size topology. Finite-size topology of four- or higher-dimensional systems is therefore possible in experimental settings without recourse to thermodynamically large synthetic dimensions. © 2023 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/"Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.