  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. 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 non-trivial 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 non-trivial magneto-electric
polarizability linked to the original 3D bulk STI. Finite-size topological
phases therefore exhibit signatures associated with the non-trivial topological
invariant of a higher-dimensional bulk. 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 1 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.
