  Quantum wells in InAs/GaSb heterostructures can be tuned to a topological
regime associated with the quantum spin Hall effect, which arises due to an
inverted band gap and hybridized electron and hole states. Here, we investigate
electron-hole hybridization and the fate of the quantum spin Hall effect in a
quasi one-dimensional geometry, realized in a core-shell-shell nanowire with an
insulator core and InAs and GaSb shells. We calculate the band structure for an
infinitely long nanowire using $\mathbf{k \cdot p}$ theory within the Kane
model and the envelope function approximation, then map the result onto a BHZ
model which is used to investigate finite-length wires. Clearly, quantum spin
Hall edge states cannot appear in the core-shell-shell nanowires which lack
one-dimensional edges, but in the inverted band-gap regime we find that the
finite-length wires instead host localized states at the wire ends. These end
states are not topologically protected, they are four-fold degenerate and split
into two Kramers pairs in the presence of potential disorder along the axial
direction. However, there is some remnant of the topological protection of the
quantum spin Hall edge states in the sense that the end states are fully robust
to (time-reversal preserving) angular disorder, as long as the bulk band gap is
not closed.
