  The simultaneous presence of seemingly incompatible properties of solids
often provides a unique opportunity to address questions of fundamental and
practical importance. The coexistence of ferroelectric and topological orders
is one such example. Ferroelectrics, which have a spontaneous macroscopic
polarization switchable by an applied electric field, usually are
semiconductors with a well-developed wide band gap with a few exceptions. On
the other hand, time-reversal symmetric $Z_2$ topological insulators (TI),
characterized by robust metallic surface states protected by the topology of
the bulk, usually are narrow-gap semiconductors ($< 0.7$ eV) which allow band
inversion induced by the spin-orbit interaction. To date, a ferroelectric
topological insulator (FETI) has remained elusive, owing to the seemingly
contradictory characters of the ferroelectric and topological orders. Here, we
report that the FETI can be realized in halide perovskite CsPbI$_3$ under
strain. Our first-principles study reveals that a non-centrosymmetric
ferroelectric structure of CsPbI$_3$ is energetically favored under a wide
range of pressures, while maintaining its topological order. The proposed FETI
is characterized by switchable polar surfaces with spin-momentum locked Dirac
cones, which allows for electric-field control of topological surface states
(TSSs) and the surface spin current. Our demonstration of a FETI in a feasible
material opens doors for future studies combining ferroelectric and topological
orders, and offers a new paradigm for diverse applications in electronics,
spintronics, and quantum information.
