  Quasi-one-dimensional (1D) materials provide a superior platform for
characterizing and tuning topological phases for two reasons: i) existence for
multiple cleavable surfaces that enables better experimental identification of
topological classification, and ii) stronger response to perturbations such as
strain for tuning topological phases compared to higher dimensional crystal
structures. In this paper, we present experimental evidence for a
room-temperature topological phase transition in the quasi-1D material
Bi$_4$I$_4$, mediated via a first order structural transition between two
distinct stacking orders of the weakly-coupled chains. Using high resolution
angle-resolved photoemission spectroscopy on the two natural cleavable
surfaces, we identify the high temperature $\beta$ phase to be the first weak
topological insulator with gapless Dirac cones on the (100) surface and no
Dirac crossing on the (001) surface, while in the low temperature $\alpha$
phase, the topological surface state on the (100) surface opens a gap,
consistent with a recent theoretical prediction of a higher-order topological
insulator beyond the scope of the established topological materials databases
that hosts gapless hinge states. Our results not only identify a rare
topological phase transition between first-order and second-order topological
insulators but also establish a novel quasi-1D material platform for exploring
unprecedented physics.
