Phonons Drive the Topological Phase Transition in Quasi-One-Dimensional Bi$_4$I$_4$
Quasi-one-dimensional bismuth halides offer an exceptional platform for exploring diverse topological phases, yet the nature of the room-temperature topological phase transition in Bi$_4$I$_4$ remains unresolved. While theory predicts the high-temperature $\beta$-phase to be a strong topological insulator (TI), experiments observe a weak TI. Here we resolve this discrepancy by revealing the critical but previously overlooked role of electron-phonon coupling in driving the topological phase transition. Using our newly developed ab initio framework for phonon-induced band renormalization, we show that thermal phonons alone drive $\beta$-Bi$_4$I$_4$ from the strong TI predicted by static-lattice calculations to a weak TI above ~180 K. At temperatures where $\beta$-Bi$_4$I$_4$ is stable, it is a weak TI with calculated surface states closely match experimental results, thereby reconciling theory with experiment. Our work establishes electron-phonon renormalization as essential for determining topological phases and provides a broadly applicable approach for predicting topological materials at finite temperatures.