Polarization Engineering of the Orbital Hall Conductivity in Two-dimensional Ferroelectric Higher-Order Topological Insulator Tl${}_2$S and SnS

Ferroelectric higher-order topological insulators (HOTIs) exhibit tunable physical properties arising from the interplay between ferroelectric polarization and band topology. This work investigates the topological origin of two classes of two-dimensional (2D) ferroelectric HOTIs with out-of-plane or in-plane polarization, revealing their distinct orbital transport behaviors and the mechanism for engineering orbital Hall conductivity (OHC) via polarization control. Our results demonstrate the unique role of polarization in modulating both the higher-order band topology and orbital transport. A strong coupling between in-plane polarization and higher-order topology is identified, establishing in-plane polarization as an intrinsic means to reversibly switch the OHC plateau within the band gap. Using Tl${}_2$S and SnS as representative models of the two HOTI types, we demonstrate persistent and electrically switchable orbital transport, respectively. Our study advances the understanding of the coupling among ferroelectricity, higher-order topology, and orbital transport, offering new avenues for controllable orbitronics.