  Berry curvature (BC) governs topological phases of matter and generates
anomalous transport. When a magnetic field is applied, phonons can acquire BC
indirectly through spin-lattice coupling, leading to a linear phonon Hall
effect. Here, we show that polar lattice distortion directly couples to a
phonon BC dipole, which causes a switchable non-linear phonon Hall effect. In a
SnS monolayer, the in-plane ferroelectricity induces a phonon BC and leads to
the phononic version of the non-volatile BC memory effect. As a new type of
ferroelectricity-phonon coupling, the phonon Rashba effect emerges and opens a
mass-gap in tilted Weyl phonon modes, resulting in a large phonon BC dipole.
Furthermore, our ab initio non-equilibrium molecular dynamics simulations
reveal that non-linear phonon Hall transport occurs in a controllable manner
via ferroelectric switching. The ferroelectricity-driven phonon BC and
corresponding non-linear phonon transports provide a novel scheme for
constructing topological phononic transport/memory devices.
