Cyclic Ferroelectric Switching and Quantized Charge Transport in CuInP$_2$S$_6$

The van der Waals layered ferroelectric CuInP$_2$S$_6$ has been found to exhibit a variety of intriguing properties arising from the fact that the Cu ions are unusually mobile in this system. While the polarization switching mechanism is usually understood to arise from Cu ion motion within the monolayers, a second switching path involving Cu motion across the van der Waals gaps has been suggested. In this work, we perform zero-temperature first-principles calculations on such switching paths, focusing on two types that preserve the periodicity of the primitive unit cell: ``cooperative" paths preserving the system's glide mirror symmetry, and ``sequential" paths in which the two Cu ions in the unit cell move independently of each other. We find that CuInP$_2$S$_6$ features a rich and varied energy landscape, and that sequential paths are clearly favored energetically both for cross-gap and through-layer paths. Importantly, these segments can be assembled to comprise a globally insulating cycle with the out-of-plane polarization evolving by a quantum as the Cu ions shift to neighboring layers. In this sense, we argue that CuInP$_2$S$_6$ embodies the physics of a quantized adiabatic charge pump.