  Ferroelectric control of two-dimensional magnetism is promising in
fabricating electronic devices with high speed and low energy consumption. The
newly discovered layered MnBi$_2$Te$_4$(Bi$_2$Te$_3$)$_n$ and their Sb
counterparts exhibit A-type antiferromagnetism with intriguing topological
properties. Here, we propose to obtain tunable magnetic multistates in their
thin films by ferroelectrically manipulating the interlayer magnetic couplings
(IMCs) based on the Heisenberg model and first-principles calculations. Our
strategy relies on that interfacing the thin films with appropriate
ferroelectric materials can switch on/off an interlayer hopping channel between
Mn-$e_g$ orbitals as the polarizations reversed, thus resulting in a switchable
interlayer antiferromagnetism-to-ferromagnetism transition. On the other hand,
the interface effect leads to asymmetric energy barrier heights for the two
polarization states. These properties allow us to build ferroelectrically
switchable triple and quadruple magnetic states with multiple Chern numbers in
thin films. Our study reveals that ferroelectrically switchable magnetic and
topological multistates in MnBi$_2$Te$_4$ family can be obtained by rational
design for multifunctional electronic devices, which can also be applied to
other two-dimensional magnetic materials.
