  Van der Waals heterostructures have recently emerged as an exciting platform
for investigating the effects of strong electronic correlations, including
various forms of magnetic or electrical orders. Here, we perform an unbiased
exact diagonalization study of the effects of interactions on topological flat
bands of twisted transition metal dichalcogenides (TMDs) at odd integer
fillings. For hole-filling $\nu_h = 1$, we find that Chern insulator phases,
expected from interaction-induced spin-valley polarization of the bare bands,
are quite fragile, and give way to spontaneous multiferroic order -- coexisting
ferroelectricity and ferromagnetism, in presence of long-range Coulomb
repulsion. We provide a simple real-space picture to understand the phase
diagram as a function of interaction range and strength. Our findings establish
twisted TMDs as a novel and highly tunable platform for multiferroicity, and we
outline a potential route towards electrical control of magnetism in the
multiferroic phase.
