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Ferroelectricity and topological vortices from molecular ordering in metal-organic frameworks

Metal-organic frameworks comprehend a wide class of hybrid organic-inorganic materials with general structure A$_m$BX$_n$, with $A$ and $X$ being organic molecules and B a metal cation. This often results in enhanced structural flexibility and new functionalities. Hybrid perovskites ABX$_3$ are a well-known example.} In an Iron-based perovskites, (DMA)Fe^{II-III}(COOH)_3, dimethylammonium (DMA) molecules are organized in a hexagonal structure. They are orientationally disordered at high temperatures, but order at around $T=100$~K in a peculiar toroidal pattern. Recent experimental and theoretical study suggest the appearance of ferroelectric polarization in this phase, although the measured polarization is small, and the mechanism of ferroelectricity is still debated. We formulate a Landau-type theory that clarifies the connection between the electric polarization, molecular pattern, and distortive modes of the inorganic lattice. We find a remarkable mechanism of improper ferroelectricity, analogue to the trimerization process in inorganic hexagonal ferrites and manganites, but here driven by the ordering of organic molecules in a metal-organic framework. Our study reveals an extremely rich phase diagram with the prediction of topological domain walls, where the ferroelectricity arise from tripling the unit cells due to molecular ordering. Wide domain walls with inner structure are predicted.