Kagome materials have attracted much attention recently due to the emergence of various exotic physical phenomena. A typical type of structural distortion in the kagome lattice is the trimerization of cations, known as the breathing mode, which may have a significant impact on the electronic and magnetic properties of kagome magnets. Here, we reveal the role of the breathing mode in the band topology based on the tight-binding model, and then investigate the influence of the breathing mode on the electronic property, ferroelectricity, and magnetism in a selected material family with a breathing kagome lattice ${\mathrm{Zr}}_3{X}_8$ ($X=\text{Cl}$, Br, I), using first-principles calculations. Although the breathing mode can only open a topologically trivial band gap at the Dirac point, its gradual enhancement can induce a topological phase transition from a nontrivial to a trivial insulator due to the competitive effect between the breathing mode and spin-orbit coupling in modulating the band gap. The breathing mode induces out-of-plane ferroelectric polarization through a linear coupling effect with the polar displacement mode, and the ferroelectric switching is accomplished by the simultaneous reversal of the two modes. The magnetic ground states change from a stripe antiferromagnetic to ferromagnetic phase with increasing anion radius. This work facilitates the exploration of physical phenomena driven by the breathing mode in the kagome lattice.

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• Ferroelectricity
• Topological insulators
• Kagome lattice
• Multiferroics
• First-principles calculations
• Tight-binding model