The kagome materials with tunable magnetism and nontrivial topological state are appealing in both the theoretical study and the applications, such as the spintronics device and topological quantum calculations. In this work, we study the magnetism and topological states of the shandite kagome materials family $M_3{X}_2{\mathrm{S}}_2(M=\mathrm{Rh},\mathrm{Ni};X=\mathrm{Sn},\mathrm{Pb})$ comprehensively. Under the nonmagnetic states, ${\mathrm{Rh}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ is a strong topological insulator after doping one electron in the primitive cell and ${\mathrm{Rh}}_3{\mathrm{Pb}}_2{\mathrm{S}}_2$ is the Dirac semimetal. ${\mathrm{Rh}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ has flat bands and a sharp density of state peak near the Fermi level, which induces itinerant ferromagnetism. In the ferromagnetic state, ${\mathrm{Rh}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ shows a Weyl semimetal phase and an axion insulator phase under different occupations. Due to the different hybridization between $d$ orbitals of Rh and $s$ orbitals of Sn/Pb, the isomorphic material ${\mathrm{Rh}}_3{\mathrm{Pb}}_2{\mathrm{S}}_2$ has no magnetic order, which is different from that of ${\mathrm{Rh}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. Meanwhile, ${\mathrm{Ni}}_3{\mathrm{Pb}}_2{\mathrm{S}}_2$ has three more electrons in the primitive cell and no magnetization in our calculations. It shows a weak topological insulator state. Our work reveals the magnetism and nontrivial topological phases in the shandite kagome materials family $M_3{X}_2{\mathrm{S}}_2(M=\mathrm{Rh},\mathrm{Ni};X=\mathrm{Sn},\mathrm{Pb})$. The tunable magnetism and topological states also provide a material platform for the further study of novel physical phenomena related to the magnetic nontrivial topological properties and further applications.

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• First-principles calculations
• Magnetism
• Topological materials
• Magnetic systems
• Topological insulators