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Quantum states of matter combining non-trivial topology and magnetism attracta lot of attention nowadays; the special focus is on magnetic topologicalinsulators (MTIs) featuring quantum anomalous Hall and axion insulator phases.Feasibility of many novel phenomena that \emph{intrinsic} magnetic TIs may hostdepends crucially on our ability to engineer and efficiently tune theirelectronic and magnetic structures. Here, using angle- and spin-resolvedphotoemission spectroscopy along with \emph{ab initio} calculations we reporton a large family of intrinsic magnetic TIs in the homologous series of the vander Waals compounds (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_m$ with $m=0, ..., 6$.Magnetic, electronic and, consequently, topological properties of thesematerials depend strongly on the $m$ value and are thus highly tunable. Theantiferromagnetic (AFM) coupling between the neighboring Mn layers stronglyweakens on moving from MnBi2Te4 (m=0) to MnBi4Te7 (m=1), changes toferromagnetic (FM) one in MnBi6Te10 (m=2) and disappears with further increasein m. In this way, the AFM and FM TI states are respectively realized in the$m=0,1$ and $m=2$ cases, while for $m \ge 3$ a novel and hitherto-unknowntopologically-nontrivial phase arises, in which below the correspondingcritical temperature the magnetizations of the non-interacting 2D ferromagnets,formed by the \MBT\, building blocks, are disordered along the third direction.The variety of intrinsic magnetic TI phases in(MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_m$ allows efficient engineering of functionalvan der Waals heterostructures for topological quantum computation, as well asantiferromagnetic and 2D spintronics.

Variety of magnetic topological phases in the (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_m$ family