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

Quantum states of matter combining non-trivial topology and magnetism attract a lot of attention nowadays; the special focus is on magnetic topological insulators (MTIs) featuring quantum anomalous Hall and axion insulator phases. Feasibility of many novel phenomena that \emph{intrinsic} magnetic TIs may host depends crucially on our ability to engineer and efficiently tune their electronic and magnetic structures. Here, using angle- and spin-resolved photoemission spectroscopy along with \emph{ab initio} calculations we report on a large family of intrinsic magnetic TIs in the homologous series of the van der Waals compounds (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_m$ with $m=0, ..., 6$. Magnetic, electronic and, consequently, topological properties of these materials depend strongly on the $m$ value and are thus highly tunable. The antiferromagnetic (AFM) coupling between the neighboring Mn layers strongly weakens on moving from MnBi2Te4 (m=0) to MnBi4Te7 (m=1), changes to ferromagnetic (FM) one in MnBi6Te10 (m=2) and disappears with further increase in 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-unknown topologically-nontrivial phase arises, in which below the corresponding critical 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 functional van der Waals heterostructures for topological quantum computation, as well as antiferromagnetic and 2D spintronics.