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Magnetic kagome metals, in which topologically non-trivial band structuresand electronic correlation are intertwined, have recently emerged as anexciting platform to explore exotic correlated topological phases, that areusually not found in weakly interacting materials described within thesemi-classical picture of electrons. Here, via a comprehensive single-crystalneutron diffraction and first-principles density functional theory study of thearchetypical topological kagome metal Mn$_3$Sn, which is also a magnetic Weylfermion material and a promising chiral magnet for antiferromagneticspintronics, we report the realisation of an emergent spin-density wave (SDW)order, a hallmark correlated many-body phenomenon, that is engineered by theFermi surface nesting of topological flat bands. We further reveal that thephase transition, from the well-known high-temperature coplanar andnon-collinear k = 0 inverse triangular antiferromagnetic order to a double-$k$non-coplanar modulated incommensurate magnetic structure below $T_1$ = 280 K,is primarily driven by the SDW instability. The double-$k$ nature of thiscomplex low-temperature magnetic order, which can be regarded as an intriguingsuperposition of a longitudinal SDW with a modulation wavevector k$_L$ and atransverse incommensurate helical magnetic order with a modulation wavevectork$_T$, is unambiguously confirmed by our observation of the inter-modulationhigh-order harmonics of the type of 2k$_L$+k$_T$. This discovery not onlysolves a long-standing puzzle concerning the nature of the phase transition at$T_1$, but also provides an extraordinary example on the intrinsic engineeringof correlated many-body phenomena in topological matter. The identifiedmulti-$k$ magnetic state can be further exploited for the engineering of thenew modes of magnetization and chirality switching in antiferromagneticspintronics.

Flat band-engineered spin-density wave and the emergent multi-$k$ magnetic state in the topological kagome metal Mn$_{3}$Sn