Incommensurate Spin Density Wave state in metamagnetic Fe$_3$Ga$_4$

Fe$_3$Ga$_4$ displays a rich competition between magnetic states without structural transitions: a ferromagnetic(FM) ground state transitions to an antiferromagnetic(AFM) intermediate state above 68 K followed by a reemergence of the FM state above room temprature(T). The reentrance of the FM state hints of a coupling of the magnetic degrees of freedom to other modes. To explore the nature of the magnetic states, we have performed extensive single crystal neutron diffraction measurements over a wide range of T and pressure. These measurements revealed two very different magnetic states with the low T FM state having magnetic moments along the c-axis while we discovered that the AFM state is in an incommensurate spin density wave(SDW) order with moments mostly along the a-axis. However, there is still considerable non-collinear and non-coplanar contributions along the b- and c-axial directions. This non-coplanar moment is likely to be the origin of the very large anomalous Hall effect(HE) including a substantial topological HE that we discovered in Fe$_3$Ga$_4$. Study of the effect of hydrostatic pressure indicates a reduction of the T$_c$ and a destabilization of the SDW phase.