Real-space and reciprocal-space topology in the Eu(Ga_1-xAl_x)₄ square net system

Magnetotransport measurements of the centrosymmetric square net Eu(Ga1-xAlx)4 compounds reveal evidence of both reciprocal- and real-space topology. For compositions 0.50 x 0.90, several intermediate field phases are found by magnetization measurements when H II c, where a maximum in the topological Hall effect is observed, pointing to the existence of topological (real-space topology) or noncoplanar spin textures. For 0.25 x 0.39, magnetization measurements reveal an intermediate field state, but no transition is visible in the Hall measurements. For x = 0.15, only one magnetic transition occurs below the Neel temperature TN, and no intermediate field spin reorientations are observed. The Hall effect varies smoothly before the spin-polarized (SP) state. However, in the SP state, Hall measurements reveal a large anomalous Hall effect (AHE) for all compositions, a consequence of reciprocal-space topology. Density functional theory calculations in the paramagnetic state indeed reveal a Dirac point that lies very near the Fermi energy, which is expected to split into Weyl nodes in the SP state, thereby explaining the observed AHE. These results suggest the Eu(Ga1-xAlx)4 family is a rare material platform where real- and reciprocal-space topology exist in a single material platform.