Title: Quasi-one-dimensional spin transport in antiferromagnetic $Z^3$ nodal net metals

Abstract: In three dimensions, the quasi-one-dimensional (Q1D) transport is commonly thought to only appear in the systems with Q1D chain structure. Here, based on first-principle calculations, we go beyond the common belief to show that the Q1D transport can also be realized in many 3D antiferromagnetic (AFM) metals with topological node structure but without Q1D chain structure, including the existing compounds $\beta$-Fe$_2$(PO$_4$)O and Co$_2$(PO$_4$)O, and LiTi$_2$O$_4$. All these materials have an AFM ground state and exhibit an ideal crossed $Z^3$ Weyl nodal line in each spin channel, formed by three straight and flat nodal lines traversing the whole Brillouin zone. These nodal lines eventually lead to an AFM $Z^3$ nodal net. Surprisingly, we find that the longitudinal conductivity $\sigma_{xx}$ in these topological nodal net metals is dozens of times larger than $\sigma_{yy}$ and $\sigma_{zz}$ in the up-spin channel, while $\sigma_{yy}$ dominates the transport in the down-spin channel. This means that each spin channel has a Q1D transport signature, and the principal moving direction for the two spin channels is orthogonal, leading to Q1D direction-dependent spin transport. This novel phenomenon can not be found in both conventional 3D bulk materials and Q1D chain materials, and may solely be induced by the $Z^3$ nodal net, as it gradually disappears when the Fermi level moves away from the nodal net. Our work not only enhances the comprehension of topological physics in antiferromagnets, but also opens a new direction for the exploration of topological spintronics.