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

Abstract: In three dimensions, quasi-one-dimensional (Q1D) transport has traditionally been associated with systems featuring a 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 three-dimensional (3D) altermagnetic (AM) metals with a topological nodal net in momentum space but lacking Q1D chain structure in real space, including the existing compounds $\beta$-Fe$_2$(PO$_4$)O, Co$_2$(PO$_4$)O, and LiTi$_2$O$_4$. These materials exhibit an AM ground state and feature an ideal crossed $Z^3$ Weyl nodal line in each spin channel, formed by three straight and flat nodal lines traversing the entire Brillouin zone. These nodal lines eventually lead to an AM $Z^3$ nodal net. Surprisingly, 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 transport in the down-spin channel. This suggests a distinctive Q1D transport signature in each spin channel, with orthogonal principal moving directions for the two spin channels, resulting in Q1D direction-dependent spin transport. This novel phenomenon cannot be found in both conventional 3D bulk materials and Q1D chain materials. In particular, it gradually disappears as the Fermi level moves away from the nodal net, further confirming its topological origin. Our work not only enhances the comprehension of topological physics in altermagnets but also opens a new direction for the exploration of topological spintronics.