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

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 this understanding to show that the Q1D transport can also be realized in certain 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 around Fermi level, 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, the electronic 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, and the principal moving directions for the two spin channels are orthogonal, 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, the Q1D spin transport gradually disappears as the Fermi energy 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.