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We use density-functional band-structure calculations to explore the originof the up-up-down-down (UUDD) magnetic order in Cu$_2$GeO$_4$ with thefrustrated $J_1-J_2$ spin chains coupled into layers within the spinel-likecrystal structure. In contrast to earlier studies, we find that thenearest-neighbor coupling $J_1$ may be negligibly small, owing to a nearlyperfect compensation of the ferromagnetic direct exchange and antiferromagneticsuperexchange. Under this condition, weak symmetric anisotropy of the exchangecouplings gives rise to the UUDD order observed experimentally and alsoelucidates the non-trivial ordering pattern between the layers, whereas a smallDzyaloshinsky-Moriya interaction causes a spin canting that may generate localelectric polarization. We argue that the buckling of the copper chains plays acrucial role in the suppression of $J_1$ in Cu$_2$GeO$_4$ and sets thiscompound apart from other $J_1-J_2$ chain magnets.

The origin of up-up-down-down magnetic order in Cu$_2$GeO$_4$