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

## TL;DR

This study uses density-functional calculations to explain the unique up-up-down-down magnetic order in Cu$_2$GeO$_4$, highlighting the role of nearly canceled exchange interactions and structural buckling in its magnetic behavior.

## Contribution

It reveals that weak symmetric anisotropy and structural buckling are key to the magnetic order, differing from previous assumptions about exchange interactions.

## Key findings

- Negligibly small nearest-neighbor coupling J_1 due to exchange compensation.
- Weak anisotropy induces the UUDD magnetic order.
- Buckling of copper chains suppresses J_1, distinguishing Cu$_2$GeO$_4$ from similar magnets.

## Abstract

We use density-functional band-structure calculations to explore the origin of the up-up-down-down (UUDD) magnetic order in Cu$_2$GeO$_4$ with the frustrated $J_1-J_2$ spin chains coupled into layers within the spinel-like crystal structure. In contrast to earlier studies, we find that the nearest-neighbor coupling $J_1$ may be negligibly small, owing to a nearly perfect compensation of the ferromagnetic direct exchange and antiferromagnetic superexchange. Under this condition, weak symmetric anisotropy of the exchange couplings gives rise to the UUDD order observed experimentally and also elucidates the non-trivial ordering pattern between the layers, whereas a small Dzyaloshinsky-Moriya interaction causes a spin canting that may generate local electric polarization. We argue that the buckling of the copper chains plays a crucial role in the suppression of $J_1$ in Cu$_2$GeO$_4$ and sets this compound apart from other $J_1-J_2$ chain magnets.