Pressure-tuned spin chains in brochantite, Cu$_4$SO$_4$(OH)$_6$

Using high-pressure single-crystal x-ray diffraction combined with thermodynamic measurements and density-functional calculations, we uncover the microscopic magnetic model of the mineral brochantite, Cu$_4$SO$_4$(OH)$_6$, and its evolution upon compression. The formation of antiferromagnetic spin chains with the effective intrachain coupling of $J\simeq 100$\,K is attributed to the occurrence of longer Cu--Cu distances and larger Cu--O--Cu bond angles between the structural chains within the layers of the brochantite structure. These zigzag spin chains are additionally stabilized by ferromagnetic couplings $J_2$ between second neighbors and moderately frustrated by several antiferromagnetic couplings that manifest themselves in the reduced N\'eel temperature of the material. Pressure tuning of the brochantite structure keeps its monoclinic symmetry unchanged and leads to the growth of antiferromagnetic $J$ with the rate of 3.2\,K/GPa, although this trend is primarily caused by the enhanced ferromagnetic couplings $J_2$. Our results show that the nature of magnetic couplings in brochantite and in other layered Cu$^{2+}$ minerals is controlled by the size of the lattice translation along their structural chains and by the extent of the layer buckling.