Data Underpinning: Incommensurate and multiple-q magnetic misfit order in the frustrated quantum spin ladder material antlerite

This contains the data underpinning our recent paper on antlerite, Cu₃⁢SO₄(OH)₄, published in Phys. Rev. B 106, 174431 (2022). The abstract of the article is reproduced here: In frustrated magnetic systems, the competition amongst interactions can introduce extremely high degeneracy and prevent the system from readily selecting a unique ground state. In such cases, the magnetic order is often exquisitely sensitive to the balance among the interactions, allowing tuning among novel magnetically ordered phases. In antlerite, Cu3⁢SO4⁢(OH)4, Cu2+ (𝑆=1/2) quantum spins populate three-leg zigzag ladders in a highly frustrated quasi-one-dimensional structural motif. We demonstrate that at zero applied field, in addition to its recently reported low-temperature phase of coupled ferromagnetic and antiferromagnetic spin chains, this mineral hosts an incommensurate helical+cycloidal state, an idle-spin state, and a multiple-𝑞 phase which is the magnetic analog of misfit crystal structures. The antiferromagnetic order on the central leg is reentrant. The high tunability of the magnetism in antlerite makes it a particularly promising platform for pursuing exotic magnetic order.