Anisotropic magnetic couplings and structure-driven canted to collinear transitions in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>IrO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math>by magnetically constrained noncollinear DFT

We study the canted magnetic state in ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$ using fully relativistic density functional theory (DFT) including an on-site Hubbard $U$ correction. A complete magnetic phase diagram with respect to the tetragonal distortion and the rotation of ${\mathrm{IrO}}_{6}$ octahedra is constructed, revealing the presence of two types of canted to collinear magnetic transitions: a spin-flop transition with increasing tetragonal distortion and a complete quenching of the basal weak ferromagnetic moment below a critical octahedral rotation. Moreover, we put forward a scheme to study the anisotropic magnetic couplings by mapping magnetically constrained noncollinear DFT onto a general spin Hamiltonian. This procedure allows for the simultaneous account and direct control of the lattice, spin, and orbital interactions within a fully ab initio scheme. We compute the isotropic, single site anisotropy and Dzyaloshinskii-Moriya (DM) coupling parameters, and clarify that the origin of the canted magnetic state in ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$ arises from the structural distortions and the competition between isotropic exchange and DM interactions.