Strain control of a bandwidth-driven spin reorientation in Ca${}_3$Ru${}_2$O${}_7$

The layered-ruthenate family of materials possess an intricate interplay of structural, electronic and magnetic degrees of freedom that yields a plethora of delicately balanced ground states. This is exemplified by Ca${}_3$Ru${}_2$O${}_7$, which hosts a coupled transition in which the lattice parameters jump, the Fermi surface partially gaps and the spins undergo a ${90}^{\circ }$ in-plane reorientation. Here, we show how the transition is driven by a lattice strain that tunes the electronic bandwidth. We apply uniaxial stress to single crystals of Ca${}_3$Ru${}_2$O${}_7$, using neutron and resonant x-ray scattering to simultaneously probe the structural and magnetic responses. These measurements demonstrate that the transition can be driven by externally induced strain, stimulating the development of a theoretical model in which an internal strain is generated self-consistently to lower the electronic energy. We understand the strain to act by modifying tilts and rotations of the RuO${}_6$ octahedra, which directly influences the nearest-neighbour hopping. Our results offer a blueprint for uncovering the driving force behind coupled phase transitions, as well as a route to controlling them.