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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.
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