Giant lattice softening at a Lifshitz transition in Sr$_{2}$RuO$_{4}$

The interplay of electronic and structural degrees of freedom in solids is a topic of intense research. Experience and intuition suggest that structural changes drive conduction electron behavior, because the large number of valence electrons dominate the structural properties. As part of a seminal paper written over sixty years ago, Lifshitz discussed an alternative possibility: lattice softening driven by conduction electrons at topological Fermi surface transitions. The effect he predicted, however, was small, and has not been convincingly observed. Using measurements of the stress-strain relationship in the ultra-clean metal Sr$_{2}$RuO$_{4}$, we reveal a huge softening of the Young's modulus at a Lifshitz transition of a two-dimensional Fermi surface, and show that it is indeed entirely driven by the conduction electrons of the relevant energy band.