Changes in elastic moduli as evidence for quadrupolar ordering in the rare-earth frustrated magnet Tb 2 Ti 2 O 7

Numerous materials feature unexplained phases with invisible or hidden order of electronic origin. A particularly mysterious case is that of ${\\mathrm{Tb}}_{2}{\\mathrm{Ti}}_{2}{\\mathrm{O}}_{7}$, which avoids magnetic order to the lowest temperatures, but nevertheless has an unexplained second-order phase transition near $T=0.5$ K. Our ultrasound measurements of ${\\mathrm{Tb}}_{2}{\\mathrm{Ti}}_{2}{\\mathrm{O}}_{7}$ provide direct evidence of a huge softening followed by strong hardening of the structural lattice below $T=0.5$ K. In the absence of magnetic order at this temperature, our results provide conclusive evidence for the proposed quadrupolar order and emphasize the importance of higher-order multipolar interactions in rare-earth frustrated magnets.