Spontaneous Strains in the Ground State of the Frustrated Pyrochlore Tb$_2$Ti$_2$O$_7$

The geometrically frustrated pyrochlore antiferromagnet Tb$_2$Ti$_2$O$_7$ has been the focus of much recent attention due to the spin liquid physics it displays at low temperatures. We have carried out high resolution x-ray scattering measurements of the principal Bragg peaks of Tb$_2$Ti$_2$O$_7$ which show substantial longitudinal broadening below approximately 20 K. Measurements on the non-magnetic pyrochlore Y$_2$Ti$_2$O$_7$ show no such broadening, suggesting this effect is driven by strong magnetoelastic coupling, as had been concluded from earlier anomolous low temperature Young's modulus measurements [Mamsurova et.al,JETP Letters,1986]. We compare our x-ray scattering results to similar measurements in TbVO$_4$, which undergoes a cooperative Jahn-Teller phase transition near 34 K, and conclude that the longitundinal broadening is due to cubic-tetragonal fluctuations near a zero temperature phase transition, which acts to relieve the magnetic frustration.