Lattice distortion in the spin liquid phase of Tb$_2$Ti$_2$O$_7$

The Tb ions in the pyrochlore antiferromagnet Tb$_2$Ti$_2$O$_7$ form a corner-shared tetrahedral network. It has been shown that the quantum anti-ferromagnetic ground state of a single tetrahedron accounts for short wavelength correlations observed in inelastic neutron scattering experiments. A bulk $k=0$ lattice distortion, specifically an $A_{2u}$ displacement of the Tb ions, can account for the isolation of single tetrahedra in the tetrahedral network. Such a distortion results in a space group reduction $Fd\bar{3}m \rightarrow F\bar{4}3m$ which is consistent with experimental observations.