Quantum Spin Ordering in Tb$_2$Ti$_2$O$_7$

Group theoretical methods are used to analyse quantum spin states in the geometrically frustrated pyrochlore Tb$_2$Ti$_2$O$_7$. The magnetic rare earth spins, with $J=6$, have a ground state doublet due to local crystal electric field. With 4 Tb ions per unit cell, there is therefore a 16-fold degeneracy of the ground state. Symmetry considerations predict a lifting of the degeneracy into a singlet, three doublets and three triplets. One of the triplet configurations is found to be responsible for the $[0,0,2]$ peak in diffuse neutron scattering.