Spin correlations of quantum spin liquid and quadrupole-ordered states of Tb_2+x Ti_2-x O_7+y

Spin correlations of the frustrated pyrochlore oxide Tb_2+x Ti_2-x O_7+y have been investigated by using inelastic neutron scattering on single-crystalline samples (x = -0.007, 0.000, and 0.003), which have the putative quantum-spin-liquid (QSL) or electric-quadrupolar ground states. Spin correlations, which are notably observed in nominally elastic scattering, show short-range correlations around L points $\left[q=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right],$ tiny antiferromagnetic Bragg scattering at L and Γ points, and pinch-point-type structures around Γ points. The short-range spin correlations were analyzed using a random-phase approximation (RPA) assuming the paramagnetic state and two-spin interactions among Ising spins. These analyses have shown that the RPA scattering intensity well reproduces the experimental data using temperature- and x-dependent coupling constants of up to tenth-neighbor site pairs. This suggests that no symmetry breaking occurs in the QSL sample and that a quantum treatment beyond the semiclassical RPA approach is required. Implications of the experimental data and the RPA analyses are discussed.