Continuum excitation and pseudospin wave in quantum spin-liquid and quadrupole ordered states of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$

Abstract: The ground states of the frustrated pyrochlore oxide Tb${2+x}$Ti${2-x}$O${7+y}$ have been studied by inelastic neutron scattering experiments. Three single-crystal samples are investigated; one shows no phase transition ($x=-0.007<x{\text{c}}\sim -0.0025$), being a putative quantum spin-liquid (QSL), and the other two ($x=0.000, 0.003$) show electric quadrupole ordering (QO) below $T_{\text{c}} \sim 0.5$ K. The QSL sample shows continuum excitation spectra with an energy scale 0.1 meV as well as energy-resolution limited (nominally) elastic scattering. As $x$ is increased, pseudospin wave of the QO state emerges from this continuum excitation, which agrees with that of powder samples and consequently verifies good $x$ control for the present single crystal samples.