Spin and quadrupole correlations by three-spin interaction in the frustrated pyrochlore magnetTb2+xTi2−xO7+y

We have investigated the origin of the magnetic dipole correlations $\ensuremath{\langle}{\ensuremath{\sigma}}_{\mathbit{Q}}^{z}{\ensuremath{\sigma}}_{\ensuremath{-}\mathbit{Q}}^{z}\ensuremath{\rangle}$ characterized by the modulation wave vector $\mathbit{k}\ensuremath{\sim}(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ observed in the frustrated pyrochlore magnet ${\mathrm{Tb}}_{2+x}{\mathrm{Ti}}_{2\ensuremath{-}x}{\mathrm{O}}_{7+y}$. This magnetic short-range order cannot be accounted for by adding further-neighbor exchange interactions to the nearest-neighbor pseudospin-$\frac{1}{2}$ Hamiltonian for quantum pyrochlore magnets. Using classical Monte Carlo simulation and quantum simulation based on thermally pure quantum (TPQ) states we have shown that the spin correlations with $\mathbit{k}\ensuremath{\sim}(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ are induced at low temperatures by a three-spin interaction of a form ${\ensuremath{\sigma}}_{\mathbit{r}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\sigma}}_{{\mathbit{r}}^{\ensuremath{'}}}^{z}{\ensuremath{\sigma}}_{{\mathbit{r}}^{\ensuremath{'}\ensuremath{'}}}^{z}$, which is a correction to the Hamiltonian due to the low crystal-field excitation. Simulations using TPQ states have shown that the spin correlations coexist with electric quadrupole correlations $\ensuremath{\langle}{\ensuremath{\sigma}}_{\mathbit{Q}}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{\ensuremath{-}\mathbit{Q}}^{\ensuremath{\beta}}\ensuremath{\rangle}$ ($\ensuremath{\alpha},\ensuremath{\beta}=x,y$) with $\mathbit{k}\ensuremath{\sim}\mathbit{0}$. These results suggest that the putative quantum spin liquid state of ${\mathrm{Tb}}_{2+x}{\mathrm{Ti}}_{2\ensuremath{-}x}{\mathrm{O}}_{7+y}$ is located close to phase boundaries of the spin-ice, quadrupole-ordered, and magnetic-ordered states in the classical approximation, and that the three-spin interaction brings about a quantum disordered ground state with both spin and quadrupole correlations.