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Spin and quadrupole correlations by three-spin interaction in the frustrated pyrochlore magnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$

We have investigated the origin of the magnetic dipole correlations $\langle σ_{\boldsymbol{Q}}^z σ_{\boldsymbol{-Q}}^z \rangle$ characterized by the modulation wave vector $\boldsymbol{k} \sim (\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ observed in the frustrated pyrochlore magnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$. This magnetic short-range order cannot be accounted for by adding further-neighbor exchange interactions to the nearest-neighbor pseudospin-$\tfrac{1}{2}$ 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 $\boldsymbol{k} \sim (\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ are induced at low temperatures by a three-spin interaction of a form $σ_{\boldsymbol{r}}^{\pm} σ_{\boldsymbol{r}^{\prime}}^z σ_{\boldsymbol{r}^{\prime \prime}}^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 $\langle σ_{\boldsymbol{Q}}^α σ_{\boldsymbol{-Q}}^β \rangle$ ($α, β= x, y$) with $\boldsymbol{k} \sim \boldsymbol{0}$. These results suggest that the putative quantum spin liquid state of Tb$_{2+x}$Ti$_{2-x}$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.