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

Abstract

We have investigated the origin of the magnetic dipole correlations $⟨{\sigma }_{\mathbf{Q}}^z{\sigma }_{-\mathbf{Q}}^z⟩$ characterized by the modulation wave vector $\mathbf{k}\sim (\frac{1}{2},\frac{1}{2},\frac{1}{2})$ 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-$\frac{1}{2}$ 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 $\mathbf{k}\sim (\frac{1}{2},\frac{1}{2},\frac{1}{2})$ are induced at low temperatures by a three-spin interaction of a form ${\sigma }_{\mathbf{r}}^{\pm }{\sigma }_{{\mathbf{r}}^{\prime }}^z{\sigma }_{{\mathbf{r}}^{\prime \prime }}^z$, 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 $⟨{\sigma }_{\mathbf{Q}}^{\alpha }{\sigma }_{-\mathbf{Q}}^{\beta }⟩$ ($\alpha ,\beta =x,y$) with $\mathbf{k}\sim 0$. 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.

arXiv comment28 pages, 25 figures

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