Double-layered monopolar order in the Tb${}_2$Ti${}_2$O${}_7$ spin liquid

Ho${}_2$Ti${}_2$O${}_7$ and Dy${}_2$Ti${}_2$O${}_7$ spin ices exhibit elementary excitations akin to magnetic monopoles. Here, we focus on Tb${}_2$Ti${}_2$O${}_7$ spin liquid, where correlated magnetic moments keep fluctuating down to very low temperatures. Using a monopole picture, we have reanalyzed the field-induced magnetic structure previously determined by neutron diffraction in Tb${}_2$Ti${}_2$O${}_7$. We show that under a high field applied along a [110] direction, Tb${}_2$Ti${}_2$O${}_7$ orders as a three-dimensional arrangement of monopole and antimonopole double layers. In contrast, Ho${}_2$Ti${}_2$O${}_7$ spin ice in the same conditions behaves as a monopole-free state. By symmetry analysis, we derived the distortions compatible with the observed magnetic structure of Tb${}_2$Ti${}_2$O${}_7$, which can be related to the appearance of the double-layered monopolar order.

Figure 1

Magnetic monopoles in spin ice. Left panel: Spins of two adjacent tetrahedra obeying the “2-in, 2-out” ice rule. Right panel: Monopole-antimonopole pair induced by flipping of the spin connecting the tetrahedra.

Figure 2

Neutron scattering of Tb${}_2$Ti${}_2$O${}_7$ in 0 T (left panel), Tb${}_2$Ti${}_2$O${}_7$ in 5 T (middle panel), and Ho${}_2$Ti${}_2$O${}_7$ in 2 T (right panel) at 1.6 K with $\mathit{H}\parallel \left[110\right]$. Intensity is given in a logarithmic scale. Selected magnetic reflections are highlighted. In zero field, only the peaks of the crystal structure are seen.

Figure 3

A typical fit for Tb${}_2$Ti${}_2$O${}_7$ at 1.6 K and 7 T. The calculated integrated intensities ($F_{\mathrm{calc}}^2$) are plotted against the experimental ones ($F_{\mathrm{obs}}^2$). Left panel: Both magnetic $\mathit{k}=\mathbf{0}$ and nuclear intensities. Right panel: Pure magnetic intensities of the $\mathit{k}=(0,0,1)$ structure.

Figure 4

Magnetic structures of Tb${}_2$Ti${}_2$O${}_7$ (left panel) and Ho${}_2$Ti${}_2$O${}_7$ (right panel) in a [110] field. Blue and red balls correspond to opposite magnetic charges. The orientations of the $\beta$ moments inside a chain are AFM in Tb${}_2$Ti${}_2$O${}_7$ and FM-like in Ho${}_2$Ti${}_2$O${}_7$.

Figure 5

Left panel: Double-layered monopolar structure of Tb${}_2$Ti${}_2$O${}_7$ with vacuum pair excitations. Right panel: Magnetically vacuum state of Ho${}_2$Ti${}_2$O${}_7$ with monopole pair excitation.

Figure 6

Left panel: The structural distortion in Tb${}_2$Ti${}_2$O${}_7$ compatible with the observed high-field magnetic structure. The directions of displacements of the axial oxygens are shown by white arrows. Right panel: The undistorted structure of Ho${}_2$Ti${}_2$O${}_7$.