Dynamics of electrically polarized magnetic monopoles in spin ice

We propose a simple mechanism through which elementary excitations in spin ice, namely, magnetic monopoles, lead to spontaneous macroscopic electric polarization by breaking the space inversion symmetry in the spin configuration. We show that while the magnetic monopole charge is conserved in hopping, the corresponding electric dipole moment cannot be a constant of motion. The possibilities as well as the limitations involved in electrical manipulation of the monopole dynamics are also discussed.

Figure 1

The Dzyaloshinskii-Moriya (DM) interaction in ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ exists due to the noncollinearity of spins contributed by the rare earth atom Dy, which leads to the nonzero local spin current $j_s\propto \overrightarrow{s_i}\times \overrightarrow{s_j}$ and the electric dipole moment.

Figure 2

The time relaxation of the linear and quadratic magnetoelectric coupling coefficient as the system approaches the equilibrium configuration is shown in the low-temperature regime (arbitrary scale); ${\tau }_0$ is the relaxation time and $C_1$ is a constant dependent on temperature.

Figure 3

In the ordered two-in, two-out spin configuration, the local spontaneous electric polarization in each tetrahedron is 0. Once a monopole-antimonopole pair is created by flipping any one of the spins [thick (red) arrows], spontaneous electric polarization emerges. The electric field could be applied in the (111) direction. The four vertices in the tetrahedron constitute six nearest-neigbor pairs, each contributing an electric dipole moment shown at the middle of the bond by the thin black arrows and directed either away or towards the center of the tetrahedron. The component of electric polarization along the (111) direction can be mapped onto a 2D “Kagome” vector diagram [a combination of the two overlapping dashed (pink) triangles] as shown separately for the two cases. The perpendicular components cancel each other due to symmetry. The circular color coding attached to each polarization vector signifies the subgroup of spins it originates from. For example, $S_3$ spin is associated with three color balls, i.e., yellow, sky blue, and red. The flipping of the $S_3$ spin would imply that the electric dipole moments associated with these colors have to be flipped.

Figure 4

Monopole dynamics in a network of four corner-sharing tetrahedra: We show that while the monopole charge is conserved as it moves from one tetrahedron to another, the corresponding electric dipole moment can undergo sign reversal in the absence of a strong electric field $E$. Let us start from the spin ice state, which we denote as the zeroth step. The net electric dipole moment in each tetrahedra is 0 as shown earlier. In the first step a monopole-antimonopole pair is created in tetrahedra B and C by flipping the spin $S_6$. The dipole moments connected to $S_6$ in tetrahedra B and C would flip accordingly as shown in the adjacent Kagome vector diagram. The resultant electric dipole moment is shown by the large (gray) arrowhead. The second step involves the flipping of $S_1$. The one-in, three-out configuration now shifts from B to A, while the dipole moment attached to the antimonopole is reversed.

Figure 5

The scenario when the movement of the monopole accompanies a change in magnitude of the dipole moment but no sign reversal. We refer again to the initial spin-ice state in Fig. 4. In the first step a one-in, three-out configuration is created in tetrahedra C by flipping the spin $S_8$. The dipole moments connected to $S_8$ in tetrahedra C would flip accordingly as shown in the adjacent Kagome vector diagram, with the resultant electric dipole moment represented by the large (gray) arrowhead. The second step involves the flipping of $S_6$. The one-in, three-out configuration now shifts from C to B, while the dipole moment attached to the antimonopole has undergone a change in magnitude but no change in sign.

Figure 6

Top: Possible polarization states corresponding to a monopole/antimonopole. $\pm 2P_0$ states correspond to a larger number (i.e., 15) of possible spin configurations, hence are most probable. Bottom: Monopole path consisting of unit building blocks. Two adjacent blocks correspond to possible initial and final polarization states as the magnetic charge moves to the second block from the first one reversibly. All movement is governed by the selection rule $\pm 4P_0$. In the absence of an external magnetic field all paths (a–c) are permissible. In the presence of $H$ along the [111] or [001] direction, (a) and (b) are the only possible paths.

Figure 7

The monopole-antimonopole “pinned state”; flipping of the shared spin would lead to a four-in, four-out configuration. Such a state is always associated with head-to-head electric dipole moments, each with magnitude $6P_0$.