Towards a microscopic theory of toroidal moments in bulk periodic crystals

We present a theoretical analysis of magnetic toroidal moments in periodic systems, in the limit in which the toroidal moments are caused by a time and space reversal symmetry breaking arrangement of localized magnetic dipole moments. We summarize the basic definitions for finite systems and address the question of how to generalize these definitions to the bulk periodic case. We define the “toroidization” as the toroidal moment per unit cell volume, and we show that periodic boundary conditions lead to a multivaluedness of the toroidization, which suggests that only differences in toroidization are meaningful observable quantities. Our analysis bears strong analogy to the “modern theory of electric polarization” in bulk periodic systems, but we also point out some important differences between the two cases. We then discuss the instructive example of a one-dimensional chain of magnetic moments, and we show how to properly calculate changes of the toroidization for this system. Finally, we evaluate and discuss the toroidization (in the local dipole limit) of four important example materials: $\mathrm{Ba}\mathrm{Ni}{\mathrm{F}}_4$, $\mathrm{Li}\mathrm{Co}\mathrm{P}{\mathrm{O}}_4$, $\mathrm{Ga}\mathrm{Fe}{\mathrm{O}}_3$, and $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$.

Figure 1

Simple arrangements of magnetic moments which can lead to toroidal moments. (a) and (b) have equal and opposite toroidal moments. The antiferromagnetic arrangement in (c) has a toroidal moment, whereas that in (d) does not.

Figure 2

Decomposition of a ferrimagnetic arrangement of two localized moments (left) into its fully compensated component (middle) and its uncompensated “ferromagnetic” component (right). The ferromagnetic component at each site is the total magnetic moment divided by the total number of moments ${\stackrel{\mathrm{̃}}{\mathit{m}}}_{\alpha }=\frac{1}{2}({\mathit{m}}_1+{\mathit{m}}_2)$, and the compensated part is the difference between the magnitude of the original local moment and the uncompensated contribution.

Figure 3

(Color online) Calculation of the toroidization for two different one-dimensional antiferromagnetic periodic arrangements of magnetic moments. Our choice of unit cell is indicated by the shaded area in each case. (a) shows a nontoroidal state, which is space-inversion symmetric with respect to each moment site. (b) is a toroidal state.

Figure 4

(Color online) Allowed values of the toroidization for the antiferromagnetic chain of Fig. 3 as a function of displacement $d$ from the nontoroidal case $(d=0)$. The cartoons at the bottom indicate the corresponding positions of the magnetic moments within the unit cell.

Figure 5

(Color online) Effect on the magnetic moment configuration of Fig. 3b (center) of a reversal of all magnetic moments (right) and of a reversal of the noncentrosymmetric distortion $d$ (left). Note that the right and left final states are identical, with the moments on the left translated by half a unit cell compared to those on the right.