Electric activity at magnetic moment fragmentation in spin ice

Spin ice systems display a variety of very nontrivial properties, the most striking being the existence in them of magnetic monopoles. Such monopole states can also have nontrivial electric properties: there exist electric dipoles attached to each monopole. A novel situation is encountered in the moment fragmentation (MF) state, in which monopoles and antimonopoles are perfectly ordered, whereas spins themselves remain disordered. We show that such partial ordering strongly modifies the electric activity of such systems: the electric dipoles, which are usually random and dynamic, become paired in the MF state in (d, −d) pairs, thus strongly reducing their electric activity. The electric currents existing in systems with noncoplanar spins are also strongly influenced by MF. We also consider modifications in dipole and current patterns in magnetic textures (domain walls, local defects) and at excitations with nontrivial dynamics in a MF state, which show very rich behaviour and which could in principle allow to control them by electric field.

1. Creation and motion of defects in systems with moment fragmentation and their dipole character.
Point defects and the corresponding excitations in MF states can be created by some perturbations, e.g. by external defects etc., but they can also be thermally excited. In the later case they are created in pairs by reversing some spins. As discussed in the main text if one reverses the "special" spin say in Figs. 2(a) and 2(c), one creates a pair of (all-in) and (all-out) sites (triangles in kagome, tetrahedra in pyrochlore systems). The charges of these objects are, respectively, ±3Q and ±4Q. Note right away that by making this pair of defects we simultaneously "destroy" two electric dipoles which originally formed a (d, −d) dimer pair on these sites. Another possibility is to reverse a "normal", not a special spin. In kagome systems this will interchange the monopole and antimonopole, i.e. create a (μ, µ) pair at the wrong places, in wrong sublattices. In pyrochlore systems such reversal of the usual spin will create two tetrahedra of the (2-in)-(2-out) type. As discussed in the main text, in both cases the defects created by that have some unpaired dipoles, though for example in pyrochlores the (2-in)-(2-out) sites themselves have no dipoles, see Fig. 3(c), (d).
By making such spin reversal we either create "supermonopoles" ((3-in) states in kagome, (4-in) states in pyrochlores), with the increased charge, or we instead decrease the charges at sites, e.g. by making non-monopole (2-in)-(2-out) tetrahedra in pyrochlores. One can speak in this case not of the charge of the defect itself, but of the excess charge of an excitation: one gets e.g. the "double monopole" of (4-in) type with charge 4Q instead of the original state of a monopole with charge 2Q, i.e. the excess charge of such excitation is +2Q. This language is very convenient when considering the motion of such excitations.
Once created, these defects can start to move in a crystal by consecutively reversing some spins one after the other. But, in contrast to a similar motion of monopoles in regular spin ice [5], here these excitations move on the background of ordered monopoles. This makes their motion more complicated and more interesting.
If the motion of such defects would occur via simultaneous flipping of two spins [31], the defects can remain in their own sublattice. But the usual motion occurs via flipping of 1 one spin at a time. As shown in Fig. S1, in this case at each step the excitations moves from one sublattice to the other and by that always change their character (including their dipole structure). But they carry their excess charge with them. Thus, the excitation having the form of a tri-pole of Fig. S1(b) with charge 3Q and excess charge +2Q moves to the neighbouring site, Fig. S1(c), and at this site its charge is −Q + (+2Q) = +Q, i.e. instead of the original antimonopole at this site (the blue triangle in Fig. S1(a), (b)) it becomes a monopole with charge +Q (the magenta triangle in Fig. S1(c)). Simultaneously, as is seen form this figure, one creates the defect with three unpaired dipoles, as in Fig. 3(c) of the main text.
After that the defect has two options. Either it can move forward by reversing again the "special" spin, Fig. S1(d), or by reversing the remaining usual spin. In the first case one creates the tri-pole at a site of a monopole sublattice, i.e. one effectively moves the tripole from one site of this sublattice, red triangle in Fig. S1(b), to another site of the same sublattice, Fig. S1(d). This motion occurs via the intermediate "virtual" state when this excitation is on the "wrong" sublattice, Fig. S1(c), but in effect one can move in this way the excitation in its own sublattice. And one sees that by such motion we change the spins, and also electric dipoles, on the trajectory of the defect, but, similar to the case of the usual spin ice, all monopole states on such trajectory return to their original state, i.e. such string has no tension and there is no confinement. However if on the second step one would reverse not the special spin, as is done in going from (c) to (d), but the remaining usual spin of the magenta triangle in Fig. S1(c), moving the defect "to the right" in  whereas such kagome ice state itself has here no dipoles (every tetrahedron in this case has the (two-in)-(two-out) configuration without dipoles).) All in all, this is a very interesting situation deserving special attention (see also [31]), which, however, lies beyond the scope of the present paper.

Defects in pyrochlores with moment fragmentation.
Similarly to the case of kagome systems discussed in the main text, one can see that in pyrochlores the defects have in principle very similar properties, as to their dipole activity.
In Fig. S3(a) we show the structure of the first type of point defects: the "supermonopole" state of the type (4-in) or (4-out). One sees from this figure that by recommuting the remaining spins one can form such defect without any free unpaired dipoles. But this is not the case for the other types of defect. One of them, the (2-in)-(2-out) state, shown in For other types of defects or textures, e.g. for domain walls, very important is the situation with two neighbouring tetrahedra both of monopole or antimonopole type. As one sees from out-spin (i.e. our "special" spin) for the other tetrahedron. In this second tetrahedron the electric dipole would point to this special spin, and it would not form a (d, −d) pair -it would definitely be an unpaired dipole.
Using this picture it is easy to understand that e.g. the isolated point defect -monopole in place of an antimonopole, would lead to the formation of even four such unpaired dipoles, 3. Currents and orbital moments in a moment fragmentation state.
As is clear from Eqs.
(2), (3), nontrivial effects such as spontaneous orbital currents and corresponding orbital moments exist only for magnetic structures with noncoplanar spins. Consequently they are absent e.g. in kagome systems with spins in the xyplane. There exists however a very interesting group of kagome systems -"kagome-frompyrochlores" (or "tripod kagome") obtained by nonmagnetic dilution of spin-ice pyrochlores, According to Eqs.
(2), (3) the scalar spin chiralities and corresponding currents and orbital moments are of one sign (e.g. currents clockwise, orbital moments up) for monopoles, and are opposite for antimonopoles. In regular kagome spin ice ("kagome-from-pyrochlores") they are random, but in the moment fragmentation state with full (µ,μ) ordering these currents and orbital moments are also fully ordered, Fig. S6. In that sense currents and orbital moments in MF kagome systems are different from electric dipoles in those: dipoles form pairs, but are random and dynamic, but currents and orbital moments are fully ordered together with monopoles themselves.
The situation with spontaneous currents and orbital moments in pyrochlores, with and without MF, is a bit more tricky than that in kagome systems, but conceptually similar.
One can show that for spin ice tetrahedra there would exist nonzero currents even in a pure spin-ice case (2-in)-(2-out) without monopoles. Indeed, using the expressions (2) (2)). Note that in both cases orbital magnetic moments are parallel (or antiparallel) to the net spin moment of corresponding tetrahedra.
(the currents on the other three edges of such tetrahedra, each belonging to two triangles, would cancel). And the corresponding orbital moment would point again in the direction of the total spin of corresponding tetrahedra.
The situation with currents and orbital moments in pyrochlores with moment fragmentation is different from that in a kagome ice. When we look at a pair of neighbouring tetrahedra with opposite monopole charges, Fig. S8, and with the "special" spin at the common site between those, we indeed see that the electric dipoles of these tetrahedra are (L, L) pairs move and would change orientation. This is in contrast to the case of kagome systems (kagome-from-pyrochlore), in which in the fragmentation state currents and orbital moments are long-range ordered together with monopoles themselves, whereas the (d, −d) pairs fluctuate together with spins.