the resultant relativistic contribution to electric polarization in M1-O-M2
clusters. At present, it is a difficult and, probably, hopeless task to
propose a more reliable and so physically clear estimation. Taking into
account the typical value of $\Delta g/g\sim 0.1$ we can estimate the maximal
value of $|\Pi_{ij}|$ as $10^{-3}|e|\AA$($\sim 10^{2}\mu C/m^{2}$) that points
to the exchange-relativistic mechanism to be a weak contributor to a giant
multiferroicity with ferroelectric polarization of the order of $10^{3}\mu
C/m^{2}$ as in TbMnO3,[128] though it may be a noticeable contributor in,
e.g., Ni3V2O8 [129].

Concluding the section, let us pay attention to the magnetoelectric effect in
orthoferrites RFeO3. Their $Pbnm$ structure is particularly interesting in
terms of non-collinear magnetism, however, the perovskite $Pbnm$ phase is not
polar and thus none of the compounds crystallizing within this space group
will exhibit a spontaneous electric polarization. It can be shown, from pure
symmetry arguments, that neither the spontaneous polarization nor the linear
or quadratic magnetoelectric effect is possible if the magnetic order involves
the Fe-site spins only. At the same time, spin-canted structures of rare-earth
orthoferrites can produce ferroelectricity, if there exists some R-site
orderings enabling, by themselves, the linear magnetoelectric effect. For
instance, GdFeO3 and DyFeO3 represent two important examples of such low-
temperature multiferroics [130]. Interestingly, the substitution of some of
the dysprosium ions in DyFeO3 for bismuth ions leads to the appearance of a
strong quadratic magnetoelectric effect, the nature of which is associated
with the anomalously high polarizability of Bi3+ ions leading to the formation
of extended clouds of local electric polarization near Bi3+ ions [131].

##  11 Conclusion

The DM coupling being simple in form, can result in very different magnetic
phenomena: weak ferro- antiferro- and weak transversal ferrimagnetism in a
wide number of magnetic 3$d$ oxides, multiferroism, helimagnetism in CsCuCl3 ,
helical and skyrmion structures in MnSi-type crystals etc. In the paper we
performed an overview of the microscopic theory of the DM coupling and other
related exchange-relativistic effects such as exchange anisotropy, electron-
nuclear antisymmetric supertransferred hyperfine interactions, antisymmetric
magnetogyrotropic effects, and antisymmetric magnetoelectric coupling in
orthoferrites RFeO3 and several typical weak ferromagnets. Most attention in
the paper focused on the comprehensive generalization of the Moriya’s theory
and derivation of the Dzyaloshinskii vector, its value, orientation, and sense
under different types of the (super)exchange interaction and crystal field.
Microscopically derived expression for the dependence of the Dzyaloshinskii
vector on the superexchange geometry made it possible to find all the overt
and hidden canting angles, that is weak ferro- and antiferromagnetic modes in
orthoferrites RFeO3. The theoretical predictions have been successfully
confirmed by various experimental techniques. Being based on the theoretical
predictions regarding the sign of the Dzyaloshinskii vector we have predicted
and studied in detail a novel magnetic phenomenon, weak ferrimagnetism in
mixed weak ferromagnets such as RFe1-xCrxO3 with competing signs of the
Dzyaloshinskii vectors. In contrast to the end compositions, weak ferrimagnets
possess a complex of unusual magnetic properties, including concentration and
temperature compensation points, new spin-reorientation transitions, including
the newly discovered transition to the angular phase with the spatial
orientation of the antiferromagnetism vector and emergence of the
$b$-component of the magnetic moment. These new materials have broad prospects
for practical application, including thermally assisted magnetic random access
memories, thermomagnetic switches and other multifunctional devices.

The ligand NMR measurements in weak ferromagnets are shown to be an effective
tool to inspect the effects of DM coupling in an external magnetic field and
determine the mutual orientation of the vectors of ferro- and
antiferromagnetism, and therefore establish the sign of the Dzyaloshinskii
vector. We considered a number of exchange-relativistic interactions, which in
one way or another have a common nature with the spin-bilinear DM coupling. As
a result of a detailed analysis of the tensor structure of the exchange-
relativistic two-ion anisotropy, the emergence of new nondipole contributions
with an unconventional temperature dependence has been established. It is
shown that an analysis of the field dependencies of the 57Fe NMR frequencies
in orthoferrites indicates the existence of a noticeable antisymmetric
supertransferred hyperfine interaction as an electron-nuclear analogue of the
Dzyaloshinskii interaction. A new unusual analogue of the DM coupling, the
antisymmetric spin-other-orbit interaction can make a decisive contribution to
the circular magneto-optics of weak ferromagnets. We considered the exchange-
relativistic antisymmetric contribution to the spin-dependent electric
polarization for 3$d$ magnets and established the dependence of the dipole
moment of superexchange-coupled ions on the bonding geometry.

The DM coupling as well as related exchange-relativistic effects are nowadays
believed to play furthermore a prominent role in many strongly correlated
materials.

I am grateful to I. E. Dzyaloshinskii for supporting my work and stimulating
discussions. I consider it my duty to note that most of the work was done in
close collaboration with A. M. Kadomtseva and her laboratory staff. I thank E.
V. Sinitsyn and I. G. Bostrem for very fruitful multi-year collaboration, S.
V. Maleev, A. K. Zvezdin, B. Z. Malkin, M. V. Eremin, A. A. Mukhin, B. S.
Tsukerblatt, S.-L. Drechsler, R. E. Walstedt, and V. E. Dmitrienko for
stimulating and encouraging discussions.

This research was funded by Act 211 Government of the Russian Federation,
agreement No 02.A03.21.0006 and by the Ministry of Education and Science,
project No FEUZ-2020-0054.

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