Dzyaloshinskii-Moriya coupling in 3d insulators

We present an overview of the microscopic theory of the Dzyaloshinskii-Moriya (DM) coupling in strongly correlated 3d compounds. Most attention in the paper centers around the derivation of the Dzyaloshinskii vector, its value, orientation, and sense (sign) under different types of the (super)exchange interaction and crystal field. We consider both the Moriya mechanism of the antisymmetric interaction and novel contributions, in particular, that of spin-orbital coupling on the intermediate ligand ions. We have predicted a novel magnetic phenomenon, {\it weak ferrimagnetism} in mixed weak ferromagnets with competing signs of the Dzyaloshinskii vectors. We revisit a problem of the DM coupling for a single bond in cuprates specifying the local spin-orbital contributions to Dzyaloshinskii vector focusing on the oxygen term. We predict a novel puzzling effect of the on-site staggered spin polarization to be a result of the on-site spin-orbital coupling and the the cation-ligand spin density transfer. The intermediate ligand NMR measurements are shown to be an effective tool to inspect the effects of the DM coupling in an external magnetic field. We predict the effect of a $strong$ oxygen weak antiferromagnetism in edge-shared CuO$_2$ chains due to uncompensated oxygen Dzyaloshinskii vectors. We revisit the effects of symmetric spin anisotropy directly induced by the DM coupling. A critical analysis will be given of different approaches to exchange-relativistic coupling based on the cluster and the DFT based calculations. Theoretical results are applied to different classes of 3d compounds from conventional weak ferromagnets ($\alpha$-Fe$_2$O$_3$, FeBO$_3$, FeF$_3$, RFeO$_3$, RCrO$_3$,.. ) to unconventional systems such as weak ferrimagnets (e.g., RFe$_{1-x}$Cr$_x$O$_3$), helimagnets (e.g., CsCuCl$_3$), and parent cuprates (La$_2$CuO$_4$,...).


I. INTRODUCTION
More than a hundred years have passed since T. Smith [1] in 1916 found a weak, or parasitic ferromagnetism in an "international family line" of different natural hematite α-Fe 2 O 3 single crystalline samples from Italy, Hungary, Brasil, and Russia (Schabry, Ural Mountains, small settlement near Ekaterinburg) that was first assigned to ferromagnetic impurities. Later the phenomenon was observed in many other 3d compounds, such as fluoride NiF 2 with rutile structure, orthorhombic orthoferrites RFeO 3 (where R is a rareearth element or Y), rhombohedral antiferromagnets MnCO 3 , NiCO 3 , CoCO 3 , and FeBO 3 .
However, only in 1954 L.M. Matarrese and J.W. Stout for NiF 2 [2] and in 1956 A.S. Borovik-Romanov and M.P. Orlova for very pure synthesised carbonates MnCO 3 and CoCO 3 [3] have firmly established that the connexion between the weak ferromagnetism and any impurities or inhomogeneities seems very unlikely as weak ferromagnetism is observed in chemically pure antiferromagnetic materials and therefore it is a specific intrinsic property of some antiferromagnets. Furthermore, Borovik-Romanov and Orlova assigned the uncompensated moment in MnCO 3 and CoCO 3 to an overt canting of the two magnetic sublattices in almost antiferromagnetic matrix. The model of a canted antiferromagnet became generally adopted model of the weak ferromagnet.
A theoretical explanation and first thermodynamic theory for weak ferromagnetism in α-Fe 2 O 3 , MnCO 3 , and CoCO 3 was provided by Igor Dzyaloshinskii (Dzialoshinskii, Dzyaloshinsky) [4] in 1957 on the basis of symmetry considerations and Landau's theory of the second kind phase transitions. Free energy of the two-sublattice uniaxial weak ferromagnet such as α-F e 2 O 3 , MnCO 3 , CoCO 3 , F eBO 3 was shown to be written as follows In this expression m 1 and m 2 are unit vectors in the directions of the sublattice moments, M is the sublattice magnetization, m = 1 2 (m 1 + m 2 ) and l = 1 2 (m 1 − m 2 ) are the ferro-and antiferromagnetic vectors, respectively, H 0 is the applied field, H E is the exchange field, is now called the Dzyaloshinskii interaction, H D > 0 is the Dzyaloshinskii field. The anisotropy energy E A is assumed to have the form: where H A is the anisotropy field. The choice of sign for the anisotropy field H A assumes that the c axis is a hard direction of magnetization. In a general sense the Dzyaloshinskii interaction implies the terms that are linear both on ferro-and antiferromagnetic vectors. For instance, in orthorhombic orthoferrites and orthochromites the Dzyaloshinskii interaction consists of the antisymmetric and symmetric terms while for tetragonal fluorides NiF 2 and CoF 2 the Dzyaloshinskii interaction consists of the only symmetric term. Despite Dzyaloshinskii supposed that weak ferromagnetism is due to relativistic spin-lattice and magnetic dipole interaction, the theory was phenomenological one and did not clarify the microscopic nature of the Dzyaloshinskii interaction that does result in the canting. Later on, in 1960, Toru Moriya [5] suggested a model microscopic theory of the exchange-relativistic antisymmetric exchange interaction to be a main contributing mechanism of weak ferromagnetism where r i,j are unit radius vectors for O -M i,j bonds with presumably equal bond lenghts.
Valerii Ozhogin et al. [17] in 1968 first raised the issue of the sign of the Dzyaloshinskii vector, however, only in 1990 the reliable local information on the sign of the Dzyaloshinskii vector, or to be exact, that of the Dzyaloshinskii parameter d 12 , was first extracted from the 19 F ligand NMR data in weak ferromagnet F eF 3 [18]. In 1977 we have shown that the Dzyaloshinskii vectors can be of opposite sign for different pairs of S-type ions [10] that allowed us to uncover a novel magnetic phenomenon, weak ferrimagnetism, and a novel class of magnetic materials, weak ferrimagnets, which are systems such as solid solutions YFe 1−x Cr x O 3 with competing signs of the Dzyaloshinskii vectors and the very unusual concentration and temperature dependence of the magnetization [19]. The relation between Dzyaloshinskii vector and the superexchange geometry (6) allowed us to find numerically all the overt and hidden canting angles in the rare-earth orthoferrites [9] that was nicely confirmed in 57 F e NMR [20] and neutron diffraction [21] measurements.
The stimulus to a renewed interest to the subject was given by the cuprate problem, in particular, by the weak ferromagnetism observed in the parent cuprate La 2 CuO 4 [13] and many other interesting effects for the DM systems, in particular, the "field-induced gap" phenomena [22]. At variance with typical 3D systems such as orthoferrites, the cuprates are characterised by a low-dimensionality, large diversity of Cu-O-Cu bonds including cornerand edge-sharing, different ladder configurations, strong quantum effects for s = 1/2 Cu 2+ centers, and a particularly strong Cu-O covalency resulting in a comparable magnitude of hole charge/spin densities on copper and oxygen sites. Several groups (see, e.g., Refs. [23][24][25]) developed the microscopic model approach by Moriya for different 1D and 2D cuprates, making use of different perturbation schemes, different types of the low-symmetry crystalline field, different approaches to the intra-atomic electron-electron repulsion. However, despite a rather large number of publications and hot debates (see, e.g., Ref. [26]) the problem of exchange-relativistic effects, that is of the DM coupling and related problem of spin anisotropy in cuprates remains to be open (see, e.g., Refs. [27,28] for experimental data and discussion). Common shortcomings of current approaches to DM coupling in 3d oxides concern a problem of allocation of the Dzyaloshinskii vector and respective "weak" (anti)ferromagnetic moments, and full neglect of spin-orbital effects for "nonmagnetic" oxy-gen O 2− ions, which are usually believed to play only indirect intervening role. From the other hand, the oxygen 17 O NMR-NQR studies of weak ferromagnet La 2 CuO 4 [29] seem to evidence unconventional local oxygen "weak-ferromagnetic" polarization whose origin cannot be explained in frames of current models.
In recent years interest has shifted towards other manifestation of the DM coupling, such as the magnetoelectric effect [30,31], so-called flexoelectric effect in multiferroic bismuth ferrite BiF eO 3 with coexisting spin canting and the spin cycloidal ordering [32], and skyrmion states [33], where reliable theoretical predictions have been lacking.
It was shown the particular importance of this interaction for magnetic nanostructures, In fact, it is known for a long time that the DM coupling can produce long-period magnetic spiral structures in ferromagnetic and antiferromagnetic crystals lacking inversion symmetry.
This effect was suggested for MnSi and other crystals with B20 structure and it has been carefully proved that the sign of the DM coupling, hence the sign of the spin helix, is determined by the crystal handedness.
Phenomenologically antisymmetric DM coupling in a continual approximation gives rise to so-called Lifshitz invariants, or energy contributions linear in first spatial derivatives of the magnetization m(r) [34] m i ∂m j ∂x l − m j ∂m i ∂x l (9) (x l is a spatial coordinate). These chiral interactions derived from the DM coupling stabilize localized (vortices) and spatially modulated structures with a fixed rotation sense of the magnetization [33]. In fact, these are the only mechanism to induce nanosize skyrmion structures in condensed matter.
In this paper we present an overview of the microscopic theory of the DM coupling in strongly correlated compounds such as 3d oxides. The rest part of the paper is organized as follows. In Sec. 2 The modern microscopic theory of the (super)exchange coupling had been elaborated by many physicists starting with well-known papers by P. Anderson [35], especially intensively in 1960-70th (see review articles [36] crystal-field effects [40], off-diagonal exchange [41], exchange in excited states [43], angular dependence of the superexchange coupling [7]. The irreducible tensor operators (the Racah algebra) were shown to be very instructive tool both for description and analysis of the exchange coupling in the 3d-and 4f-systems [7,[37][38][39][40].
First poor man's microscopic derivation for the dependence of the superexchange integral on the bonding angle (see Fig. 1) was performed by the author in 1970 [7] under simplified assumptions. As a result, for S-ions with configuration 3d 5 (F e 3+ , Mn 2+ ) where parameters a, b, c depend on the cation-ligand separation. A more comprehensive analysis has supported validity of the expression. Interestingly, the second term in (11) is determined by the ligand inter-configurational 2p-ns excitations, while other terms are related with intra-configurational 2p-, 2s-contributions.
Later on the derivation had been generalized for the 3d ions in a strong cubic crystal field [11]. Orbitally isotropic contribution to the exchange integral for pair of 3d-ions with configurations t n 1 2g e n 2 g can be written as follows where g γ i , g γ j are effective "g-factors" of the γ i , γ j subshells of ion 1 and 2, respectively: Kinetic exchange contribution to partial exchange parameters I(γ i γ j ) related with the electron transfer to partially filled shells can be written as follows [11,40] I(e g e g ) = (t ss + t σσ cosθ) 2 2U ; where t σσ > t πσ > t ππ > t ss are positive definite d-d transfer integrals, U is a mean dd transfer energy (correlation energy). All the partial exchange integrals appear to be positive or "antiferromagnetic", irrespective of the bonding angle value, though the combined effect of the ss and σσ bonds ∝ cosθ in I(e g e g ) yields a ferromagnetic contribution given bonding angles π/2 < θ < π. It should be noted that the "large" ferromagnetic potential contribution [42] has a similar angular dependence [43].
Some predictions regarding the relative magnitude of the I(γ i γ j ) exchange parameters can be made using the relation among different d-d transfer integrals as follows t σσ : t πσ : t ππ : t ss ≈ λ 2 σ : λ π λ σ : λ 2 π : λ 2 s , where λ σ , λ π , λ s are covalency parameters. The simplified kinetic exchange contribution (14) where ∆E (35) is the energy separation between 3 E g and 5 E g terms for t 3 2g e g configuration (Cr 2+ ion). Obviously, these contributions have a ferromagnetic sign. Furthermore, the exchange integral I(CrCr) can change sign at θ = θ cr : Microscopically derived angular dependence of the superexchange integrals does nicely describe the experimental data for exchange integrals I(F eF e), I(CrCr), and I(F eCr) in orthoferrites, orthochromites, and orthoferrites-orthochromites [44] (see Fig. 2). The fitting allows us to predict the sign change for I(CrCr) and I(F eCr) at θ 12 ≈ 133 • and 170 • , respectively. In other words, the Cr however, this relation yields the exchange integrals that can be one and a half or even twice less than the values obtained by other methods [11,45].
Above we addressed only typically antiferromagnetic kinetic (super)exchange contribution as a result of the second order perturbation theory. However, actually this contribution does compete with typically ferromagnetic potential (super)exchange contribution, or Heisenberg exchange, which is a result of the first order perturbation theory. The most important contribution to the potential superexchange can be related with the intra-atomic ferromagnetic Hund exchange interaction of unpaired electrons on orthogonal ligand orbitals hybridized with the d-orbitals of the two nearest magnetic cations.
Strong dependence of the d − d superexchange integrals on the cation-ligand-cation separation is usually described by the Bloch's rule [46]:

A. Moriya's microscopic theory
First microscopic theory of weak ferromagnetism, or theory of anisotropic superexchange interaction was provided by Moriya [5], who extended the Anderson theory of superexchange to include spin-orbital coupling V so = i ξ(l i · s i ). Moriya started with the one-electron Hamiltonian for d-electrons as followŝ where is a spin-orbital correction to transfer integral, m and m ′ are orbitally nondegenerate ground states on sites f and f ′ , respectively. Then Moriya did calculate the generalized Anderson kinetic exchange that contains both conventional isotropic exchange and anisotropic symmetric and antisymmetric terms, that is quasidipole anisotropy and DM coupling, respectively.
We emphasize that the expression for the Dzyaloshinskii vector has been obtained by Moriya assuming orbitally nondegenerate ground states m and m ′ on sites f and f ′ , respectively. It is worth noting that the spin-operator form of the DM coupling follows from the relation: which is a simple consequence of the spin algebra, in particular, of the commutation relations for the spin projection operators.
Moriya found the symmetry constraints on the orientation of the Dzyaloshinskii vector d ij . Let two ions 1 and 2 are located at the points A and B, respectively, with C point bisecting the AB line: 1. When C is a center of inversion: d=0.
2. When a mirror plane ⊥AB passes through C, d mirror plane or d ⊥ AB.
3. When there is a mirror plane including A and B, d ⊥ mirror plane. 4. When a twofold rotation axis ⊥ AB passes through C, d ⊥ twofold axis.

5.
When there is an n-fold axis (n≥2) along AB, d AB.
Despite its seeming simplicity the operator form of the DM coupling (4) raises some questions and doubts. First, at variance with the scalar product (S 1 · S 2 ) the vector product of the spin operators [S 1 × S 2 ] changes the spin multiplicity, that is the net spin S 12 = S 1 +S 2 , that underscores the need for quantum description. Spin nondiagonality of the DM coupling implies very unusual features of the d-vector somewhat resembling vector orbital operator whose transformational properties cannot be isolated from the lattice [47]. It seems the d-vector does not transform as a vector at all.
Another issue that causes some concern is the structure and location of the d vector and corresponding spin cantings. Obviously, the d 12 vector should be related in one or another way to spin-orbital contributions localized on sites 1 and 2, respectively. These components may differ in their magnitude and direction, while the operator form (4) implies some averaging both for d 12 vector and spin canting between the two sites.
Moriya did not take into account the effects of the crystal field symmetry and strength and did not specify the character of the (super)exchange coupling, that, as we'll see below, can crucially affect the direction and value of the Dzyaloshinskii vector up to its vanishing.
Furthermore, he made use of a very simplified form (21) of the spin-orbital perturbation correction to the transfer integral (see Exp. (2.5) in Ref. [5]). The fact is that the structure of the charge transfer matrix elements implies the involvement of several different on-site . Hence, the perturbation correction has to be more complicated than (21), at least, it should involve the spin-orbital matrix elements (and excitation energies!) for one-and two-particle configurations. As a result, it does invalidate the author's conclusion about the equivalence of the two perturbation schemes, based on the V SO corrections to the transfer integral and to the exchange coupling, respectively.
Another limitation of the Moriya's theory is related to a full neglect of the ligand spinorbital contribution to DM coupling. Despite these shortcomings the Moriya's estimation for the ratio between the magnitudes of the Dzyaloshinskii vector d = |d| and isotropic exchange J: d/J ≈ ∆g/g, where g is the gyromagnetic ratio, ∆g is its deviation from the free-electron value, respectively, in some cases may be helpful, however, only for a very rough estimation.
B. Microscopic theory of the DM coupling: direct exchange interaction of the

S-type ions
We start with a derivation of the DM coupling in the pair of the exchange coupled free ions with valent n 1 l N 1 1 and n 2 l N 2 2 shells to be a result of the second-order perturbation theory as a combined effect of the exchange and spin-orbital couplings when schematicallŷ where excited states |ES are the terms which are allowable one by the spin-orbital selection rules ∆L ≤ 1, ∆S ≤ 1. Spin-orbit interaction has a fairly simple form V so = i ξ nl (l i · s i ), whereas for the exchange interaction Hamiltonian one has to use a complex expression in terms of irreducible tensor operators [7,10,[37][38][39]48]. The task seems to be more limited to academic interest, however, it is of a great importance from methodological point of view.
After some routine though rather intricate procedure we arrive at the Dzyaloshinskii vector to be a complicated "multistory" irreducible orbital operator as follows [10] where we make use of standard notations for 6j-symbols, irreducible matrix elements, spectroscopic coefficients, and irreducible tensor products [48][49][50][51]. Matrix elements of irreducible tensor operatorsV b (L) are defined by the Wigner-Eckart theorem [49,50] as follows For the exchange parameters we have a simple dependence on the pair radius-vector: Here in (25) we took into account V so (1) while the contribution of V so (2) has the same expression with the minus sign and the 1↔2 permutation. In addition, we restrict ourselves by the DM coupling operator which is diagonal on the spin and orbital moments. Obviously, nonzero DM coupling is only at even value of (b 1 + b ′ 1 ) and |b 1 is also should be an even number. Thus we should conclude that for the pair of equivalent free S-ions (F e 3+ , Mn 2+ ) when b 2 = b ′ 1 = 0 we have no DM coupling [11]. We arrive at the same conclusion, if to take into account that the exchange parameters I(101q) and I(011q) specifying the appropriate contribution turn into zero [11]. The appearance of the DM coupling in such a case can be driven by the inter-configurational or crystal field effects.
As the most illustrative example we consider a pair of 3d 5 ions such as F e 3+ , or Mn 2+ with the ground state 6 S in an intermediate octahedral crystal field which does split the 2S+1 L terms into crystal 2S+1 LΓ terms and mix the crystal terms with the same octahedral symmetry, that is with the same Γ's [52]. Spin-orbital coupling does mix the 6 S ground state with the 4 P T 1g term, however the 4 P T 1g term has been mixed with other 4 T 1g terms, 4 F T 1g and 4 GT 1g . Namely the latter effect is believed to be a decisive factor for appearance of the DM coupling. The |4(L)T 1g wave functions can be easily calculated by a standard technique [52] as follows [11]: given the crystal field and intra-atomic correlation parameters [52] typical for orthoferrites [53]: 10Dq = 12200 cm −1 ; B = 700 cm −1 ; C = 2600 cm −1 .
The huge expression (25) reduces to a more compact form as follows: where V 6 S 4 G is the conventional spectroscopic Racah coefficient [49], α4 P , α4 G are the mixing coefficients for the 4 T 1g term, I(404T 1 q) = β α T 1 q 4β I(404β) are the T 1 -symmetry combinations of the exchange parameters. It is worth noting the conclusive effect of the 4 P − 4 G mixing.
For the direct exchange we have a simple expression for the parameters where C 4T 1 q is the T 1 -symmetry combination, or cubic harmonics. Finally we arrive at a remarkable relation: where the T 1 -symmetry combinations of spherical harmonics are taken in local coordinate systems for the first and second ions, respectively, d 0 (12) ∝ J(404) and d 0 (21) ∝ J(044) are determined by the spin-orbital coupling on the sites 1 and 2, respectively. For locally equivalent F e 3+ centers J(404) = J(044) and d 0 (12) = d 0 (21). In the coordinate axes with and where θ and ϕ are polar and azimuthal angles of the R 12 vector. Obviously, the Dzyaloshinskii vector turns into zero, if local crystal field axes coincide for the both ions. In addition, d(12) = 0, if R 12 C 2 , C 3 , C 4 , that is to any symmetry axis for the first and second site. If and n 2 l N 2 2 via intermediate nonmagnetic ligand ion has the same general expression as for direct exchange [11], however, with a specific dependence of the exchange parameters on the superexchange geometry: In the local coordinate system for the site 1 with O z R 10 we can write out the superexchange parameter I(404T 1 q) as follows where   k 1 k 2 4 0 q 2 q 2   is the Clebsch-Gordan coefficient [49,50]. Obviously, for the superexchange mechanisms related with a particular ligand 2s or 2p electrons we have for k 2 : k 2 = 0 or k 2 = 0; 2, respectively. For mechanisms related with the ligand inter-configurational 2p → 3s excitations k 2 = 1. Taking into account the properties of the α T 1 q 4q 2 coefficients (30) we see that since |q 2 | ≤ 2 it follows that the terms with k 2 = 1 and k 2 = 2 in (33) can be expressed in terms of the vector product [ Obviously, final expression for the Dzyaloshinskii vector can be written as follows with where the first and the second terms are determined by the superexchange mechanisms related with the ligand inter-configurational 2p → 3s excitations and intra-configurational 2p − 2p effects, respectively. It should be noted that given θ = θ cr , where cosθ cr = −d 1 /d 2 , the Dzyaloshinskii vector changes its sign. Hereafter we address the DM coupling for the S-type magnetic 3d ions with orbitally nondegenerate high-spin ground state in a strong cubic crystal field, that is for the 3d ions with half-filled shells t 3 2g , t 3 2g e 2 g , t 6 2g e 2 g and ground states 4 A 2g , 6 A 1g , 3 A 2g , respectively. The strong crystal field approximation seems to be more appropriate for the most part of 3d ions in crystals. In particular, for the 4 T 1g terms of the 3d 5 ion in a strong cubic crystal field approximation instead of expressions (26) we arrive at a superposition of the wave functions for different t n 1 2g e n2 g configurations (n 1 + n 2 = 5) [52]. Using the same crystal field and correlation parameters as in Exp. (26) we get a triplet of new functions as follows with a more clearly defined contribution of a particular configuration compared with the intermediate crystal field scheme.
Making use of expressions for spin-orbital coupling V so [48] and main kinetic contribution to the superexchange parameters, that define the DM coupling, after routine algebra we have found that the DM coupling can be written in a standard form (36), where d 12 can be written as follows [10,11] where the X and Y factors do reflect the exchange-relativistic structure of the second-order perturbation theory and details of the electron configuration for S-type ion. The exchange factors X are where g t 2g are effective g-factors for e g , t 2g subshells, respectively, t σσ > t πσ > t ππ > t ss are positive definite d -d transfer integrals, U is the d -d transfer energy (correlation energy).
The dimensionless factors Y are determined by the spin-orbital constants and excitation energies as follows where W (1T 1 ) are spectroscopic coefficients for cubic point group [48] and summation runs on all the terms 2S+1 Γ, mixed by the spin-orbital coupling with the ground state term 2S i +1 Γ i . It should be noted that the nonzero DM coupling for S-type ions can be obtained only due to inter-configurational t 2g − e g interaction. The factors X and Y are presented in Table II for S-type 3d-ions. There ξ 3d is the spin-orbital parameter, ∆E2S+1 Γ is the energy of the 2S+1 Γ crystal term.
The signs for X and Y factors in Table II are predicted for rather large superexchange bonding angles |cosθ 12 | > t ss /t σσ which are typical for many 3d compounds such as oxides and a relation ∆E4 T 1g (41) < ∆E4 T 1g (32) which is typical for high-spin 3d 5 configurations.
It is worth noting that while working with the paper we have detected and corrected a casual and unintentional error in sign of the X i parameters having made both in our earlier papers [10,11] and very recent paper Ref. [54]. Hereafter we present correct signs for X i in (40) and Table II.
Rather simple expressions (40) and (41) for the factors X i and Y i do not take into account the mixing/interaction effects for the 2S+1 Γ terms with the same symmetry and the contribution of empty subshells to the exchange coupling (see Ref. [11]). Nevertheless, the data in Table II allow us to evaluate both the numerical value and sign of the d 12 parameters.
It should be noted that for critical angle θ cr , when the Dzyaloshinskii vector changes its Making use of different experimental data for covalency parameters (see, e.g., Ref. [55]) we arrive at d 1 /d 2 ∼ 1 5 − 1 3 and θ cr ≈ 100 • − 110 • for F e 3+ − F e 3+ pairs in oxides. Relation among different X's given the superexchange geometry and covalency parameters typical for orthoferrites and orthochromites [11] is however, it should be underlined its sensitivity both to superexchange geometry and covalency parameters. Simple comparison of the exchange parameters X (see (40) and Table II) with exchange parameters I(γ i γ j ) (14) evidences their close magnitudes. Furthermore, the relation (15) allows us to maintain more definite correspondence.
Given typical values of the cubic crystal field parameter 10Dq ≈ 1.5 eV we arrive at a relation among different Y 's [11] The highest value of the d 12 factor is predicted for d 8 − d 8 pairs, while for d 5 − d 5 pairs one expects a much less (may be one order of magnitude) value. The d 12 factor for d 3 − d 3 pairs is predicted to be somewhat above the value for d 5 − d 5 pairs. For different pairs: Puzzlingly, that despite strong isotropic exchange coupling for d 5 − d 5 and d 5 − d 8 pairs, the DM coupling for these pairs is expected to be the least one among the S-type pairs.
For d 5 − d 5 pairs, in particular, F e 3+ − F e 3+ we have two compensation effects. First, the σ-bonding contribution to the X parameter is partially compensated by the π-bonding contribution, second, the contribution of the 4 T 1g term of the t 4 2g e 1 g configuration is partially compensated by the contribution of the 4 T 1g term of the t 2 2g e 3 g configuration. Theoretical predictions of the corrected sign of the Dzyaloshinskii vector in pairs of the S-type 3d-ions with local octahedral symmetry (the sign rules) are presented in Table III. superexchange pathes for the two corner shared F eO 9− 6 octahedrons to the net Dzyaloshinskii vector strictly turns into zero. Exactly the same result will be obtained, if we consider the TABLE II. Expressions for the X and Y parameters that define the magnitude and the sign of the Dzyaloshinskii vector in pairs of the S-type 3d-ions with local octahedral symmetry. Signs for X i correspond to the bonding angle θ > θ cr .
− F e 3+ exchange in the system of two ideal F eO 9− 6 octahedrons bonded through the three common oxygen ions when R 12 C 3 . Obviously, it is precisely this fact that caused a tiny spin canting in hematite being an order of magnitude smaller than, e.g., in orthoferrites RF eO 3 or borate F eBO 3 . So what was the real reason of weak ferromagnetism in α-F e 2 O 3 as "opening a new page of weak ferromagnetism"? What is a microscopic origin of nonzero Dzyaloshinskii vector which should be directed along the C 3 symmetry axis according Moriya rules? First of all we should consider trigonal distortions for the F eO 9− 6 octahedrons which have a T 2 symmetry and give rise to a mixing of the 4 T 1g terms with 4 A 2g and 4 T 2g terms. The best way to solve the problem in principle is to proceed with a coordinate system where O z axis is directed along the C 3 symmetry axis rather than with the usually applied O z C 4 geometry.
In the coordinate axes with O z C 3 the nonzero coefficients α T 1 q 4β have another expres-sion [56]. Instead of (30) we arrive at It is easy to see that for , that means that all the components of the two contributions to Dzyaloshinskii vector (29) turn into zero.
However, the situation changes under axial (trigonal) distortion of the F eO 9− 6 octahedrons that can be described by a simple effective Hamiltonian as followŝ is the only irreducible tensor operator permitted by the symmetry of the distortion, B trig is a trigonal field parameter.
Such a distortion gives rise to a mixing of the 4 T 1g terms with 4 A 2g , 4 E g , and 4 T 2g terms.
As a result the bare | 4 T 1g functions are transformed as follows: where c 0 is a normalization coefficient, c A 2g 2 δ q0 are the mixing coefficients. The Dzyaloshinskii vector now has a representation as follows where we again suggest the possibility of nonequivalent centers 1,2. Cubic harmonics C 4T 2 µ one can find, if make use of data from Ref. [56] α We see that given R 12 C 3 the only nonzero cubic harmonic is Trigonal hematite α-F e 2 O 3 has the same crystal symmetry R3c − D 6 3d as weak ferromagnet F eBO 3 . Furthermore, the borate can be transformed into hematite by the F e 3+ ion substitution for B 3+ with a displacement of both "old" and "new" iron ions along trigonal axis. As a result we arrive at emergence of an additional strong isotropic (super)exchange coupling of three-corner-shared non-centrosymmetric F eO 6 (g m,n are the Lande factors) can arise for f − f superexchange only due to a spin-orbital contribution on intermediate ligands. Obviously, for the rare-earth-3d-ion (super)exchange we have an additional contribution of the 3d-ion spin-orbital interaction. The rare-earth- has been theoretically and experimentally considered in Ref. [57] for GdFeO 3 .

A. Overt and hidden canting in orthoferrites
At variance with isotropic superexchange coupling the DM coupling has a much more complicated structural dependence. In Table IV we present structural factors [r 1 × r 2 ] x,y,z for the 3. Basic vectors of magnetic structure for 3d sublattice in orthoferrites and orthochromites [   Fig.3), where main G-type antiferromagnetic order is accompanied by both overt canting characterized by ferromagnetic vector F (weak ferromagnetism!) and two types of a hidden canting, A and C (weak antiferromagnetism!): where a, b, c are unit cell parameters, x 1,2 , y 1,2 , z 2 are oxygen (O I,II ) parameters [58], l is a mean cation-anion separation. These relations imply an averaging on the F e bonds in ab plane and along c-axis. It is worth noting that |A x,y | > |F x,z | > |C y,z |.
First of all we arrive at a simple relation between crystallographic parameters and magnetic moment of the Fe-sublattice: in units of G · g/cm 3 where ρ and V are the unit cell density and volume, respectively. The overt canting F x,z can be calculated through the ratio of the Dzyaloshinskii (H D ) and exchange (H E ) fields as If we know the Dzyaloshinskii field we can calculate the d(θ) parameter in orthoferrites as follows that yields |d(θ)| ∼ = 3.2 K in YFeO 3 given H D = 140 kOe [15]. It is worth noting that despite F z ≈ 0.01 the d(θ) parameter is only one order of magnitude smaller than the exchange integral in YFeO 3 .
Our results have stimulated experimental studies of the hidden canting, or "weak antiferromagnetism" in orthoferrites. As shown in Table I theoretically predicted relations between overt and hidden canting nicely agree with the experimental data obtained for different orthoferrites by NMR [20] and neutron diffraction [21,59].

B. The DM coupling and effective magnetic anisotropy
Hereafter we demonstrate a contribution of the DM coupling into effective magnetic anisotropy in orthoferrites. The classical energies of the three spin configurations in ortho- The energies allow us to find the constants of the in-plane magnetic anisotropy E an = k 1 cos2θ (ac, bc planes), E an = k 1 cos2ϕ (ab plane): . Detailed analysis of different mechanisms of the magnetic anisotropy of the orthoferrites [60] points to a leading contribution of the DM coupling. Indeed, for all the orthoferrites RFeO 3 (R = Y, or rare-earth ion) this mechanism does predict a minimal energy for Γ 4 configuration which is actually realized as a ground state for all the orthoferrites, if one neglects the R-Fe interaction. Furthermore, predicted value of the constant of the magnetic anisotropy in ac-plane for YFeO 3 k 1 (ac) =2.0·10 5 erg/cm 3 is close enough to experimental value of 2.5·10 5 erg/cm 3 [15]. Interestingly, the model predicts a close energy for Γ 1 and Γ 2 configurations so that |k 1 (bc)| is about one order of magnitude less than |k 1 (ac)| and |k 1 (ab)| for most orthoferrites [60].
It means the anisotropy in bc-plane will be determined by a competition of the DM coupling with relatively weak contributors such as magneto-dipole interaction and single-ion anisotropy. It should be noted that the sign and value of the k 1 (bc) is of a great importance for the determination of the type of the domain walls for orthoferrites in their basic Γ 4 configuration (see, e.g., Ref. [61]).    Fig. 4a with experimental data for the low-temperature net magnetization [19] and the MFA calculations with δ = -2 ( Fig. 4b) points to a reasonably well agreement everywhere except x ∼ 0.5, where δ parameter seems to be closer to δ = -3. In Fig. 4d  given constant value of δ = -1.5 (Fig. 5a,b [65]). Temperature-dependent magnetization studies from 4.2 to 600 K have been made for the [66]. A rapid decrease is observed in the saturation magnetization with increasing x at 4.2 K up to 0.40, after which a broad compositional minimum is found up to x = 0.60. Compositions in the range of 0.40≤ x ≤0.60 display unusual magnetization behavior as a function of temperature in that maxima and minima are present in the curves below the Curie temperatures. Fig. 6b shows a nice agreement between experimental data [66] and our MFA calculations [64]. It should be noted that just recently Dmitrienko et al. [68] have first discovered experimentally that in accordance with our theory (see Table III Cr-Cr. Weak ferrimagnets can exhibit the tunable exchange bias effect [73] and have potential applications in electromagnetic devices [70]. Combining magnetization reversal effect with magnetoelectronics can exploit tremendous technological potential for device applications, for example, thermally assisted magnetic random access memories, thermomagnetic switches and other multifunctional devices, in a preselected and convenient manner. Nowadays a large body of magnetic materials might be addressed as systems with competing antisymmetric exchange [74], including novel class of mixed helimagnetic B20 alloys such as How to measure the sign of the DM interaction in weak ferromagnets? According to Ref. [17], an answer to this question can be given by determining experimentally the direction of rotation of the antiferromagnetism vector l around the magnetic field H in the geometry H d easy axis. However, as was pointed out later (see Ref. [76]), the Mössbauer experiment on easy-axis hematite did not give an unambiguous result.
According to Dmitrienko et al. [77], first of all, a strong enough magnetic field should be applied to obtain the single domain state where the DM coupling pins antiferromagnetic ordering to the crystal lattice. Next, single crystal diffraction methods sensitive both to oxygen coordinates and to the phase of antiferromagnetic ordering should be used. In other words, one should observe those Bragg reflections hkl where interference between magnetic scattering on Mn atoms and nonmagnetic scattering on oxygen atoms is significant. There are three suitable techniques: neutron diffraction, Mössbauer γ-ray diffraction, and resonant x-ray scattering. The sign of the DM vector in weak ferromagnetic FeBO 3 was deduced from observed interference between resonant X-ray scattering and magnetic X-ray scattering [77].
The authors in Ref. [76] claimed that the character of the field-induced transition from an antiferromagnetic phase to a canted phase in cobalt fluoride CoF 2 is due to the "sign" of the Dzyaloshinskii interaction, and this affords an opportunity to determine experimentally the "sign" of the Dzyaloshinskii interaction. However, in fact they addressed a symmetric Dzyaloshinskii interaction that is magnetic anisotropy rather than antisymmetric DM coupling.
A. Ligand NMR in weak ferromagnets and first determination of the sign of the Dzyaloshinskii vector As was firstly shown in our paper [18] reliable local information on the sign of the Dzyaloshinskii vector, or to be exact, that of the Dzyaloshinskii parameter d 12 , can be extracted from the ligand NMR data in weak ferromagnets. The procedure was described in details for 19 F NMR data in weak ferromagnet FeF 3 [18].
The F − ions in the unit cell of F eF 3 occupy six positions [78]. In a trigonal basis these are in an orthogonal basis with O z C 3 and O x C 2 . Each F − ion is surrounded by two F e 3+ from different magnetic sublattices. Hereafter we introduce basic ferromagnetic F and antiferromagnetic G vectors: where F e 3+ 1 and F e 3+ 2 occupy positions (1/2,1/2,1/2) and (0,0,0), respectively. F eF 3 is an easy plane weak ferromagnet with F and G lying in (111) plane with F⊥G. The two possible variants of the mutual orientation of the F and G vectors in the basis plane, tentatively called as "left" and "right", respectively, are shown in Fig. 7. The DM energy per F e 3+ −F − −F e 3+ bond can be written as follows In other words, the "left" and "right" orientations of basic vectors are realized at d(θ) < 0 and d(θ) > 0, respectively.
If we know the Dzyaloshinskii field we can calculate the d(θ) parameter as follows that yields |d(θ)| ∼ = 1.1 K that is three times smaller than in YFeO 3 .
The local field on the nucleus of the nonmagnetic F − anion in weak ferromagnet FeF 3 , induced by neighboring magnetic S-type ion (F e 3+ , Mn 3+ , . . .) can be written as follows [79] (γ n is a gyromagnetic ratio, γ n =4.011 MHz/kOe, S is the spin moment of the magnetic ion), where the tensor of the transferred hyperfine interactions (HFI)Â consists of two terms: i) ii) anisotropic term with where n is a unit vector along the nucleus-magnetic ion bond and the A p parameter includes the dipole and covalent contributions Here f s,π,σ are parameters for the spin density transfer: magnetic ion -ligand along the proper s−, σ−, π-bond; |ϕ 2s (0)| 2 is a probability density of the 2s-electron on nucleus; 1 is a radial average.
The A s and A p parameters we need to calculate parameter a F and the HFI anisotropy tensor ↔ a that is to calculate the "ferro-" and "antiferro-" contributions to H one can find in the literature data for the pair 19 F −F e 3+ . For instance, in KMgF 3 : F e 3+ (R M gF = 1.987Å) [80] A s = +72, A p = +18 MHz, in K 2 NaF eF 6 (R F eF = 1.91Å), in K 2 NaAlF 6 : F e 3+ A s = +70.17, In the absence of an external magnetic field the NMR frequencies for 19 F in positions 1,

2, 3 can be written as follows
where the a xy , a yz components are taken for 19 F 1 in position 1; ϕ is an azimuthal angle for ferromagnetic vector F in the basis plane. The formula (70) and and magnetic (F, ϕ, ±) structures. As of particular importance one should note a specific dependence of the 19 F NMR frequencies on mutual orientation of the ferro-and antiferromagnetic vectors or the sign of the Dzyaloshinskii vector: upper signs in (70) correspond to "right orientation" while lower signs do to "left orientation" as shown in Fig.7.
For minimal and maximal values of the 19 F NMR frequencies we have Taking into account smallness of isotropic HFI contribution, signs of a F and A xy we arrive at estimations Thus By using the A s and A p values, typical for 19 F − F e 3+ bonds [80,81] we get ν + min = 57.6, ν + max = 75.7, (ν max − ν min ) + = 18.1MHz (74) given "right orientation" of F and G (Fig.7) given "left orientation" of F and G (Fig.7) .
The zero-field 19 F NMR spectrum for single-crystalline samples of F eF 3 we simulated on assumption of negligibly small in-plane anisotropy [83] is shown in Fig. 8 for two different mutual orientations of F and G vectors. For a comparison in Fig. 8 we adduce the experimental NMR spectra for polycrystalline samples of F eF 3 [84,85], which are characterized by the same boundary frequencies despite rather varied shape. Obviously, the theoretically simulated NMR spectrum does nicely agree with the experimental ones only for "right" mutual orientations of F and G vectors, or d(F eF e) > 0, in a full accordance with our theoretical sign predictions (see Table III).
The same result, d(F eF e) > 0 follows from the the magnetic x-ray scattering amplitude measurements in the weak ferromagnet FeBO 3 [77].
where a = 4.626Å , b = 8.012Å are parameters of the orthohexagonal unit cell, x h = 0.2981 oxygen parameter, l = 2.028Å is a mean Fe-O separation [86].
Similarly to FeF 3 the DM energy per F e 3+ − O 2− − F e 3+ bond can be written as follows In other words, the "left" and "right" orientations of basic vectors are realized at d(θ) < 0 and d(θ) > 0, respectively.
If we know the Dzyaloshinskii field we can calculate the d 12 (θ) parameter as follows that yields |d(θ)| ∼ = 1.5 K that is two times smaller than in YFeO 3 . The difference can be easily explained, if one compares the superexchange bonding angles in FeBO 3 (θ ≈ 125 • ) and  Table III). cuprates the three-center (Cu 2+ 1 -O-Cu 2+ 2 ) two-hole system with tetragonal Cu on-site symmetry and ground Cu 3d x 2 −y 2 hole states (see Fig. 9) whose conventional bilinear effective spin Hamiltonian is written in terms of copper spins as follows [87,88] where J 12 > 0 is an exchange integral, D 12 is the Dzyaloshinskii vector, where sum runs on the holes 1 and 2 rather than sites 1 and 2. This form implies not only both copper and oxygen hole location, but allows to account for purely oxygen two-hole configurations. Moreover, such a form allows us to neatly separate both the one-center and two-center effects. Two-hole spin Hamiltonian (81)

The p-d hopping for Cu-O bond implies a conventional Hamiltonian
wherep † α creates a hole in the α state on the ligand site, whiled β annihilates a hole in the β state on the copper site; t pαdβ is a pd-transfer integral.
For basic 101 configuration with two d x 2 −y 2 holes localized on its parent sites we arrive at a perturbed wave function as follows where the summation runs both on different configurations and different orbital Γ states,  [35] with two main contributions of so-called kinetic and potential exchange, respectively. Then he took into account the spinorbital corrections to the effective d-d transfer integral and potential exchange. However, the Moriya's approach seems to be improper to account for purely ligand effects. In later papers (see, e.g. Refs. [24,89]) the authors made use of the Moriya scheme to account for spin-orbital corrections to p-d transfer integral, however, without any analysis of the ligand contribution.
It is worth noting that in both instances the spin-orbital renormalization of a single hole transfer integral leads immediately to a lot of problems with correct responsiveness of the on-site Coulomb hole-hole correlation effects. Anyway the effective DM spin-Hamiltonian evolves from the high-order perturbation effects that makes its analysis rather involved and leads to many misleading conclusions.
At variance with the Moriya approach we consider the DM couplinĝ to be a result of a projection of the spin-orbital operatorV so =V so (Cu 1 ) +V so (O) +V so (Cu 2 ) on the ground state singlet-triplet manifold [87]. Then we calculate the singlet-triplet mixing amplitude due to the three local spin-orbital terms to find the local contributions to Dzyaloshinskii vector: Remarkably that the net Dzyaloshinsky vector D 12 has a particularly local structure to be a superposition of partial contributions of different ions (i = 1, 0, 2) and ionic configurations {n} = 101, 110, 011, 200, 020, 002. Local spin-orbital coupling is taken as follows: with a single particle constant ξ nl > 0 for electrons and ξ nl < 0 for holes. Herê We make use of orbital matrix elements: for Cu 3d holes d x 2 −y 2 |l x |d yz = d x 2 −y 2 |l y |d xz = i, d x 2 −y 2 |l z |d xy = −2i, i|l j |k = −iǫ ijk with Cu 3d yz =|1 , 3d xz =|2 3d xy =|3 , and for the ligand np-holes p i |l j |p k = iǫ ijk . Free cuprous Cu 2+ ion is described by a large spin-orbital coupling with |ξ 3d | ∼ = 0.1 eV (see, e.g., Ref. [90]), though its value may be significantly reduced in oxides, chlorides... due to covalency effects.
Information regarding the ξ np value for the ligand np-orbitals is scant if any. Usually one considers the spin-orbital coupling on the oxygen in oxides to be much smaller than that on the copper, and therefore may be neglected [91,92]. However, even for a free oxygen atom the electron spin orbital coupling turns out to reach of appreciable magnitude: ξ 2p ∼ = 0.02 eV [93] while for the oxygen O 2− ion in oxides one expects the visible enhancement of spinorbital coupling due to a larger compactness of 2p wave function [94]. If to account for ξ nl ∝ r −3 nl and compare these quantities for the copper ( r −3 3d ≈ 6 − 8 a.u. [94]) and the oxygen ( r −3 2p ≈ 4 a.u. [29,94]) we arrive at a maximum factor two difference in ξ 3d and ξ 2p . However, for other ligands the spin-orbital effects can be of comparable value with that of Cu 2+ . For a free chlorine atom the electron spin-orbital coupling turns out to reach of appreciable magnitude: ξ 3p ∼ = 0.07 eV [93] close to ξ 3d while for the chlorine Cl − ion in chlorides one expects the visible enhancement of spin-orbital coupling due to a larger compactness of 3p wave function.
As for the DM interaction we deal with two competing contributions [87,88]. The first one is determined by the inter-configurational mixing effect and is derived as a first order contribution which does not take account of Cu 1,2 3d-orbital fluctuations for a ground state 101 configuration. Projecting the spin-orbital coupling (86) onto states (83) we see that Λ V ·V term is equivalent to a spin DM coupling with local contributions to Dzyaloshinskii vector .
It should be noted that at variance with the original Moriya approach [5] both spinless and spin-dependent parts of exchange Hamiltonian contribute additively and comparably to DM coupling, if one takes account of the same magnitude and opposite sign of the singlet-singlet and triplet-triplet exchange matrix elements on the one hand and orbital antisymmetry of spin-orbital matrix elements on the other hand.
It is easy to see that the contributions of 002 and 200 configurations to Dzyaloshinskii vector bear a similarity to the respective second type (∝ V so H ex ) contributions, however, in the former we deal with spin-orbital coupling for two-hole Cu 1,2 configurations, while in the latter with that of one-hole Cu 1,2 configurations.

Copper contribution
First we address a relatively simple example of strong rhombic crystal field for the intermediate ligand ion with the crystal field axes oriented along global coordinate x, y, z-axes, respectively. It is worth noting that in such a case the ligand np z orbital does not play an active role both in symmetric and antisymmetric (DM) exchange interaction as well as Cu 3d yz orbital appears to be inactive in DM coupling due to a zero overlap/transfer with ligand np orbitals.
For illustration, hereafter we address the first contribution (88) of two-hole on-site 200, 002 d 2 x 2 −y 2 , d x 2 −y 2 d xy , and d x 2 −y 2 d xz configurations, which do covalently mix with ground state configuration [87,88]. Calculating the singlet-triplet mixing matrix elements in the global coordinate system we find all the components of the local Dzyaloshinskii vectors. The Cu 1 contribution turns out to be nonzero only for 200 configuration, and may be written as a sum of several terms. First we present the contribution of the singlet (d 2 x 2 −y 2 ) 1 A 1g state: where where ǫ x,y are the ligand p x,y -hole energies. In a vector form we arrive at where r 1 is an unit vector directed along Cu 1 -O bond, z is the unit vector in xyz system.
Taking into account that c 002 we see that the Cu 2 contribution to the Dzyaloshinskii vector can be obtained from Exps. (90), if θ, δ 1 replace by −θ, δ 2 , respectively.
Both collinear (θ = π) and rectangular (θ = π/2) superexchange geometries appear to be unfavorable for copper contribution to antisymmetric exchange, though in the latter the result depends strongly on the relation between the energies of the ligand np x and np y orbitals.
Contribution of singlet (d x 2 −y 2 d xy ) 1 A 2g and (d x 2 −y 2 d xz ) 1 E g states to the Dzyaloshinskii vector yields Here we deal with a vector directed along the Cu 1 -O bond whose magnitude is determined by a partial cancellation of two terms.
It is easy to see that the copper V so (1) contribution to the Dzyaloshinskii vector for two-site 110 and 011 configurations is determined by a dp-exchange.

Ligand contribution
In frames of the same assumption regarding the orientation of rhombic crystal field axes for the intermediate ion the local ligand contribution to the Dzyaloshinskii vector for onesite 020 configuration appears to be oriented along local O z axis and may be written as follows [87,88] This vector can be written as where r 1,2 are unit radius-vectors along Cu 1 -O, Cu 2 -O bonds, respectively, and where E s (p 2 x,y ) are the two-hole singlet energies. It is worth noting that D (0) does not depend on the δ 1 , δ 2 angles. The D O (θ) dependence is expected to be rather smooth without any singularities for collinear and rectangular superexchange geometries.
The local ligand contribution to the Dzyaloshinskii vector for the two-site 110 and 011 configurations is determined by a direct dp-exchange and may be written similarly to (96) with where we take account only of the dpσ exchange (I dpσ ∝ t 2 pdσ ).

DM coupling in La 2 CuO 4 and related cuprates
The DM coupling and magnetic anisotropy in La 2 CuO 4 and related copper oxides has attracted considerable interest in 90-ths (see, e.g., Refs. [23][24][25][26]), and is still debated in the literature [27,28]. In the low-temperature tetragonal (LTT) and orthorhombic (LTO) phases of La 2 CuO 4 , the oxygen octahedra surrounding each copper ion rotate by a small tilting angle (δ LT T ≈ 3 0 , δ LT O ≈ 5 0 ) relative to their location in the high-temperature tetragonal (HTT) phase. The structural distortion allows for the appearance of the antisymmetric DM coupling. In terms of our choice for structural parameters to describe the Cu 1 -O-Cu 2 bond we have for LTT phase: θ = π; δ 1 = δ 2 = π 2 ± δ LT T for bonds oriented perpendicular to the tilting plane, and θ = ±(π − 2δ LT T ); δ 1 = δ 2 = π 2 for bonds oriented parallel to the tilting plane. It means that all the local Dzyaloshinskii vectors turn into zero for the former bonds, and turn out to be perpendicular to the tilting plane for the latter bonds. For LTO Comparative analysis of Exps. (90), (97), and (98) given estimations for different parameters typical for cuprates [106] (t pdσ ≈ 1.5 eV, t pdπ ≈ 0.7 eV, A = 6.5 eV, B = 0.15 eV, C = 0.58 eV, F 0 = 5 eV, F 2 = 6 eV) evidences that copper and oxygen Dzyaloshinskii vectors can be of a comparable magnitude, however, in fact it strongly depends on the Cu 1 -O-Cu 2 bond geometry, correlation energies, and crystal field effects. The latter determines the single hole energies both for O 2p-and Cu 3d-holes such as ǫ x,y and ǫ xy,xz , whose values are usually of the order of 1 eV and 1-3 eV, respectively. It is worth noting that for two limiting bond geometries: θ ∼ π and θ ∼ π/2 (near collinear and near rectangular bonding, respectively) we deal with a strong "geometry reduction" of the DM coupling due to the sin θ factor for the former and the factor like for the latter. Really, the resulting effect for the near rectangular Cu 1 -O-Cu 2 bonding appears to be very sensitive to the local oxygen crystal field. A critical angle θ Cu to turn the Cu-contribution to Dzyaloshinskii vector into zero is defined as follows: while for the oxygen contribution (97) we arrive at another critical angle: Maximal value of the scalar parameter D O (θ) which determines the oxygen contribution to Dzyaloshinskii vector can be estimated to be of ≤1 meV given the above mentioned typical parameters, however, the unfavorable geometry of the Cu-O-Cu bonds in the corner-shared cuprates leads to a small value of the resulting Dzyaloshinskii vector and canting angles [13].
As a whole, our model microscopic theory is believed to provide a reasonable estimation of the direction, sense, and numerical value of the Dzyaloshinskii vectors. Seemingly more important result concerns the elucidation of the role played by Cu 1 -O-Cu 2 bond geometry, crystal field, and correlation effects.

C. DM coupled Cu 1 -O-Cu 2 bond in external fields
Application of an uniform external magnetic field h S will produce a net staggered spin polarization in the antiferromagnetically coupled Cu 1 -Cu 2 pair with antisymmetric V S-susceptibility tensor: χ V S αβ = −χ V S βα . It is worth noting that only in a classical representation the net contribution of the three local spin-orbital couplings does reduce to a conventional antiferromagnetic spin order: while in quantum representation one should say about emergence of some nonequivalence of spins for holes formally numbered as 1 and 2 on different sites. Puzzlingly, we arrive at a very unusual effect of the on-site staggered spin order to be a result of the on-site spinorbital coupling and the the cation-ligand spin density transfer. One sees that the sense of a staggered spin polarization, or antiferromagnetic vector, depends on that of Dzyaloshinskii vector. The V S coupling results in many interesting effects for the DM systems, in particular, the "field-induced gap" phenomena in 1D s=1/2 antiferromagnetic Heisenberg system with alternating DM coupling [22]. Approximately, the phenomenon is described by a so-called staggered s=1/2 antiferromagnetic Heisenberg model with the Hamiltonian which includes the effective uniform field h u and the induced staggered field h s ∝ h u perpendicular both to the applied uniform magnetic field and Dzyaloshinskii vector.
The DM copling for ferromagnetically coupled Cu 1 -Cu 2 pair does also produce a net staggered spin polarization oriented perpendicular both to the net magnetic moment and Dzyaloshinskii vector. It should be noted that all the partial contributions to the net staggered spin polarization can, in general, have distinct orientations.
Application of a staggered field h V for an antiferromagnetically coupled Cu 1 -Cu 2 pair will produce a local spin polarization both on copper and oxygen sites which can be detected by different site-sensitive methods including neutron diffraction and, mainly, by nuclear magnetic resonance. It should be noted that SV -susceptibility tensor is the antisymmetric one: χ SV αβ = −χ SV βα . Strictly speaking, the both formulas (99) and (102)  to oxygens in the local Cu-O-Cu bonds whose axis is perpendicular to the in-plane external field. It is worth noting once more, that the most part of experimental data was collected in paramagnetic state for temperatures well above T N where there is no frozen moments! The data were first interpreted as an indication of a direct oxygen spin polarization due to a local DM antisymmetric exchange coupling. However, it demands unphysically large values for such a polarization, hence the puzzle remained to be unsolved [29].
Our interpretation of the ligand NMR data in low-symmetry systems as La 2 CuO 4 implies a thorough analysis both of the spin canting effects and of the transferred hyperfine interactions with a revisit of some textbook results being typical for the model high-symmetry systems [88]. First, we start with spin-dipole hyperfine interactions for O 2p-holes which are main participants of Cu 1 -O-Cu 2 bonding. Making use of a conventional formula for a spin-dipole contribution to local field and calculating appropriate matrix elements on oxygen 2p-functions we present the local field on the 17 O nucleus in Cu 1 -O-Cu 2 system as a sum of ferro-and antiferromagnetic contributions as follows [88] where along with a conventional textbook ferromagnetic (∝ Ŝ ) transferred hyperfine contribution to local field which simply mirrors a sum total of two Cu-O bonds, we arrive at an additional unconventional antiferromagnetic (∝ V ) contribution whose symmetry and magnitude strongly depend on the orientation of the oxygen crystal field axes and Cu 1 -O-Cu 2 bonding angle. In the case of Cu 1 -O-Cu 2 geometry shown in Fig.9 we arrive at a diagonal ↔ A S tensor: and the only nonzero components of ↔ A V tensor: where f σ is the parameter of a transferred spin density and we made use of a simple approximation E s,t (dp x,y ) ≈ ǫ p . Thus, the ligand 17 O NMR provides an effective tool to inspect the spin canting effects in oxides with DM coupling both in paramagnetic and ordered phases.
The two-term structure of the oxygen local field (103) implies a two-term S-V structure of the 17 O Knight shift p ≈ 100 kG/spin [29], | sin θ| ≈ 0.1, and f σ ≈ 20% we obtain ≈ 6 kG as a maximal value of a low-temperature antiferromagnetic contribution to hyperfine field which is equivalent to a giant 17 O Knight shift of the order of almost ∼10%. Nevertheless, this value agrees with a low-temperature extrapolation of high-temperature experimental data by Walstedt et al. [29]. Interestingly, the sizeable effect of anomalous negative contribution to 17 O Knight shift has been observed in La 2 CuO 4 well inside the paramagnetic state for temperatures T ∼ 500 K that is essentially higher than T N ≈ 300 K. It points to the close relation between the magnitude of field-induced staggered magnetization and spin-correlation length which goes up as one approaches T N .  Fig.9 given θ ≤ π, δ 1 = δ 2 ≈ π/2. It should be emphasized that the above effect is determined by the net Dzyaloshinskii vector in Cu 1 -O-Cu 2 triad rather than by a local oxygen "weak-ferromagnetic" polarization as it was firstly proposed by Walstedt et al. [29].
Similar effect of anomalous ligand 13 C Knight shift has recently been observed in copper pyrimidine dinitrate [CuPM(NO 3 ) 2 (H 2 O) 2 ] n , a one-dimensional S=1/2 antiferromagnet with alternating local symmetry, and was also interpreted in terms of the field-induced staggered magnetization [98]. However, the authors did take into account only the inter-site magnetodipole contribution to ↔ A V tensor that questions their quantitative conclusions regarding the "giant" spin canting.

VIII. DM COUPLING IN HELIMAGNETIC CsCuCl 3
All the systems described above were somehow or other connected with weak ferromagnets where DM coupling manifests itself in the canting of a basic antiferromagnetic structure. Caesium cupric chloride, CsCuCl 3 is a unique screw antiferroelectric crystal with a low-temperature helimagnetically distorted ferromagnetic order. CsCuCl 3 possesses the hexagonal CsNiCl 3 -type D 4 6h (P 6 3 /mmc) structure [99] above the transition temperature T c (≈ 423 K) and is distorted through a first-order phase transition to low symmetry by a cooperative Jahn-Teller effect below T c [100]. In the high-temperature phase CuCl 6 octahedra are linked together by sharing faces, thus forming a one-dimensional chain structure along the c axis. The octahedra are not regular but trigonally compressed along the c axis, with all Cu-Cl distances remaining equal. At T c all of the constituent atoms are displaced from the normal position along the c-axis to form a helix whose period is three times the lattice constant c of the high-temperature phase. The room-temperature structure was determined by x-ray diffraction [101]. The space group is one of two enantiomorphous groups D 2 6 (P 6 1 22) or D 3 6 (P 6 5 22) without a center of symmetry, corresponding to the right and left helixes, respectively, with six formula units in a unit cell. Deformation of each CuCl 6 octahedron associated with the transition at T c is the Cu 2+ displacement and tetragonal elongation with the directions of their longest axes alternating by 120 • in adjacent octahedra lying along the chain.
In addition, CsCuCl 3 has a peculiar magnetic property. It is a quantum frustrated magnetic system with a triangular lattice of antiferromagnetically coupled s=1/2 spins of Cu 2+ in the ab plane. In the magnetically ordered state, below T N (10.5-10.7 K) spins lie in the basal plane and form the 120 • -structure, while along the c-direction, a long period (about 71 triangular layers) helical incommensurate arrangement (Dzyaloshinskii helix [102]) with a slow spin spiraling (pitch angle of about 5 • ) is realized [103], due to the competition between the dominant ferromagnetic interaction and the additional DM coupling along the chain.
The DM coupling forces the spins to lie almost flat in the ab plane, so this spin system is approximately an XY-system. In fact, from the structure determination, the spins are known to be slightly canted out of the ab plane [103]. CsCuCl 3 is the first example having a helical magnetic structure due to the antisymmetric exchange interaction. The reduction (to 0.58 µ B ) of the ordered moment of the s = 1/2 spin of the Cu 2+ ion is not uncommon in frustrated triangular-lattice antiferromagnetic systems. It is worth noting that Plakhty et al. [102] have revealed a modulation of the CsCuCl 3 crystal structure with the periodicity of the incommensurate long-period Dzyaloshinskii helix.
Despite numerous experimental and theoretical studies many details of spin structure in CsCuCl 3 remain to be answered. The NMR data do not support incommensurability in CsCuCl 3 , the 63,65 Cu NMR spectra clearly indicate that the Cu 2+ moments refer regularly to a local symmetry axis rather than to a spin spiral arrangement [104], or the in-plane spin projection forms a commensurate spiral with the pitch angle 60 • [105]. According to Ref. [103] the Dzyaloshinskii vector appears to be parallel to the vector between the nearest along c helically displaced Cu 2+ ions. However, Plakhty et al. [102] argued that the vector should be directed perpendicular to a plane formed by the Cu-Cl-Cu triad. Another point of a great importance for a detailed spin structure determination in CsCuCl 3 is a nonzero chlorine spin polarization whose accounting can strongly influence the interpretation of magnetic, neutron, and NMR data. To the best of our knowledge the neutron diffraction data for chain cuprate Li 2 CuO 2 by Chung et al. To make a semiquantitative analysis of the Cu-Cl(1)-Cu DM coupling, hereafter we assume a tetragonal symmetry at Cu sites with local coordinate systems as shown in Fig. 9.
The net Dzyaloshinskii vector D for the Cu 1 -Cl(1)-Cu 2 superexchange is a superposition of three contributions D = D (1) + D (O) + D (2) attached to the respective sites. In general, all the vectors are oriented differently. In other words, the direction of the net Dzyaloshinskii vector D nn+1 seems to be more complicated that it is suggested in Refs. [102,103]. Interestingly, the x-component of the Dzyaloshinskii vector, or its projection onto the Cu n -Cu n+1 direction gives rise to a helical spin ordering along c-axis with spins in ab-plane, while y and z components compete for the spin canting upward and downward from the ab-plane with a periodicity of six Cu 2+ ion spacings along the c-axis.
Comparative analysis of Exps. (90), (97), and (98) given estimations for different parameters typical for cuprates [106] evidences that copper and chlorine Dzyaloshinskii vectors can be of a comparable magnitude, however, in fact it strongly depends on the Cu 1 -Cl-Cu 2 bond geometry, correlation energies, and crystal field effects. Maximal value of the scalar parameter D O (θ) which determines the chlorine contribution to Dzyaloshinskii vector can be estimated to be of ∼ 1 meV given the above mentioned typical parameters. As a whole, our model microscopic theory is believed to provide a reasonable estimation of the direction, sense, and numerical value of the Dzyaloshinskii vectors and the role of the Cu 1 -Cl-Cu 2 bond geometry, crystal field, and correlation effects.

IX. EFFECTIVE TWO-ION SYMMETRIC SPIN ANISOTROPY DUE TO DM COUPLING
When one says about an effective spin anisotropy due to DM coupling one usually addresses a simple classical two-sublattice weak ferromagnet where the free energy has a minimum when both ferro-(∝ Ŝ ,Ŝ =Ŝ 1 +Ŝ 2 ) and antiferromagnetic (∝ V ,V =Ŝ 1 −Ŝ 2 ) vectors, being perpendicular to each other, lie in the plane perpendicular to the Dzyaloshinskii vector D. However, the issue is rather involved and appeared for a long time to be hotly debated. In our opinion, first of all we should define what the spin anisotropy is. Indeed, the description of any spin system implies the free energy Φ depends on a set of vector order parameters (e.g., Ŝ , V ) under kinematic constraint, rather than a single magnetic moment as in a simple ferromagnet, that can make the orientational dependence of the free energy Φ extremely involved. Such a situation needs in a careful analysis of respective spin Hamiltonian with a choice of proper approximations.
Effective symmetric spin anisotropy due to DM interaction can be easily derived as a second order perturbation correction due to DM coupling. For antiferromagnetically coupled spin 1/2 pairĤ DM an may be written as follows [87,88]: We see that in frames of a simple MFA approach this anisotropy stabilizes a Néel state with V ⊥ D. However, in fact we deal with a MFA artefact. Indeed, let examine the second order perturbation correction to the ground state energy of an antiferromagnetically coupled spin 1/2 pair in a Néel-like staggered field h V n (Ψ α,0 = cos α|00 + sin α|1n , tan 2α = 2h V J ): where E ⊥ = J; E = J cos 2 α + h V sin 2α; E g = J sin 2 α −h V sin 2α. First term in (108) stabilizes n D configuration while the second one does the n ⊥ D configuration. Interestingly that (E − E g ) cos 2 α = (E ⊥ − E g ), that is for any staggered field E DM an does not depend on its orientation, if to account for: |D · n| 2 + |D × n| 2 = |D| 2 . In other words, at variance with a simple MFA approach, the DM contribution to the energy of anisotropy for an exchange coupled spin 1/2 pair in a staggered field turns into zero. Anyway, theĤ DM an term has not to be included into an effective spin anisotropy Hamiltonian for quantum 1/2 spins. However, for large spins S ≫ 1/2 the MFA, or classical approach to anisotropy induced by the DM coupling can be more justified.

X. "FIRST-PRINCIPLES" CALCULATIONS OF THE DM COUPLING
The electronic states in strongly correlated 3d oxides manifest both significant localization and dispersional features. One strategy to deal with this dilemma is to restrict oneself to small many-electron clusters embedded to a whole crystal, then creating model effective lattice Hamiltonians whose spectra may reasonably well represent the energy and dispersion of the important excitations of the full problem. Despite some shortcomings the method did provide a clear physical picture of the complex electronic structure and the energy spectrum, as well as the possibility of a quantitative modeling.
However, last decades the condensed matter community faced an expanding flurry of papers with ab initio calculations of electronic structure and physical properties for strongly correlated systems such as 3d compounds based on density functional theory (DFT). The modern formulation of the DFT originated in the work of Hohenberg and Kohn [107], on which based the other classic work in this field by Kohn and Sham [108]. The Kohn-Sham equation, has become a basic mathematical model of much of present-day methods for treating electrons in atoms, molecules, condensed matter, solid surfaces, nanomaterials, and man-made structures [109].
However, DFT still remains, in some sense, ill-defined: many of the DFT statements were ill-posed or not rigorously proved. Most widely used DFT computational schemes start with a "metallic-like" approaches making use of approximate energy functionals, firstly LDA (local density approximation) scheme, which are constructed as expansions around the homogeneous electron gas limit and fail quite dramatically in capturing the properties of strongly correlated systems. The LDA+U and LDA+DMFT (DMFT, dynamical meanfield theory) [110] methods are believed to correct the inaccuracies of approximate DFT exchange correlation functionals, however, these preserve many shortcomings of the DFT-LDA approach. All efforts to account for the correlations beyond LDA encounter an insoluble problem of double counting (DC) of interaction terms which had just included into Kohn-Sham single-particle potential. In a certain sense the cluster based calculations seem to provide a better description of the overall electronic structure of insulating 3d oxides and its optical response than the DFT based band structure calculations, mainly due to a clear physics and a better account for correlation effects (see, e.g., Refs. [106,111]).
Basic drawback of the spin-polarized DFT approaches is that these start with a local density functional in the form (see, e.g. Ref. [112]) v(r) = v 0 [n(r)] + ∆v[n(r), m(r)](σ · m(r) |m(r)| ) , where n(r), m(r) are the electron and spin magnetic density, respectively, σ is the Pauli matrix, that is these approaches imply presence of a large fictious local one-electron spinmagnetic field ∝ (v ↑ −v ↓ ), where v ↑,↓ are the on-site LSDA spin-up and spin-down potentials.
Magnitude of the field is considered to be governed by the intra-atomic Hund exchange, while its orientation does by the effective molecular, or inter-atomic exchange fields. Despite the supposedly spin nature of the field it produces an unphysically giant spin-dependent rearrangement of the charge density that cannot be reproduced within any conventional technique operating with spin Hamiltonians. Furthermore, a direct link with the orientation of the field makes the effect of the spin configuration onto the charge distribution to be unphysically large. However, magnetic long-range order has no significant influence on the redistribution of the charge density. The DFT-LSDA community needed many years to understand such a physically clear point.
In general, the LSDA method to handle a spin degree of freedom is absolutely incompatible with a conventional approach based on the spin Hamiltonian concept. There are some intractable problems with a match making between the conventional formalism of a spin Hamiltonian and LSDA approach to the exchange and exchange-relativistic effects. Visi-bly plausible numerical results for different exchange and exchange-relativistic parameters reported in many LSDA investigations (see, e.g., Refs. [113]) evidence only a potential capacity of the LSDA based models for semiquantitative estimations, rather than for reliable quantitative data. It is worth noting that for all of these "advantageous" instances the matter concerns the handling of certain classical Néel-like spin configurations (ferro-, antiferro-, spiral,...) and search for a compatibility with a mapping made with a conventional quantum spin Hamiltonian. It's quite another matter when one addresses the search of the charge density redistribution induced by a spin configuration as, for instance, in multiferroics. In such a case the straightforward application of the LSDA scheme can lead to an unphysical overestimation of the effects or even to qualitatively incorrect results due to an unphysically strong effect of a breaking of spatial symmetry induced by a spin configuration (see, e.g. Refs. [114] and references therein).
As a typical starting point for the "first-principles" calculation of the exchange interactions and DM coupling one makes use of a predetermined classical spin configuration and classical Hamiltonian as follows where e i is a unit vector in the direction of the ith site magnetization, J ij is the exchange interaction, and D ij is the Dzyaloshinskii vector. It should be noted that this oversimplification together with an exceptionally one-particle nature of the LDA approach bounds all the efforts to account of intricate quantum effects that perturbatively define the DM coupling for many-electron ions, though keeps a possibility of a plausible estimation. We'd like to remind that classical approximation for the singlet-triplet exchange splitting in the pair of quantum s=1/2 spins yields the three times smaller value than the quantum result.
Obviously, the LDA based approaches cannot provide a comprehensive description of the DM coupling and other anisotropic interactions that are derived from the higher than the isotropic exchange perturbation orders and imply an intricate interplay of different manyelectron quantum fluctuations. It is worth noting that at variance with isotropic exchange the DM coupling does mix spin multiplicity that cannot be distinctly reproduced in classical approach. The so-called LDA+U+SO approach that attempts (and fails) to incorporate spin-orbit coupling within LDA+U scheme leads to unphysical results such as an "intraatomic noncollinear magnetic ordering" when the spins of different orbitals appear to be noncollinear to each other or an appearance of the single-ion anisotropy for s=1/2 ions (Cu 2+ ) [115]. The LDA+U+SO calculations [113] show appearance of unphysical on-site contribution in the magnetic torque and DM coupling, moreover this false term gives the main contribution to Dzyaloshinskii vector (!?). Recently a distinct approach for calculations of DM coupling and other anisotropic interactions in molecules and crystals has been proposed [116,117]. The authors derive a set of equations expressing the parameters of the magnetic interactions characterizing a strongly correlated electronic system in terms of single-electron Green's functions and self-energies. This allows to establish a mapping between the initial electronic system and a classical spin model (110) including up to quadratic interactions between the effective spins, with a general interaction (exchange) tensor that accounts for DM coupling, single-and two-ion anisotropy. As they argue, the scheme leads to a simple and transparent analytical expression for the Dzyaloshinskii vector with a natural separation into spin and orbital contributions, though they do not present physical explanation for such a separation. However, the mere possibility of such a mapping seems to be unacceptable, as any ions with a bare spin and orbital degeneracy are characterized by a number of multicomponent spin and orbital order parameters that cannot be reduced to the only vector order parameter. The application of inappropriate techniques makes it often hard to compare results obtained by different "first-principles" calculations even for the same weak ferromagnet. For instance, for the spin canting angle in La 2 CuO 4 one obtains 0.7·10 −3 [113] and 5·10 −3 [116] as compared with experimental value of (2-3)·10 −3 .
In our opinion, any comprehensive physically valid description of the exchange and exchange-relativistic effects for strongly correlated systems should combine simple physically clear cluster ligand-field analysis with a numerical calculation technique such as LDA+MLFT [118] with a regular appeal to experimental data. In all cases, the magnitude of the Dzyaloshinskii vector d 12 is anticorrelated with the magnitude of the superexchange integral J 12 in the sense that the superexchange geometry, favorable for the former, is unfavorable for the latter. As a typical example, parent cuprates can be cited, where the small value of the Dzyaloshinsky vector is determined by only a small tilting of the CuO 6 octahedra from the CuO 2 planes, which practically does not affect the large value of the exchange integral, J 12 ≥ 0.1 eV [13]. The specific supersensitivity of the DM coupling to the superexchange geometry and the energy of orbital excitations for Cu and O ions allows us to consider this interaction, first of all, the value and orientation of the Dzyaloshinskii vector, as one of the most important indicators determining the role of structural factors, in particular, the tilts and bond disproportions in the CuO 2 lattice network associated with "lattice strain" [119][120][121][122][123], and different orbital excitations [124,125] in the formation of an unusual electronic structure of the normal and superconducting state of HTS cuprates.
The work clearly shows advantages of the cluster method as compared with the DFTbased technique to provide an adequate description of the DM coupling and related exchangerelativistic effects in strongly correlated 3d compounds such as exchange anisotropy [126], spin-other-orbit interaction [127][128][129], antisymmetric magnetoelectric coupling [114], and electron-nuclear antisymmetric supertransferred hyperfine interactions [130,131]. However, it should be noted that the DFT with functionals more advanced than LDA can be effective in calculating correctly the sign and strength of the DM coupling in non-correlated materials [132][133][134].