A High-Temperature Expansion Method for Calculating Paramagnetic Exchange Interactions

A high-temperature expansion method for calculating paramagnetic exchange interactions

O.M. Sotnikov1, V.V. Mazurenko1 1Theoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Str.19, 620002 Ekaterinburg, Russia (Dated: October 11, 2018) The method for calculating the isotropic exchange interactions in the paramagnetic phase is proposed. It is based on the mapping of the high-temperature expansion of the spin-spin correlation function calculated for the Heisenberg model onto Hubbard Hamiltonain one. The resulting expression for the exchange interaction has a compact and transparent formulation. The quality of the calculated exchange interactions is estimated by comparing the eigenvalue spectra of the Heisenberg model and low-energy magnetic part of the Hubbard model. By the example of quantum rings with diﬀerent hopping setups we analyze the contributions from the diﬀerent part of the Hubbard model spectrum to the resulting exchange interaction.

I. INTRODUCTION TABLE I. List of methods for calculation of the isotropic exchange interaction. φi(x) is a wave function centered at the The magnetic properties of a correlated system can lattice site i. tij and U are the hopping integral and the on- be fully described by its magnetic susceptibility char- site Coulomb interaction, respectively. z is the number of acterizing the response of the system to an external the nearest neighbors. EFM and EAF M are the energies of magnetic field.1 Modern numerical methods of the dy- the ferromagnetic and antiferromagnetic solutions obtained (1) namical mean-field theory for solving realistic electronic by using a mean-field electronic structure approach. Γij is models provide the most reliable information concern- the first order term in the high-temperature expansion of the ing the electronic and magnetic excitation spectra of the spin-spin correlation function. strongly correlated materials. Importantly, by using the Method Expression 2 dynamical mean-field theory (DMFT) the frequency- ∗ ∗ 0 0 12 R φi (x)φj (x)φj (x )φi(x ) 0 and momentum-dependent susceptibilities of a correlated Heitler-London’s exchange Jij = |x−x0| dxdx material can be directly calculated at different external 2 parameters (temperatures and magnetic fields) and com- kin 4tij Anderson’s superexchange Jij = U pared with those measured in the experiment. However, theory13 the solution and reproduction of the experimentally observed susceptibilities do not mean a truly microscopic EFM −EAF M Total energies method J = 4zS2 understanding of the magnetic properties formation. In this respect the determination of the individual magnetic 6 ∂2E Local force theorem Jij = ~ ~ ∂Si∂Sj interactions, Jij of the Heisenberg model is of crucial im- (1) portance. The corresponding Hamiltonian is given by Γ HTE method (this work) J = − ij ij ( 1 S(S+1))2 X ˆ ˆ 3 HˆHeis = JijS~iS~j. (1) ij The development of the methods for calculating the interactions taking the dynamical Coulomb correlations exchange interactions Jij between magnetic moments in 7–11 modern materials is an active research field.3–6 Some im- into account. Recently, a general technique to extract portant examples of methods are listed in Table I. The the complete set of the magnetic couplings by taking into density-functional exchange formula proposed in Ref.6 account the vertices of two-particle Green’s functions and is based on the idea about infinitesimal rotation of the non-local self-energies was developed in Ref.5. By construction the methods reviewed above assume

arXiv:1601.03205v1 [cond-mat.str-el] 13 Jan 2016 magnetic moments from the collinear ground state. The resulting exchange interaction is the response of the sys- some type of the magnetic ordering in the system. How- tem on this perturbation. Being formulated in terms of ever, there are examples when the resulting exchange in- the Green’s function of the system such an approach has teractions are very sensitive to the particular magnetic 10 a number of important options, for instance, it is possi- configuration. Thus one may obtain different sets of ble to calculate the orbital contributions to the total ex- the magnetic couplings for the same system. change interaction. The latter opened a way for a truly Another important methodological problem in this re- microscopic analysis of the magnetic couplings. search field concerns the determination of the magnetic Then in Ref.3 the method for calculating magnetic couplings in a system being in a disordered magnetic couplings within the LDA+DMFT scheme was reported. phase. Numerous magnetic experiments14–16 revealed Such an approach facilitates the analysis of the exchange quantum spin systems which, due to the low-dimensional 2 crystal structure, do not exhibit any sign of the mag- If the system in question can be described by the netic ordering even at very low temperatures. Since the Heisenberg Hamiltonian with localized magnetic mo- electron hopping integral, tij in these materials is much ments then in zero order on β one obtains smaller than the on-site Coulomb interaction, U, then the z z 1 magnetic coupling can be associated with the Anderson’s Tr(Sˆ Sˆ ) = S(S + 1)Nδij, (5) 2 i j 13 4tij 3 superexchange interaction, Jij = U . For intermediate values of t/U (∼ 0.1) one can still use the pure spin model which simply means that the spins are independent at with parameters defined from the high-order strong cou- high temperatures. Here N = (2S + 1)L is the number of pling expansion within perturbative continuous unitary the eigenstates of the Heisenberg Hamiltonian (L denotes transformations.17,18 the number of sites in the model). In turn, the simulation of the exchange interactions in The same idea is used when analyzing the contribution high-temperature paramagnetic phases can be performed of the first order on β to the spin-spin correlation func- by means of the dynamical mean-field theory and its ex- tion that carries the information concerning the exchange tension. For instance, in case of the γ-iron the authors of interaction between the spins, Ref.19 compared the magnetic susceptibilities obtained ˆz ˆz ˆ ˆz ˆz X ~ˆ ~ˆ for Heisenberg model within 1/z expansion and that cal- Tr(Si Sj HHeis) = Tr(Si Sj JmnSmSn) = culated in DMFT approach. To describe the formation of m6=n the local magnetic moment and exchange interaction in 2 ˆz ˆz ˆz ˆz 1 the α-iron, a spin-fermion model was proposed in Ref.20. = JijTr(S S S S ) = JijN S(S + 1) . (6) i j i j 3 Here we report on a distinct method, high-temperature expansion (HTE) method for calculating the isotropic This high-temperature decomposition of the spin-spin exchange interactions in the paramagnetic phase. It is correlation function was used by the authors of Ref.21 to based on the mutual mapping of the high-temperature obtain the expression for the Curie-Weiss temperature. spin-spin correlation functions calculated in Hubbard and As we will show below it can be also used for calculating Heisenberg models. Being formulated for finite clusters Jij. our method can be applied to the investigation of the In the seminal work by Anderson13 the Heisenberg ex- magnetic couplings in magnetic molecules or nanostruc- change interaction is defined in terms of the Hubbard tures deposited on the insulating and metallic surfaces. model parameters, tij and U. For that the author con- It can be also expanded on the calculation of the high- sidered the limit tij U, in which one can obtain the order couplings such as ring exchange. We have used the 4t2 famous superexchange expression, J = ij . developed approach to study the magnetic interactions ij U in quantum spin rings with different hopping configura- Our method for calculating Jij is also based on the tions. using of the Hubbard model that in the simplest one- band form can be written as X U X X Hˆ = t cˆ+ cˆ + nˆ nˆ − µ nˆ ,(7) II. METHODS Hubb ij iσ jσ 2 iσ i−σ iσ ijσ iσ iσ

The main focus in our approach is concentrated on the where σ is the spin index, tij is the hopping integral spin-spin correlation function, between ith and jth sites, U is the on-site Coulomb in- Tr(SˆzSˆze−βHˆ ) teraction and µ is the chemical potential. i j Since our aim is to deﬁne the parameters of the Heisen- Γij = ˆ , (2) Tr(e−βH ) berg model with localized spins, on the level of the Hub- where β is the inverse temperature and Hˆ is the Hamil- bard model it is naturally to start with the atomic limit tonian describing the system in question. Since the para- in which the hopping integral is much smaller than the magnetic regime is of our interest, then we can consider Coulomb interaction, U t. In this case the spectrum of the z-component of the spin operator. Following Ref.21 the eigenvalues can be divided onto low- and high-energy parts that are related to the magnetic excitations of the we consider Γij in the high temperature limit in which ˆ Heisenberg type and charge excitations of the order of the exponent is expanded as e−βH = 1 − βHˆ . Thus one U, respectively. Our method is based on the comparison obtains of the magnetic observables such as spin-spin correlation ˆz ˆz ˆz ˆz ˆ (0) (1) Tr(Si Sj ) − βTr(Si Sj H) functions calculated in Hubbard model and Heisenberg Γij ≈ Γ + βΓ = , (3) ij ij Tr(1) − βTr(Hˆ ) model approaches in the high temperature limit, β → 0. In general, the trace over spin operators, Eq.(4) dif- ˆ where Tr(A) is the trace that corresponds to the summa- fers in the case of the Heisenberg and Hubbard models. tion over all eigenstates of the Hamiltonian of the system, For instance, one should perform the summation over all ˆ H, eigenstates for the Heisenberg model. At the same time X in the case of the Hubbard model one should exclude the Tr(A)ˆ = hΨn|Aˆ|Ψni. (4) n high-energy eigenstates with doubly occupied sites from 3

1 the consideration. The energies of these states are of for the S > 2 we obtain the following expression for the order of U. paramagnetic exchange interaction In the limit of the localized spins t U that we con- z z P PN−1 ˆz ˆz sider, the traces Tr(Sˆ Sˆ ) (in the numerator of Eq.(3)) 0 En hΨn|S S 0 |Ψni i j J = mm n=0 i,m j,m . (10) ˆ ij 1 2 and Tr(H) (in the denominator of Eq.(3)) are similar to N( 3 S(S + 1)) that defined for the Heisenberg model. We are interested in the first order term on the inverse One important problem when calculating the exchange temperature for which one obtains interaction is how to estimate and control the quality of the obtained exchange interactions. It can be done by N−1 solving the corresponding Heisenberg model and by cal- ˆz ˆz ˆ X ˆz ˆz ˆ Tr(Si Sj HHubb) = hΨn|Si Sj HHubb|Ψni = culating the experimentally observed quantities (such as n=0 magnetic susceptibility, magnetization and other). The N−1 comparison of the calculated theoretical dependencies X ˆz ˆz = EnhΨn|Si Sj |Ψni, (8) with the available experimental data is standard way to n=0 define the reability of the constructed Heisenberg model. In our study the exchange interactions for the spin model here E is the eigenvalue of the Hubbard model, Ψ is n n estimated on the basis of the electronic Hubbard Hamil- the corresponding eigenvector and N is the number of tonain. Thus it is natural to estimate the quality of the the eigenstates of the Heisenberg Hamiltonian. constructed Heisenberg model by comparing the eigen- Comparing Eq.(6) and Eq.(8) one can derive the fol- value spectra of the spin model and parent electronic lowing expression for the Heisenberg’s exchange interac- Hamiltonian at different degree of the localization. tion

PN−1 z z En hΨn|Sˆ Sˆ |Ψni J = n=0 i j . (9) ij 1 2 III. EXACT SOLUTION FOR DIMER N( 3 S(S + 1)) Let us analyze the obtained expression for the para- The electronic and magnetic excitation spectra of the magnetic exchange interaction. First of all, it contains dimer that can be obtained analytically is the classical the summation over all eigenstates belonging to the mag- test in the ﬁeld of the strongly correlated materials. Im- netic part of the full Hubbard spectrum. The high-energy portantly, there are a lot of examples of the real low- part of the Hubbard spectrum describing the charge ex- dimensional materials that have the dimer motif.23–25 citations is excluded from the consideration. For each The superexchange interaction in the dimer can be also eigenstate we measure the correlation between two spins. simulated within the experiments with ultracold atoms Such a correlation can be positive or negative depending in optical lattice.26. In such experiments the hopping in- on the spin conﬁguration encoded in the eigenstate and tegral and on-site Coulomb interaction can be varied in a is multiplied by the excitation energy with respect to the wide range. For instance, the authors of Ref.27 explored ground state with E0 = 0. the ratios ranging from the metallic (t/U ∼ 0.1) to insu- The expression for the exchange interaction Eq.(9) was lating (t/U 1) regimes when performed the quantum obtained by comparing the spin-spin correlation func- simulations on the two-dimensional optical lattice. tions of the Heisenberg and Hubbard models in the limit Within the proposed method we are interested in t U. Despite of this, in some cases the calculated N = 4 lowest magnetic eigenstates of the Hubbard exchange interactions, as we will show below, lead to model, they are presented in Table II, where the fol- good agreement of the Heisenberg and Hubbard eigen- 2 1 U(1−γ) lowing notations are used C0 = 2(1+2 ) , − = 4t , value spectra even for t ∼ 1. Importantly, one can ana- − U q 16t2 lyze the magnetic interactions in the strongly correlated γ = 1 + U 2 . The eigenvalues are the following: regime. U E0 = 2 (1 − γ) − 2µ, E1 = E2 = E3 = −2µ. For transition metal oxides the typical ratio between hopping integral and on-site Coulomb interaction is of order of 0.03. It was shown that in case of 5d irid- TABLE II. Four lowest eigenstates of the Hubbard model for ium oxides22 this value can be about two times larger, the dimer. 0.07 and the implementation of the ordinary superex- n Ψn change theory is questionable. The simulation of the U(1−γ) 0 [(| ↓ ↑ i − | ↑ ↓ i) + (| ↓↑ i + | ↓↑ i)] · C0 magnetic interaction in metallic systems is another com- 4t plicated problem, we deal with the situation when the 1 | ↑ ↑ i hopping integrals are of the same order of magnitude as 2 | ↓ ↓ i the Coulomb interaction. 3 √1 · (| ↑ ↓ i + | ↓ ↑ i) In the case of the many-band Hubbard model the spin 2 operator of the ith site in Eq.(9) can be written as the ˆz P ˆz sum of the orbital contributions, Si = m Si,m. Thus By using the developed method Eq.(9), we obtain the 4

FIG. 1. Comparison of the excitation spectra of the Hubbard FIG. 2. (a) Comparison of the eigenvalue spectra obtained model (solid lines) and Heisenberg models with exchange in- from the solution the Hubbard model (black solid line) and teraction calculated by using the developed method (green the Heisenberg models with parameters calculated by the de- triangles) and superexchange Anderson approach (blue rhom- veloped approach Eq.(9) (green triangles), local force theorem bus) method Eq.(13) (red circles) and Anderson’s superexchange theory (blue rhombus). (b) Magnetization as a function of the localization. following exchange interaction in the dimer U J = − (1 − γ) . (11) According to the local force theorem the exchange inter- 2 action is given by This value is exactly the excitation energy from the sin- EF glet to triplet state of the dimer, E1 − E0. Thus our 1 Z J = Im(V˜ G↓ V˜ G↑ ) dε, (13) method can be used to construct a Heisenberg model ij 2πS2 i ij j ji reproducing the Hubbard model spectrum for any rea- −∞ sonable ratio of kinetic and Coulomb interaction param- tij here E is the Fermi level, V˜ = V ↑ − V ↓ denotes the eters, U . On the other hand the Heisenberg model con- F i i i structed by means of the Anderson’s superexchange the- spin-dependent Hartree-Fock potential calculated self- 2 ↑,↓ ↑,↓ −1 4tij consistently and G (ε) = (ε − H ) is the Green’s ory, Jij = results in the spectrum deviating from HF U function of the Hartree-Fock Hamiltonian. Unlike our that of the original Hubbard model at tij > 0.2. U approach Eq.(9), this formula requires presence of a magnetic order in the system. A. Comparison with the Hartree-Fock solution In the case of the dimer, the exchange interactions obtained by using local force approach give excellent agreement with the spectrum of the Hubbard model for One of the important results of modern magnetism t 6 U <0.2 (Fig. 2). For larger values of the hopping in- theory was the development of the local force theorem tegrals the averaged magnetic moment is strongly sup- for calculating the exchange interactions. Such an pressed and becomes almost zero at t/U ∼ 0.5. These approach give reliable results and is widely used for results indicate the limits of the applicability of the mean- simulation magnetic properties of the transition metal 7–10,22 field Green’s function approach for calculating the ex- compounds. Thus the next step of our investiga- change interaction in strongly correlated systems. tion was to compare the results of the high-temperature expansion method we developed and those obtained by using the density-functional exchange formula. For these purposes we have chosen the dimer system. IV. SOLUTIONS FOR TRIANGLE AND TRIMER Since the method based on the local force theorem requires a non-zero magnetization of the system we used the Hartree-Fock approximation to solve the Hubbard Triangle is another example of the model for which model, Eq.(7) we obtain excellent agreement of the electronic and spin eigenvalue spectra. The Heisenberg model spectrum for ˆ X + X the triangle consists of four-fold degenerate ground and HHF = tijcîσcˆjσ + U hnî−σinîσ. (12) ijσ iσ four-fold degenerate excited states. As in the case of the 5

Hubb Hubb Heis Heis En −En0 = En −En0 and obtained the following expressions for the magnetic couplings in the trimer: 2 J = (E − E ) nn 3 4 0 (14) Jnnn = Jnn − (E2 − E0)

where Jnn and Jnnn are exchange interactions between nearest neighbors and next-nearest neighbors in the trimer, respectively. One can see that the leading exchange interaction Jnn between the nearest neighbours is related to the energy splitting between ground state and highest excited state belonging to the magnetic part of the whole electronic spectrum. The situation with the next nearest-neighbour coupling is more complicated. In addition to the E4 − E0 that is related to the leading exchange interaction, it also has the ferromagnetic contribution from the intermediate excited state, E2 − E0. FIG. 3. (a) The calculated exchange interactions between As we will show below the similar picture is observed in nearest neighbours, Jnn and next-nearest neighbours, Jnnn in quantum spin rings. the trimer. (b) Comparison of the eigenvalue spectra for the trimer. Solid lines denote the Hubbard model spectrum. Blue rhombus, green triangles and red dashed lines correspond to The Heisenberg model solutions with different sets of the ex- V. QUANTUM SPIN RINGS change interactions are presented by blue rhombus (the An- derson’s superexchange) and green triangles (the developed In this section we present the results of computer simu- HTE method, Eq.(9)). lations concerning the magnetic interactions in the finite quantum clusters with ring geometry. The theoretical interest in these systems is due to the synthesis and study dimer, the exchange interaction between spins in the tri- of the magnetic properties of the molecular magnets with angle is defined by the corresponding splitting between ring geometry.28–31 Such systems demonstrate a number excited and ground state levels. From Fig.1 the Heisen- interesting and complex phenomena, quantum spin tun- berg model, which we constructed by using the HTE neling, long-time spin relaxation, topological spin phases method, precisely reproduces the magnetic part of the (Berry phases) and others. In this respect the micro- Hubbard model. scopic understanding of the intra-molecular magnetic Trimer. The situation becomes more complicated if couplings plays a crucial role.32 On the other hand the we consider the trimer with the nearest neighbor hop- spin rings are also of great practical interest, since they ping presented in Fig.3. In contrast to the triangle the can be used as building elements for novel quantum com- ground state of the trimer is two-fold degenerate. In munication technologies33 and for engineering quantum turn the highest excited state in the magnetic part of the memory that is stable against noise and imperfections.34 eigenspectrum is four-fold degenerate. As we will show In our study we have simulated the magnetic interac- below the two-fold intermediate excited level is related tions in the quantum rings describing by the Hubbard to the interaction between next nearest neighbours. models with different hopping setups presented in Fig.4. For such hopping setup, within Anderson’s superex- They can be realized in the quantum simulation experi- change theory we obtain the antiferromagnetic coupling, ments on ultracold atoms in optical lattices.35 2 4tij Jij = U between nearest neighbors in the trimer. To define the interaction between next nearest neighbors one should use the fourth-order perturbation theory on the A. Rings with nearest neighbors hoppings hopping. The situation becomes more complicated if the condition tij U is not fulfiled. On the other hand First, we analyze the results of the simulations for by using the developed method Eq.(9) we obtain anti- quantum rings describing the Hubbard model with the ferromagnetic nearest and second nearest neighbors ex- only nearest neighbor hopping integral. Similar to the changes. The solution of the corresponding Heisenberg case of the dimer and trimer our method leads to bet- model leads to perfect agreement between the spin and ter agreement between Heisenberg and low-energy Hub- Hubbard model spectra up to large values of the ratio bard model spectra than the others. Fig.5 gives the com- tij U . parison of the eigenvalues spectra calculated by different In the case of the trimer we can also explicitly relate methods in the case of the 5-site ring. One can see that the exchange interactions with the eigenvalues spectrum the high-temperature expansion method reproduces the of the Hubbard model. For that we used the condition electronic Hamiltonian spectrum up to t/U = 0.28. At 6

FIG. 4. Schematic representation of the hopping setups for the Hubbard Hamiltonian simulations. (Left) The ring model with the nearest neighbor hoppings. (Right) All-to-all conﬁg- uration in which all the hoppings between sites are the same.

FIG. 6. Contributions from the diﬀerent eigenstates of the Hubbard model to the nearest neighbor (top) and next- nearest neighbour (bottom) exchange interactions calculated for 4-site ring with t/U = 0.1.

(b), in which the Anderson’s superexchange theory with zero Jnnn leads to the eigenvalue spectrum deviating from the Hubbard model one. The expression for the paramagnetic exchange interaction, Eq.(9) that we derived contains the summation of the eigenstates belonging to the low-energy magnetic part of the Hubbard model spectrum. It is important to analyze the contribution of the individual eigenstates to the resulting exchange interaction. From Fig.6 one can see that there are ferromagnetic contributions that partially compensate the antiferromagnetic ones. Inter- estingly, the contributions from the highest excited states FIG. 5. (a) The calculated exchange interactions between are almost the same for the Jnn and Jnnn couplings. As nearest neighbours, Jnn and next-nearest neighbours, Jnnn in it was shown in the case of the trimer, the intermediate the 5-site ring. (b) Comparison of the eigenvalue spectra for excited eigenstates produce the ferromagnetic contribu- the 5-site ring. Solid lines denote the Hubbard model spectrum. Blue rhombus and green triangles correspond to the tions to Jnnn. Heisenberg model solutions with diﬀerent sets of the exchange interactions: blue rhombus (Anderson’s superexchange) and green triangles (developed method, Eq.(9)). B. Rings with all-to-all hoppings.

By the example of the results for the 5-site ring pre- this value the high- and low-energy parts of the spec- sented in Fig.5 (b) one can see that the quantum rings trum overlap, which prevents us from determining the with nearest neighbor hopping demonstrate rather com- exchange interaction. plicated spectra. However, for practical purposes, for In the hopping setup we used (Fig.4, left) there are instance, to construct a quantum logic device, we need a hopping integrals between nearest neighbors only. Never- system with the excitation spectrum as simple as possi- theless each site has non-zero antiferromagnetic exchange ble. In the case of the quantum rings that we consider the interaction with all the other sites in the ring (we denote excitation spectrum can be considerably simplified by in- them Jnnn). Fig.5(a) demonstrates the behavior of such troducing the same hopping integral for all the bonds in diagonal couplings at different t/U ratios in comparison the quantum Hamiltonian. It is so-called all-to-all hop- with the leading exchange interaction between nearest ping configuration (Fig.4). neighbors, Jnn. Despite of the fact that the coupling Jnnn The simplest Heisenberg Hamiltonian with two- growths much slower than the nearest-neighbour one it spin exchange interactions constructed by the high- cannot be neglected when constructing the Heisenberg temperature expansion method gives the eigenvalue spec- model at t/U > 0.15. It can be clearly seen from Fig.5 trum that is coincident with the Hubbard model one up 7

FIG. 7. Excitation spectra of all-to-all systems. Solid lines denote the Hubbard model spectrum. Green triangles correspond to the Heisenberg model solutions with the exchange interactions calculated by the high-temperature expansion method, Eq.9.

diate site.

VI. CONCLUSION

We propose the method for calculations of the magnetic interactions in the paramagnetic phase. Being formulated in the high-temperature and localized spin limits our approach can be used for constructing the spin Hamiltonian in a wide range of the t/U ratios. It was shown by the classical examples such as the dimer and triangle finite clusters. By using the proposed method we investigated the magnetic couplings in quantum spin rings with different hopping configurations. Our method- ring ological and calculation results will be useful for analy- Jnn FIG. 8. The calculated ratio all−to−all demostrating the Jnn sis of the data obtained in experiments with ultracold contribution of the indirect hopping processes to the two-spin fermions that provide unique possibility to measure and interaction. control the spin-spin correlation function between two sites in optical lattice.35 The proposed scheme can be also applied for simulating the magentic couplings be- to t/U = 0.07 (Fig.7). For larger values of t/U we ob- tween impurities in metallic host. For that instead of the serve the deviation of the spin and electronic models that Hubbard model one should solve two-impurity Anderson mainly concerns intermediate excited levels. The prob- model. lem may be resolved by introducing the high-order mul- tispin interactions (four-spin and six-spin).18 The pair exchange interaction between nearest neighbors in a quantum ring has the direct contributions, VII. ACKNOWLEDGMENTS. proportional to tijtji and high-order non-direct ones, P k tiktkj, where the site index k 6= i, j. In case of the We acknowledge fruitful communications with Mikhail configurations with all-to-all hoppings the non-direct pro- Katsnelson, Alexander Lichtenstein, Andrea Secchi, cesses become very efficient and strongly contribute to Alexander Tsirlin, Andrey Katanin, Sergey Brener, Igor the exchange interaction between two spins. It can be Solovyev and Vladimir Anisimov. The work is supported seen from Fig.8. For each pair in the N-site ring there by the grant program of the Russian Science Foundation are N-2 non-direct exchange path including one interme- 15-12-20021.

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