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A High-Temperature Expansion Method for Calculating Paramagnetic Exchange Interactions

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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 pro- posed. 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 expres- sion 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 ex- change 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 = jx−x0j 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 ob- served 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 ^ ^ 3 H^Heis = 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(S^ S^ ) = 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 ^z ^z ^ ^z ^z X ~^ ~^ 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 ^z ^z ^z ^z 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 H^ = t c^+ c^ + n^ n^ − µ n^ ;(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(S^zS^ze−βH^ ) teraction and µ is the chemical potential. i j Since our aim is to define the parameters of the Heisen- Γij = ^ ; (2) Tr(e−βH ) berg model with localized spins, on the level of the Hub- where β is the inverse temperature and H^ 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 ^ Heisenberg type and charge excitations of the order of the exponent is expanded as e−βH = 1 − βH^ . Thus one U, respectively. Our method is based on the comparison obtains of the magnetic observables such as spin-spin correlation ^z ^z ^z ^z ^ (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(H^ ) model approaches in the high temperature limit, β ! 0. In general, the trace over spin operators, Eq.(4) dif- ^ where Tr(A) is the trace that corresponds to the summa- fers in the case of the Heisenberg and Hubbard models.
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