Prediction of the Curie temperature considering the dependence of the phonon free energy on magnetic states
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Prediction of the Curie temperature considering the dependence of the phonon freeenergy on magnetic states
Tomonori Tanaka ∗ and Yoshihiro Gohda † Department of Materials Science and Engineering,Tokyo Institute of Technology, Yokohama 226-8502, Japan (Dated: June 5, 2020)Prediction of the Curie temperature is of significant importance for the design of ferromagneticmaterials. Even though the Curie temperature has been estimated using the Heisenberg model,magnetic exchange coupling parameters widely used is thus far based on first-principles calculationsat zero temperature. In the explicit consideration of temperature effects, it is important to minimisethe total free energy, because the magnetic and phonon free energies correlate with each other. Here,we propose a first-principles thermodynamic approach to minimise the total free energy consideringboth the influences of magnetism on phonons and the feedback effect from phonons to magnetism.By applying our scheme to bcc Fe, we find a significant reduction of the Curie temperature due tothe feedback effect. This result inevitably enforces us to change our convention as follows: we shoulduse exchange coupling constants for the disordered local moment state, not for the ferromagneticstate, in the prediction of the Curie temperature. Our results not only change the fundamentalunderstanding of finite-temperature magnetism but also provide a general framework to predict theCurie temperature more accurately.
The Curie temperature ( T C ) is one of the essentialproperties of ferromagnetic materials because it char-acterises their applicability and performance . Themethod of predicting T C is, therefore, important not onlyfor a fundamental understanding of ferromagnetic mate-rials but also for the material design for applications.A typical technique for predicting T C is a downfoldingmethod from first-principles calculations to an effectivelattice model as below: Firstly, one derives exchange cou-pling constants ( J ij ) by applying Green’s function-basedmethods or a frozen magnon approach . Secondly,one builds an effective lattice model such as the Heisen-berg model and assign J ij to the model. Finally, onesolves the model analytically or numerically and esti-mates T C . This technique is applied to a broad rangeof materials, such as 3 d transition metals and rare-earth magnets . Such many studies demonstratedthat the prediction technique has some predictive accu-racy.Such a technique usually does not include temperatureeffects on magnetic interactions. Moreover, temperature-induced interactions between magnetism and other exci-tations such as phonons sometimes make the accurateprediction of T C difficult. At a high-temperature rangearound T C , there are two kinds of interaction betweenmagnetism and phonons. One is the effect of thermalatomic displacements on J ij . The change in J ij ob-viously modifies T C . The other interaction is the effect ofmagnetic disordering on phonon frequencies. Some fer-romagnetic materials such as bcc Fe and Pd Fe showsphonon softening at elevated temperatures near T C21–23 .Some research groups approached this phenomenon bydifferent theoretical methods and achieved the sameconclusion: The phonon softening is due to magnetic dis-ordering near T C . Regarding the predictive accuracy of T C , the importance of the former interaction is easilyunderstandable, whereas the latter interaction does not apparently seem to be related to T C . However, we willrecognise the phonon softening due to magnetic disorder-ing is closely related to T C by standing a thermodynamicviewpoint.Thermal equilibrium states at finite temperature cor-respond to the minimum of the total free energy atgiven conditions. This is usually called as the mini-mum principle for the free energy. Usual proceduresto study finite-temperature magnetism is constructinga magnetic Hamiltonian and deriving thermodynamicquantities such as the magnetic energy and the magneti-sation. This series of procedures is equal to interpret thatequilibrium magnetic quantities are determined throughthe magnetic free energy only. This interpretation, how-ever, collapses in the systems that exhibit the phononsoftening due to magnetic disordering. The phonon fre-quencies are directly related to the phonon free energy.Thus the phonon softening due to magnetic disorderingmeans that magnetic states affect the phonon free energyas well as the magnetic free energy. As a result, equi-librium magnetic states should be determined throughnot only the magnetic free energy but also the phononfree energy, according to the minimum principle for thefree energy. We call this effect of phonons on equilib-rium magnetic states through the change of the phononfree energy as a thermodynamic feedback effect. Thisfeedback effect surely affects T C as a consequence of thechange of equilibrium magnetic states. However, the sig-nificance of the feedback effect on T C is unclear becausethe existence of the effect has been overlooked.In this article, we propose a thermodynamic formula-tion to treat the feedback effect from phonons to mag-netism. The formulation results in a simple optimisa-tion problem for the total free energy. The ingredientsto solve the problem are evaluated by first-principlesphonon calculations and Monte Carlo simulations basedon the Heisenberg model. By applying the formulation tobcc Fe, we demonstrate that T C of bcc Fe significantly de-creases by nearly 580 K. This result proves the feedbackeffect is crucial for accurate prediction of T C . We also dis-cuss the relationship between the predictive accuracy of T C and reference magnetic states in the derivation of J ij .Remarkably, we find a significant overestimation of T C in a paramagnetic disordered local moment (DLM) stateis rather a correct tendency. Quantitative description offinite-temperature magnetism plays an important role inboth basic and applied materials science. Therefore, ourresults have an impact on the fundamental understand-ing of magnetism and materials design for ferromagneticmaterials.We organise the following part of this paper as below.Firstly, we introduce a new thermodynamic formulationto treat the thermodynamic feedback effect. Our formu-lation based on the minimum principle for the free energyis justified through the Legendre transformation and re-sults in a simple optimisation problem. Next, we evaluatethe magnetic entropy and the phonon free energy of bccFe as functions of the magnetic energy. These functionsare needed to solve the optimisation problem. Finally,we evaluate the equilibrium magnetic energy around T C by solving the optimisation problem. The shift of T C ofbcc Fe is estimated from the results of the equilibriummagnetic energy. THERMODYNAMIC FORMULATION FORMAGNETIC MATERIALS
In conventional thermodynamic approaches for mag-netic materials, the phonon and magnetic contributionsare assessed independently. We start from this typicalcase for comparison with our formulation. The funda-mental relation is written as E tot ( S ph , S mag ) ≈ E ph ( S ph ) + E mag ( S mag ) , (1)where E is the energy and S is the entropy. The sub-scripts tot, ph and mag represent total, phonon and mag-netic, respectively. Here, we consider the Gibbs free en-ergy, G ( T, p, H ) = E − T S + pV − µ M H, (2)where T represents the temperature, p the pressure, V the volume, M the magnetisation, H the external mag-netic field and µ the vacuum permeability. In the fol-lowing, we derive the formalism for p = 0 and H = 0,but the discussion remains unchanged for the case withfinite external fields p and H . The Gibbs free energy G T = T equilibrium E mag by the Heisenberg model equilibrium E mag by the minimisation of the total free energy G ph (T , E mag )+ G mag (T , E mag ) F r ee ene r g y E mag G mag (T , E mag ) FIG. 1. Schematic image of the free energy minimisation ata temperature T . Within a common framework using theHeisenberg model, the equilibrium magnetic energy ( E mag ) iscorresponding to the minimum of the magnetic free energy, G mag (blue line). On the other hand, the equilibrium mag-netic energy in our scheme is corresponding to the minimumof the total free energy, G ph + G mag (orange line). is derived by applying the Legendre transformation. G tot ( T ) = min S ph ,S mag { E ph ( S ph ) − T S ph + E mag ( S mag ) − T S mag } (3)= min E ph ,E mag { E ph − T S ph ( E ph )+ E mag − T S mag ( E mag ) } (4)= min E ph ,E mag { G ph ( T, E ph ) + G mag ( T, E mag ) } (5)= G ph ( T ) + G mag ( T ) . (6)Here we used the one-to-one correspondence between en-ergy and entropy for fixed other thermodynamic param-eters such as V and M . The independent assessmentin conventional approaches is based on this trivial for-mulation. Next, we incorporate the dependence of thephonon free energy on magnetic states. We assume thatthe magnitude of the interaction between magnetic disor-dering and phonon frequencies can be written as thermo-dynamic quantities of the magnetic part. K¨ormann et al. proposed a solid treatment with this assumption . Theytreated the forces on each atom as a function of the mag-netic energy. As a result, the phonon frequencies, conse-quently the phonon free energy, have the dependence onthe magnetic energy (see Methods). Thermodynamicallyspeaking, their treatment means the phonon energy de-pends not only on the phonon entropy but also on themagnetic entropy. The fundamental relation thus can bewritten as E tot ( S ph , S mag ) ≈ E ph ( S ph , S mag ) + E mag ( S mag ) . (7)In principle, E mag also has a dependence on S ph . This de-pendence can be regarded as influences of thermal atomicdisplacements on J ij . If we want to incorporate thiseffect into the thermodynamic formulation, we have to −20−40−60−80−100−120−140 −160 E m ag ( m e V / a t o m ) Temperature (K) E mag (meV/atom) S mag ( E mag ) S m ag ( m e V / K / a t o m ) S m ag ( m e V / K / a t o m ) S mag ( T ) E mag ( T ) T c by Monte Carlo1522 K a b ~ ~ FIG. 2. Thermodynamic quantities of bcc Fe obtained by the rescaled Monte Carlo method. (a) Energy and entropy vs.temperature. (b) Entropy vs. energy. The theoretical Curie temperature T C was identified from the peak of the specific heat. express the magnitude of the effect as a thermodynamicquantity such as S ph . However, the correspondence be-tween the thermal displacements and S ph is not obvious.We thus focus only the dependence of E ph on S mag .We apply the Legendre transformation as before. G tot ( T ) = min S ph ,S mag { E ph ( S ph , S mag ) − T S ph + E mag ( S mag ) − T S mag } (8)= min S mag { G ph ( T, S mag ) + E mag ( S mag ) − T S mag } (9)= min E mag { G ph ( T, E mag ) + E mag − T S mag ( E mag ) } (10)= min E mag { G ph ( T, E mag ) + G mag ( T, E mag ) } . (11)Note that the entropy (or energy) and the tempera-ture can be treated as independent variables during theminimisation procedure. The thermodynamic relation-ship between the entropy and the temperature, such as ∂G/∂T = − S , holds after the minimisation, i.e. afterthe Legendre transformation. The last expression is veryintuitive from a thermodynamic viewpoint: The equilib-rium magnetic energy at a temperature T is determinedto minimise the total free energy (Fig. 1) asargmin E mag [ G ph (T , E mag ) + E mag − T S mag ( E mag )](12)= argmin E mag [ G ph (T , E mag ) + G mag (T , E mag )] . (13) EVALUATIONS OF THE MAGNETIC ENTROPYAND THE PHONON FREE ENERGY
We demonstrate the significance of the dependence ofthe phonon free energy on magnetic states for bcc Fe as an example. As a starting point, we evaluate the mag-netic entropy and the phonon free energy depending onthe magnetic energy ( S mag ( E mag ) and G ph ( T, E mag )), inorder to solve the minimisation problem in equation (12).To obtain S mag ( E mag ), we carried out the rescaledMonte Carlo method based on the Heisenberg model.This method brings thermodynamic quantities derivedfrom classical Monte Carlo simulations closer to thosefrom quantum Monte Carlo simulations. The exchangecoupling constants ( J ij ) in the Heisenberg model are de-rived from the paramagnetic disordered local moment(DLM) state (see Methods). The magnetic energyand entropy as functions of lattice-model temperature e T are shown in Fig. 2 (a). The theoretical T C (1522K) is higher than the experimental value (1043 K).Such overestimation has also been reported in previousstudies using the DLM state. The overestimationhas been recognised as a disadvantage of the DLM state,and it will be discussed later associated with our results.Since this magnetic system does not show the first-orderphase transition, the one-to-one correspondence holds be-tween not only E mag and S mag but also e T , E mag ↔ e T ↔ S mag . (14)We constructed the function S mag ( E mag ) (Fig. 2 (b)) byusing this relationship.Phonon frequencies depending on the magnetic energy( G ph ( T, E mag )) can be calculated by using first-principlesphonon calculations and Monte Carlo simulations follow-ing the previous research (see Methods). The phonondispersions and the phonon density of states of bcc Fe de-pending on the magnetic energy are shown in Fig. 3. Thedependence of the frequencies on the magnetic energyis represented through the parameter α (see Methods).The calculated phonon dispersions in the ferromagnetic(FM, α = 1) and paramagnetic DLM (PM, α = 0) lim-its are consistent with the previous research . Once the F r equen cy ( m e V ) Γ ΓΝ Η P Γ α = 1.0 (FM) α = 0.8 α = 0.6 α = 0.4 α = 0.2 α = 0.0 (PM) Phonon density of states
FIG. 3. The phonon dispersions and the phonon density ofstates of bcc Fe from the ferromagnetic state (FM, α = 1) tothe paramagnetic state (PM, α = 0). The definition of α iswritten in Methods. phonon frequencies at various magnetic energies (i.e. atvarious α ) are calculated, the phonon free energy can beevaluated from the analytical form, G ph ( T, E mag ) = k B TN q X q ,j log (cid:20) (cid:18) ~ ω q j ( E mag )2 k B T (cid:19)(cid:21) , (15)where k B represents the Boltzmann constant, ω q j ( E mag )the phonon frequency of the j -th branch at the wave num-ber q as a function of E mag and N q the total number of q points. Note that the more disordered the magneticstate is, the lower the phonon frequencies are. This ten-dency means the phonon free energies of paramagneticstates are smaller than that of the ferromagnetic statebecause of the monotonicity of the phonon free energyfor the phonon frequency. Consequently, paramagneticstates are thermodynamically stabilised by the phononsoftening effect. TOTAL FREE ENERGY MINIMISATION
We are now able to proceed to the total free en-ergy minimisation in equation (12) by using the func-tions G ph ( T, E mag ) and S mag ( E mag ). The minimisationprocedures are simple. Firstly, we fix the temperatureat T . Secondly, we calculate the total free energy( G ph (T , E mag )+ E mag − T S mag ( E mag )) for various E mag values. The variable range of E mag is from the ferro-magnetic limit to the paramagnetic limit. Thirdly, wefind E mag corresponding to the minimum total free en-ergy. The orange line in Fig. 1 is a visualisation of thesesteps. Finally, repeat these steps for a temperature rangearound T C .The equilibrium magnetic energies of bcc Fe obtainedby two difference methods are shown in Fig. 4: One isthe minimisation of the total free energy G mag + G ph ;the other is the Monte Carlo simulations based on theHeisenberg model (the result is the same as the blueline in Fig. 2 (a)). Note that the result from the lat-ter method is corresponding to that of considering only G mag in the minimisation of the free energy. The equi-librium magnetic energies obtained by the minimisationof the total free energy are larger than those of consid-ering only G mag . This is, as mentioned before, due tothe stabilisation of paramagnetic states by the phononsoftening effect, and the magnitude of the stabilisationindicates that the phonon contribution is not negligibleat all in the determination of equilibrium magnetic statesaround T C The stabilisation of paramagnetic states leads to a de-crease in T C . As shown in Fig. 4, T C in the resultsof the minimisation of the total free energy (946 K) islower than that of considering only G mag (1522 K), andthe magnitude of the decrease reached nearly 580 K. No-tably, T C of considering both G mag and G ph is dramati-cally closer to experimental value (1043 K) than that ofconsidering only G mag , i.e. T C in the Heisenberg model.Although T C = 946 K is still underestimated the ex-perimental value to some extent, the anharmonicity ofphonons probably compensates for the deviation. Heine,Hellman and Broido investigated the phonon soften-ing phenomenon in bcc Fe with including anharmonic ef-fects. They show that at 1043 K, where the anharmonic-ity is effective, the differences between the frequencies ofthe ferromagnetic and paramagnetic states are reducedcompared with those at 300 K. Thus the difference ofthe phonon free energies between the ferromagnetic andparamagnetic states is also reduced. This consequentlymakes the degree of the decrease in T C smaller than ourresult. The underestimation in our result is, therefore, E m ag ( m e V / a t o m ) −20−40−60−80−100−120−160−140 0 250 500 750 1000 1250 Temperature (K) argmin[ G mag + G ph ] argmin[ G mag ] T c1522 K T c946 K
576 K
FIG. 4. The equilibrium magnetic energy of bcc Fe as a func-tion of temperature. Orange line represents the equilibriummagnetic energies obtained by the minimisation of the totalfree energy. Blue line represents the equilibrium magnetic en-ergies by the minimisation of the magnetic free energy (thesame as the blue line in Fig. 2 (a)). The Curie temperature T C in the minimisation of the total free energy is defined asthe temperature with the same magnetic energy as the Heisen-berg model. This definition is reasonable because the samemagnetic energy gives the same magnetic ordering as long as J ij values do not vary. qualitatively correct.The substantial decrease in T C gives a doubt on theusual recognition of the accuracy in prediction techniquesfor T C . Roughly speaking, there are three reference mag-netic states in the derivation of J ij : ferromagnetic state,paramagnetic DLM state and conical spin-spiral states.The former two are used within Green’s function-basedmethods , whereas conical spin-spiral states are usedwithin the frozen magnon approach . We can sum-marise the relationship between the reference states andpredictive accuracy of T C in bcc Fe: The ferromagneticstate and spin-spiral states give T C near the experimentalvalue , whereas the paramagnetic DLM stateoverestimates T C significantly . Therefore, the fer-romagnetic state and spin-spiral states have been recog-nised to have an enough predictive accuracy of T C re-garding bcc Fe. However, our study clearly shows thatthis recognition is questionable because the contributionof the phonon free energy decreases T C of bcc Fe sig-nificantly. The substantial decrease in T C suggests theDLM state shows correct tendency regarding T C predic-tion, rather than the ferromagnetic state and spin-spiralstates. Note that this suggestion is of great importancefor theory of finite-temperature magnetism as follows. Inthe development of the theory, T C of bcc Fe has beenrecognised as a touchstone: Whether the predicted T C ofbcc Fe agrees with the experimental value or not has beenan element to examine the validity of a new theory. Ourresult, however, indicates such an examination way is in-appropriate. Instead, an appropriate judgment criterionis as follows: Without considering the phonon soften-ing, a theory that accurately describes finite-temperaturemagnetism must overestimate T C of bcc Fe.Our thermodynamic formulation becomes complete ifwe incorporate the dependence of E mag on S ph . This de-pendence may be related to the effect of thermal atomicdisplacements on J ij . Ruban and Peil studied this ef-fect by combining J ij calculations and molecular dynam-ics. They clearly showed J ij values of bcc Fe were reducedby atomic displacements, and consequently, T C was alsolargely decreased compared with the case of excludingthermal atomic displacements. The dependence of E mag on S ph is thus important and intriguing from a thermo-dynamic viewpoint. However, a concrete expression ofthis dependence is yet to be obtained. CONCLUSIONS
We have quantitatively evaluated the thermodynamicfeedback effect from phonons to magnetism on T C regard-ing bcc Fe. The phonon softening due to magnetic dis-ordering lead to the stabilisation of paramagnetic states.As a result, T C of bcc Fe was decreased by nearly 580 Kfrom the value in the case of ignoring the feedback effect,i.e. the value for the Heisenberg model. This deviationin bcc Fe is of great importance because bcc Fe is recog-nised as a touchstone for the study of finite-temperature magnetism. We stress two important knowledge regard-ing the prediction of T C : (i) An appropriate theory ofmagnetism without considering the contribution of thephonon free energy must overestimate T C in bcc Fe, con-trary to conventional understanding. (ii) We should use J ij for the DLM state rather than for the ferromagneticstate in the accurate description of T C .Finally, we mention the applicability of our thermody-namic formulation. We focused on bcc Fe in this study,but our formulation is not restricted to it. It is intrigu-ing to apply the formulation to other magnetic materialssuch as permanent magnets in which T C is critically im-portant. In addition, the core concept of the formulationcan be applied to other interacting excitation phenom-ena, not only the interaction between phonons and mag-netism: If a contribution (X) affects other contribution(Y) and changes the free energy of Y, the thermal equi-librium state of X is also affected through the minimumprinciple for the free energy. In our study, X is mag-netic states, and Y is phonons. Therefore, the concept ofour thermodynamic formulation can be applied to otherinteracting excitations if one can express the magnitudeof the interaction as thermodynamic quantities ( E mag inour study). The formulation will be helpful for a quanti-tative description of the finite-temperature properties ofmaterials. METHODSFirst-principles phonon calculations.
All of the phonon calculations were carried out withinthe harmonic approximation. To evaluate the phononfrequencies at an intermediate magnetic ordering, we em-ployed a force-averaging method . In this method, theatomic forces at an intermediate magnetic ordering aredetermined by mixing the forces at the ferromagnetic(FM) and paramagnetic (PM) DLM states. Followingthe reference , the atomic forces at an intermediate mag-netic ordering can be written as F i ≈ α F FM i + (1 − α ) F PM i , (16)where F i is the atomic force vector on i -th atom and α is a mixing parameter. They also proposed a solidexpression of α by using the magnetic energy ( E mag ) asbelow: α = E mag − E PMmag E FMmag − E PMmag , (17)where E PMmag ( E FMmag ) is the magnetic energy at high (low)temperature limit in the Heisenberg model. In the orig-inal paper , they assumed the temperature dependenceof E mag is determined by the Monte Carlo results only.Therefore, α was treated as a function of temperature( α = α ( e T )). This is equivalent to that the equilibriummagnetic energy at a temperature is determined to min-imise the magnetic free energy, not total free energy. Onthe other hand, in our study, α is not regarded as a func-tion of temperature but is interpreted as a function ofenergy ( α = α ( E )). This interpretation allows that thephonon free energy G ph can be regarded as a functionof the magnetic energy ( G ph = G ph ( T, E mag )). The tem-perature dependence of E mag is determined after the min-imisation of the total free energy in equation (12). Thisinterpretation is the most important key for solving theminimisation problem in the minimum principle for thefree energy.The paramagnetic DLM state in the phonon calcula-tions was mimicked by a special quasirandom structure on the spin configuration (up and down) as obtained fromthe ATAT package . The atomic forces were calculatedby the direct method . We used the 3 × × a = 2.86 ˚A was derived by combiningthe relaxed lattice constant and experimental lattice ex-pansion ratio at T = 1043 K . Although such determi-nation procedure of lattice constant probably gives somepressure even in the framework of the quasiharmonic ap-proximation, we assume its effect is minor and fixed thevolume. First-principles calculations were based on den-sity functional theory within the projector augmentedwave method , as implemented in the VASP code .For the exchange-correlation functional, the generalised gradient approximation parametrised by Perdew, Burkeand Ernzerhof was used. The cutoff energy 400 eV and9 × × k -point grid for the supercell were used for theforce calculations. The derivation of force constants andthe calculations of the phonon free energy were performedby using the ALAMODE code . Calculations of exchange coupling constants.
Exchange coupling constants J ij in the Monte Carlosimulations were derived with magnetic force theorem and the Korringa-Kohn-Rostoker (KKR) Green’s func-tion method along with the coherent potential ap-proximation (CPA) , implemented in the AkaiKKRcode . The exchange-correlation functional wastreated within the local density approximation . Thelattice constant was set to be the same one as in thephonon calculations. Paramagnetic DLM state wasemployed as a reference magnetic state in the derivationof J ij . Calculated J ij values are listed in Table I. Monte Carlo simulations.
To evaluate the magnetic entropy as a function of themagnetic energy, we carried out the classical Monte Carlosimulations based on the Heisenberg model H = − X ( i,j ) J ij e i · e j , (18)where J ij denotes the exchange coupling constant and e i is the unit vector on site i . We included up to thethird nearest neighbour pairs as interacting shells. Theclassical Monte Carlo simulations were performed by us-ing the ALPS code . To obtain more accurate results,we employed the rescaled Monte Carlo method whichreproduces the quantum specific heat from the classicalspecific heat. The magnetic energy and entropy werederived by integrating the specific heat. The spin quan-tum number S = 1 .
07 for the DLM condition as cal-culated by KKR-CPA was used in the rescaled MonteCarlo method. The Monte Carlo simulations were car-ried out using a 16 × ×
16 sites and involve 300,000 stepsfor equilibration and 2,700,000 steps for averaging. Tem-perature grids of 0.1 and 0.2 mRy were used in the range
TABLE I. Calculated exchange coupling constants J ij of bccFe for the paramagnetic DLM state.nearest neighbour J ij (meV)First 27 . . . . − . of near T C and other ranges, respectively. Note that theentropy in the rescaled Monte Carlo method does not goto zero at T →
0. Thus this method is not suitable todescribe thermodynamic quantities at a low-temperaturerange. Our thermodynamic formulation, however, needsonly the result at a temperature range around T C . Thusthe shortcoming does not matter in this study. Author contributions
The formulation was established by T.T and Y.G. Allof the calculations were conducted by T.T. The projectwas supervised by Y.G. All authors discussed the resultsand contributed to writing the paper.
Competing interests
The authors declare no competing interests.
ACKNOWLEDGMENTS
This work was supported in part by MEXT as Fugakuproject and the Elements Strategy Initiative Project aswell as KAKENHI Grant No. 17K04978. We are grate-ful to H. Akai for offering the unreleased version of theAkaiKKR code. The calculations were partly carriedout by using supercomputers at ISSP, The University ofTokyo, and TSUBAME, Tokyo Institute of Technology aswell as the K computer, RIKEN (Project No. hp190169). ∗ [email protected] † [email protected] Spaldin, N. A.
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