# Quantum Spin Effect and Short-Range Order above the Curie Temperature

###### Abstract

Using quantum Heisenberg model calculations with Green’s function technique generalized for arbitrary spins, we found that for a system of small spins the quantum spin effects significantly contribute to the magnetic short-range order and strongly affect physical properties of magnets. The spin dynamics investigation confirms that these quantum spin effects favor the persistence of propagating spin-wave excitations above the Curie temperature. Our investigation suggests a reconsideration of prevailing point of view on finite temperature magnetism to include quantum effects and the magnetic short-range order.

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^{†}preprint:

It is a long-standing debate over the nature of the paramagnetic (PM) state of the ferromagnetic (FM) materials, particularly in the transition metals. Early inelastic neutron experimentsmook ; lynn ; mook2 determined the persistence of spin-wave (SW) like modes above the Curie temperature () in both Ni and Fe, and these modes were interpreted as the evidence of considerable magnetic short-range order (MSRO) in the PM statekorenman . The presence of MSRO was later supported by the spin and angle-resolved photoemission studies maetz ; haines . This is rather unusual because majority of magnetism theories are based on the absence of SW excitations above . Moreover, by applying the spherical model (SM) approximation to the Heisenberg model (HM), Shastry shastry et al. concluded that in contrast with the experiment lynn , this model, with fairly long-ranged interactions, has little MSRO and no SW peaks above in Fe. A Similar conclusion was reached by Monte Carlo (MC) spin dynamics simulation of the classical HM for the same systemshastry2 . Very sophisticated techniqueDMFT also could not detect any traces of strong MSRO in Fe or Ni. Recent theoretical spin dynamics studiesMSROLDA , however, have demonstrated that a strong MSRO is a ’must have’ property of the itinerant magnets while such excitations like SW exist above in both localized and itinerant magnets.

It is puzzling that the applications of HM failed to predict the expected MSRO because BCC Fe has rather good local moments hasegawa ; hubbard and to a large extent HM should be valid edwards . What usually is omitted in the classical MC simulation is the quantum nature of spin. In quantum SM , the spin correlations and the susceptibility are proportional to while the classical coefficient scales as ; so the quantum effect contributes a factor . For small this can be unreasonably large. For example, for which leads to the unphysical correlation between nearest-neighbor (NN) spins in the case of NN coupling of the simple cubic (SC) structure. The same problem appeared in Ref.shastry where in the case of strong MSRO and . To avoid such difficulties a pragmatical approach is to scale the relevant quantities by , as was done in Ref. shastry . Then, in SM the scaled quantities are independent of so the quantitative results for MSRO and region of SW existence for and are the sameshastry . Below we will demonstrate that while the classical HM can describe some degree of MSRO above QSE properly included strongly increases MSRO and affects its influence on other physical properties.

For the HM hamiltonian in the PMstate, we use the second-order Green’s function (GF) techniquekondo ; shimahara ; barabanov ; winterfeldt . To calculate GF one applies twice the equation of motion and then decouples the high-order GF of forms and . For in one-dimensional system Kondo and Yamaji (KY) decoupled them by using a correction parameter kondo . Here we extend their method for arbitrary by introducing the following decoupling scheme (for and ,

(1) |

where and with and are the spin correlations, is the symmetric product of operators, and is a -dependent constant which will be determined later. For spin operator identities require , and Eq. (1) is reduced to the KY decoupling.

Decoupling the high order GF in the equation of motion with Eq.(1) one can obtain the following expression for the dynamic susceptibility

(2) |

where is the shell index, and are and correspondingly. with being the total number of sites on th shell and being sites on that shell.

The SW excitation spectrum is

(3) |

where with and being the Fourier transforms of and , correspondingly. At this stage is obtained by comparing Eq.(3) with the well-known result in the FM spin correlation limit .

From Eq.(2) and the spectral theorem, the spin correlation can be written as

(4) |

With the requirements and Eqs.(3) and (4) can be solved self-consistently. is determined by ( ). To check the validity of our method, in Fig. 1 we compare our calculated and for the BCC structure with the accurate results obtained by the high-temperature-expansion (HTE) methodsrushbrooke ; bowers and the SM results in the NN coupling case. Here is the Curie temperature in the mean field (MF) approximation, and are the total energies at and zero temperature, correspondingly. The parameter is a proper measure of MSRO at , and in the NN coupling case is identical to the average cosine of angles between NN spins. In the MF approximation there is no MSRO ( at and above The existence of MSRO suppresses with respect to . Such suppression exists also in the SM and is identical for all in that case. In more accurate calculations, however, is more suppressed at smaller

The MSRO parameter demonstrates the increase in MSRO for smaller . Although in the SM increases even faster (), this quantity already is not well defined owing to the appearance of e.g. for the SC structure for and in Ref.shastry for . At this stage the scaling should be introduced which leads to the elimination of real QSE.

Our formalism allows to obtain the following important result for for

(5) |

where with . This new and transparent expression provides another immediate and accurate check of applicability of our generalized GF formalism. For instance, it gives and for SC, BCC and FCC structures which are very close to the corresponding HTE results and bowers . Eq.(5) clearly indicates the importance of the correction parameter introduced above in the GF decoupling. For the parameter is the same as the one obtained in the SM.

The good agreement between our and HTE results indicates the applicability of this formalism for the case of arbitrary and NN interaction. We also studied a Heisenberg hamiltonian corresponding to a realistic material: we used extended (four NN) interactions in BCC Fe: and antropov , where mRy. For MC simulation gives and In SM and for all In our formalism, for and and and their and correspondingly. At , K is more than twice higher than the experimental K of Fe, our calculated is suppressed to the much lower value K. Comparing with Fig.1, one can see the additional suppression of with being considerably larger, thus indicating stronger MSRO than in the corresponding NN coupling case. However, the parameters in Ref.antropov have been obtained in the long-wavelength approximation and can only describe a small MSRO in classical case. The inset of Fig.2 shows directly , giving the details of the QSE enhancement of MSRO between several neighboring spins.

The spin correlation length is often used to describe the strength of MSRO. Despite the magnitude of , always tends to infinity when temperature approaches from above, so near may not be a parameter that properly reflects MSRO. Above the evaluation of in our formalism is straightforward from the long-wavelength behavior of the spin-correlation function We found that at fixed always increases as becomes smaller, in contrast to the SM where is independent of . At and for and again demonstrating the QSE enhancement of MSRO from another prospective.

Now let us analyze the SW excitations. In the standard magnetism theories such as the random phase approximationtahir2 and its various modified versionscallen , SW exist due to the magnetic long-range order, so its spectrum is renormalized to zero at In our formalism SW comes from the short-range spin correlations and the long-range order is no longer a prerequisite for its existence, so SW spectrum can be finite at In Fig.2 we plot the calculated SW spectrum obtained from Eq.(3) at . To demonstrate the -dependence of the SW renormalisation, the SW spectrum at (the FM case) is also plotted with all scaled by .

Let us estimate the renormalisation factor in the BCC Felynn where SW modes have been observed above the middle of the Brillouin zone along the (110) direction, ). The SW renormalisation factors ( is at are and for and correspondingly. Experimentally in the BCC Fe lynn and the difference between and is about lynn , so the overall SW renormalisation factor becomes and our result for is close to that. Fig. 2 also indicates that for smaller spins is less affected at elevated temperatures, implying that QSE favors the persistence of SW modes. (lattice constant

Let us now estimate the influence of dynamic effects and obtain the relaxation function . Among various analytical approximations for the three-pole approximation lovesey seems to be one of the best and it has been successfully applied to the typical Heisenberg system with large spin young ; latacz . In this approximation is expressed in terms of and , where are frequency moments of depending on the static correlation. The evaluation of is straightforward lovesey . loveluck contains four-spin correlation terms which have to be properly decoupled as a product of two-spin correlations. In the literature the conventional decoupling and appropriate for large , have been applied to obtain lovesey ; young . For small , the spin kernel effect, which is neglected in this decoupling, becomes important. This QSE can be clearly seen in case, where for or the left side of the decoupled equation vanishes while the right side is finite. To take into account this QSE we introduce the following decoupling procedure

(6) |

where If and are four different sites then Eq.(6) is the same as in the conventional decoupling. QSE occurs when two or more out of these four sites are the same. In this case Eq.(6) at is exact and is reduced to the conventional decoupling for . With these results for two opposite limits of and the introduced earlier quantum correction in one can expect that Eq.(6) will be a reasonable interpolation for arbitrary . By applying this decoupling procedure one can obtain where corresponds to the conventional decoupling lovesey ; young while is the quantum correction given by

(7) |

where is the -dependent susceptibility, , and .

As a function of the relaxation function has either one maximum at if or three maxima at and if . The latter case is often referred as the SW peak at shastry ; young ; latacz . With such a definition the criteria of the SW existence for given is . Usually is slightly larger than In the literature the SW peak was also defined as makivic which is slightly smaller than . Near the critical value the maximum of at is broad. When is decreased, the SW peak is more pronounced. In Fig. 3 we plot the magnitude of for different as a function of at . At fixed , is always decreased if becomes smaller. Along the direction, the critical values of when , are and for and , correspondingly. Our value of for agrees with the experiment result in BCC Fe, where SW modes above exist only above in (110) direction (Fig. 2 of Ref. lynn ). The SW peaks were also obtained in the SMshastry (with spin independent ), but the value of there is considerably higher. Our calculations indicate that this theory, which correspond to , will be applicable if QSE is properly taken into account. At and the ratio is approximately and which are respectively well below, close to, and well above the critical value . The corresponding dynamic structure factor as a function of is shown in the inset of Fig.3. It is clear that at this the well-defined SW exist in the case of the tendency of SW appears for and there is no SW signal at all for . Fig.3 shows that QSE favors the persistence of SW with increasing impact for smaller spins. In many real magnets is not large ( in BCC Fe and in FCC Ni) and we believe that QSE plays an important role in the MSRO and the magnetic excitations above , especially in the itinerant magnets. for

In conclusion, we analytically demonstrated the presence of MSRO in the Heisenberg model and identified the importance of quantum spin effect on MSRO for ferromagnets above . By extending the second-order Green’s function technique to arbitrary we found that for a system of small spins the quantum effects greatly contribute to the MSRO and enhance its influence. The spin dynamics investigation developed from the conventional method of moments further confirms that QSE favors the persistence of spin wave excitations. We demonstrated that this previously neglected QSE removes the long-standing controversy between theory and experiment regarding the presence of MSRO and SW in Fe and Ni above and clearly indicates that the current prevailing point of view of finite temperature magnetism should be reconsidered to properly include MSRO and quantum effects.

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Fig.1. and as a function of from SM (lines), the HTE methods (close symbols) and our formalism (open symbols) in BCC structure in the NN coupling case.

Fig.2. The calculated SW spectrum for the different at The dashed line is at (FM case). The inset shows from nearest to fifth-nearest neighbors.

Fig.3. The calculated for the different at as a function of . The criteria of SW is marked by the dashed line.The inset shows at