Bosonic and magnonic magnon dispersions

Journal of Magnetism and Magnetic Materials 502 (2020) 166533

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Research articles

Bosonic and magnonic magnon dispersions

T

U. Köbler Research Center Jülich, Institute PGI, D-52425 Jülich, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Ordered boson fields Domain configurations Dynamic dimensionality Analytical crossover events

The analytical crossover events in the magnon dispersion relations are discussed. As the available experimental data show, only a few types of magnon dispersion relations are observed for all spin- and lattice-structures. When the magnon spectrum exhibits a finite energy gap at q = 0 the dispersion consists of two q-sections with different functions of wave vector. Such analytical changes indicate that two mechanisms dominate the dispersion relation alternately. For magnon energies of larger than the gap, the dispersion exhibits over a finite q-range a single qx power function of wave vector. Within the experimental error limits, the exponent x is a rational number and is independent of the spin structure, i.e. universal, but depends on whether the spin quantum number is integer or half-integer. As we now know, this is the typical indication that the qx function is determined by the bosons of the continuous magnetic medium. The qx function holds up to the crossover to a sinefunction of wave vector for antiferromagnets but to a sine-function squared for ferromagnets. One therefore has to distinguish between the bosonic part of the magnon dispersion relation, at small q-values, and the magnonic part at large q-values. As is well known, sine functions of wave vector are the dispersion of the linear spin chain. Since the linear spin chain is not ordered at any finite temperature it follows that the observed magnons are not indicative of a long-range magnetic order. From the fact that no principal change of the magnon dispersions occurs upon crossing the magnetic ordering temperature it follows that the exchange interactions are not involved directly in the magnetic ordering process. The long-range ordered system is the boson field. At the critical temperature, the boson field orders. Ordering of the boson field is associated with the formation of domains and with the emergence of a magnon gap. In the ordered state, propagation of the bosons is restricted to the few different domain axes. The dimensionality of the ordered boson field can be recognized from the number of the in-equivalent domain orientations. For all lattice structures the boson field within each magnetic domain is perfectly one-dimensional, and aligns all spins parallel. This is the origin of the linear chain dispersion. The mass-less bosons are, however, not visible for neutrons. If the magnon-boson interaction is strong, the q-range of the sine functions is small. It then proves necessary to add a phenomenological phase shift in the argument of the sine function. As a consequence, magnon dispersions cannot be understood considering exchange interactions alone.

1. Introduction As we have already explained in considerable detail [1,2], in spite of the discrete atomic structure of the solids their dynamic properties can be as for a continuous medium [3]. Specific to the continuous solid is that there are no atoms and no spins to be considered. Instead, the bosons of the continuous solid are decisive. Neglect of atoms in the continuous solid is evidently a fundamental difference to the atomic solid and is not a question of length scale alone. In fact, we have to attribute different translational symmetries to the continuous and to the atomic solid. Associated with the two symmetries are specific excitations and therefore particular dynamic properties. The lattice structure independent dynamics of the continuous solid is called universal. The dynamics of the atomistic solid is material specific. Continuous dynamic symmetry, and therefore universality, occurs in the vicinity of a critical temperature, either at a finite ordering

temperature or at T = 0. This was evidenced for the magnetic degrees of freedom in the vicinity of the magnetic ordering temperature by Renormalization Group (RG) theory [3]. As was shown by RG theory, on approaching the magnetic ordering temperature from the paramagnetic side, the exchange interactions between the spins suddenly become unimportant for the magnetic dynamics [4]. As a consequence, the spin system is inactive and receives its dynamics from the energy degrees of freedom of the continuous magnetic medium. Since the excitations of a continuous medium are bosons, this means that the critical spin dynamics is determined by a boson field. The spins are indicators of the dynamics of the boson field only. As we could show, the bosons are essentially magnetic dipole radiation emitted by the precessing spins [5]. In other words, the sources of the bosons are atomic objects, the boson field is essentially a radiation field. When the exchange interactions between the spins are no longer relevant for the dynamics [4] it follows that there is no longer thermal

https://doi.org/10.1016/j.jmmm.2020.166533 Received 27 June 2019; Received in revised form 22 November 2019; Accepted 29 January 2020 Available online 31 January 2020 0304-8853/ © 2020 Elsevier B.V. All rights reserved.

Journal of Magnetism and Magnetic Materials 502 (2020) 166533

U. Köbler

energy in the system of the interacting spins. It then is sufficient to consider the energy degrees of freedom of the boson field. Due to the symmetry selection principle of relevance [3], thermal energy can be only either in the system with continuous translational symmetry (boson field) or in the system with discrete translational symmetry (spin system). For the magnets with a finite ordering temperature, the energy transfer from the interacting spins to the boson field occurs at the crossover from Curie-Weiss susceptibility to critical susceptibility [6]. Note that the two susceptibilities have analytically different temperature dependencies. This crossover defines the width of the boson controlled paramagnetic critical range and shifts the ordering temperature to a lower value compared to the atomistic models. Quite generally, the actual, the boson driven ordering transition is at a lower temperature than conforms to the magnon energies at the zone boundary, i.e. to the strengths of the near neighbor exchange interactions. This supports the view that the magnons are not the relevant interactions for the magnetic ordering process. Exclusion of the exchange interactions from the critical dynamics above and below the ordering temperature means that the observed critical dynamics is that of the boson field. At the critical temperature the boson field orders. The precise values of the critical exponents therefore are different from the atomistic model predictions [7]. In the ordered state of the boson field, the bosons propagate along a limited number of crystallographic directions only. Typical for ordered boson fields are domains. Propagation of the bosons is restricted to the domain axes. The dimensionality of the ordered boson field, and therefore the critical exponents, are determined essentially by the number of differently oriented domains. Within the experimental error limits, all observed critical exponents at T = 0 and at T = Tc appear to be rational numbers. This is a consequence of the fact that the dispersion relations of the bosons are power functions of wave vector with also rational exponents [1,2]. Moreover, the critical power functions hold over a finite distance from the critical temperature and are clearly limited by crossover events [6]. In all ordered magnets the widths of the critical ranges at T = 0 and at T = Tc are sufficiently large such that the two associated universal power functions of temperature of the spontaneous magnetization overlap. Universal dynamics therefore holds for all temperatures in the ordered phase [8]. Change from the critical range at T = 0 to the critical range at T = Tc is a typical crossover event. Continuum dynamics is observed not only for the magnetic degrees of freedom on approaching either T = Tc or T = 0 but also for the elastic and for the electronic degrees of freedom on approaching T = 0. The excitations of the elastic continuum are the well-known sound waves. We have called them Debye bosons [9]. The bosons of the electronic continuum of the metals as well as their dispersion relations are not yet explored [10]. The only signature of this boson field is that the heat capacity is perfectly ~T [10]. Typical for the boson fields in the vicinity of the critical temperature T = 0 is that their heat capacities are given over a finite temperature range by a single power function of absolute temperature with rational exponent. This is because the dispersion of the bosons is a single power function of wave-vector with rational exponent. However, for the disordered Debye boson field the interactions with the atomic background can be significant and can lead to moderate deviations from a perfect power function behavior of the heat capacity [9]. Apart from this restriction, the heat capacity of the Debye boson field is ~T3. The “critical” power functions at T = 0 hold over a finite distance from the critical temperature T = 0, and are limited by the crossover of the thermal energy to the corresponding atomistic excitations (phonons, electronic band states…). For the elastic and the electronic degrees of freedom this crossover commonly occurs at a temperature of 10…30 K and implies an analytical change in the temperature dependence of the heat capacity [6]. For the ordered magnets the identified exponents of the magnetic heat capacity at the critical temperature T = 0 are: ε = 3/2, 4/2, 5/2, 6/2 and 9/2 (see Table 1) [1,2]. Since the heat capacity of the boson field controls the thermal decrease of the spontaneous magnetization, these exponents

Table 1 Dispersion relations of the Goldstone bosons, qx, at the critical point T = 0 and the associated Tε functions of the heat capacities of the boson fields in dependence of the dimensionality of the global boson field and of the spin quantum number. Note that thermal decrease of the spontaneous magnetization scales with the heat capacity of the boson field [1,2,18]. Dimensionality of the boson field

Integer spin

Half-integer spin

d=3 d=2 d=1

q2; T9/2 q5/4; T2 q3/2; T3

q; T2 q2; T3/2 q3/2; T5/2

apply to the spontaneous magnetization as well. It is to be mentioned that the Tε functions dominate the heat capacity for magnets with a very low ordering temperature only (see Fig. 9 below) [17]. In this case all other contributions to the heat capacity are negligible. The Tε power functions therefore can be observed more conveniently from the spontaneous magnetization where the Tε functions hold up to the crossover to the critical range at T = Tc [17]. Note that the exponents ε do not scale with the dimensionality of the boson field. This may be due to the different magnon-boson interactions for different dimensionalities of the global boson field. The bosons of the continuous solid propagate ballistic, that is, independent of the lattice structure. This is the origin of universality. In other words, universality is the thermodynamic behavior of a boson field. Note that the ballistic propagation mode implies simple dispersion relations [1,2,11]. The excitations of the continuous magnetic solid we will call Goldstone bosons [12]. For all dimensionalities of the global Goldstone boson field, the dispersion relations of the Goldstone bosons at T = 0 are given by a single power function of wave vector (qx) with rational exponent x. This points to well defined interaction processes between the domains. The identified values of x are x = 4/4, 5/4, 6/4 and 8/4 (see Table 1) [1,2,17]. The thermodynamic observables show universality only for temperatures for which the boson field is the relevant excitation system. This is in the vicinity of critical temperatures only. Relevant means that the dispersion relation of the bosons is thermally populated. Moreover, all thermal energy is in the boson field, the dispersion relation of the non-relevant excitations is thermally not populated. In this way the dynamic symmetry is always clearly defined as continuous or discrete. If thermal energy is in the system of the atomic degrees of freedom (phonons, electronic band states, magnons…) the dynamics is nonuniversal, i.e. material specific. It is a very remarkable phenomenon that the boson fields in solids can order at a finite temperature. Note that the Debye boson field remains disordered until T → 0. In the ordered state, the bosons propagate along particular directions only and are in coherent states. Both features result from the generation process of the Goldstone bosons (magnetic dipole radiation) by the precessing spins. Generation of magnetic dipole radiation is by stimulated emission [5]. As a consequence, the basic boson field is perfectly one-dimensional. The magnetic ordering transition resembles the threshold for the onset of stimulated emission of a LASER. If stimulated emission dominates, the photons propagate precisely along one axis. The photons of the LASER beam are in a one-dimensional ordered state. A one-dimensional Goldstone boson field is realized in each magnetic domain. Note that the ferroelectric phase transition is also characterized by the formation of domains [13]. It is reasonable to assume that the bosons of the ordered ferroelectrics are essentially electric dipole radiation. For the typical ordering temperatures of the ferroelectrics, generation of electric dipole radiation seems to be dominated also by stimulated emission. The one-dimensional boson field within each magnetic domain interacts with the spin system, and provides an axial anisotropy to the spin system that is not known in the atomistic models. In the magnets with a pure spin moment, that will be considered here exclusively, the 2

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less bosons using neutron scattering [11]. Typical for bosons is that the qx functions hold over a finite q-range. A similar phenomenon is observed for the elastic degrees of freedom [9]. In many solids the dispersion of the acoustic phonons agrees initially over a large q-range with the linear dispersion of the Debye bosons (sound waves). The only difference is that the Goldstone boson field is ordered but the Debye boson field is not ordered. This seems to be of no importance on the short length-scale of the atomic excitations (phonons, magnons). Note that due to the different (translational) symmetries of bosons and atomistic excitations (magnons, phonons) their dispersion relations cannot cross but can attract each other only when there is a finite interaction between them. For a finite magnon-boson interaction the dispersions of magnons and bosons approach each other tangentially (see, however, Fig. 17 below). The functionality of the non-relevant one-dimensional basis fields within the domains is to keep the spins aligned parallel to the domain axis, while the qx function determines the dynamics. For the magnets with a pure spin moment the magnon excitation gap at q = 0 is a measure of the stability of the collinear spin order [18]. We therefore have to distinguish between the boson fields that are relevant and the boson fields that are not relevant for the dynamics (see Figs. 13, 18–20 below). Identification of the qx functions in the bosonic part of the magnon dispersions with the dispersion of the Goldstone bosons is suggested by a number of observations. First, the exponent x is universal, that is, independent of the spin structure. Second, the exponent x is different for magnets with integer and with half-integer spin quantum number. Third, the exponent x is characteristic of the dimensionality of the global boson field. In other words, the exponents x correlate with the exponents ε of the universal Tε power functions observed in the heat capacity of the boson field and in the spontaneous magnetization as well [1,2,18,19]. As a consequence, the Tε functions result from the qx power functions of the dispersion relations of the bosons. Both exponents are rational numbers to a good approximation (Table 1). The larger the magnon gap is, the larger is the boson-magnon interaction and therefore the wave vector range of the qx power function. The q-range of the sine-functions then is correspondingly small (see Figs. 20 and 21 below). It then turns out to be necessary to add a phenomenological phase shift in the argument of the sine-function. A completely equivalent behavior is observed for the interaction between acoustic phonons and Debye bosons [9,10]. Note that due to the phase shift, the zone boundary is no longer a sharp short-wavelength limit. In fact, the shortest possible wavelength of the bosons is determined by the diameter of the sources of the bosons. The sources are atomic objects. For the Goldstone bosons the shortest possible wavelength is given by the diameter of the wave functions of the magnetic electrons. As a consequence, in the compounds of the 3d-elements the shortest wavelength of the bosons can be much shorter than the lattice parameter [9,10]. The dispersion of the bosons then continues beyond the zone boundary [9,10]. The phase shift is another measure of the magnon-boson interaction strength. The phase shift can be either positive or negative. As a consequence, the observed magnon dispersions never conform to the predictions of spin wave theory [20]. This is, of course, a consequence of the fact that spin-wave theory is a local theory that does not consider boson fields and magnetic domains. Nevertheless, spin wave theory is able to predict a long-range magnetic order, at least for magnets with three-dimensional exchange interactions [21]. Characteristic for the spin order predicted by spin wave theory is the absence of domains. The collinear spin order therefore is not well stabilized. Note that on the typical length scale of the atomistic models the material specific and non-universal properties dominate. On this length scale there are considerable local exchange anisotropies that impede a collinear spin structure. Verification of spin wave theory is limited to genuine Ising magnets [5]. This is because Ising spins do not precess and therefore are unable to generate Goldstone bosons. The boson field gets not populated, and the dynamics is atomistic. In ordered Ising magnets there are

magnon gap is a measure of the stability the collinear spin order [1,2]. In other words, we owe the perfect collinear spin structures in the magnets with pure spin moments and therefore with no single particle anisotropy, to the generation process of the bosons by stimulated emission. Note that on the microscopic length scale of the exchange interactions there is no preferred anisotropy axis in the isotropic magnets with pure spin magnetism [14]. In contrary, the ubiquitous local exchange anisotropies counteract a stable collinear spin order. A three-dimensional global boson field results by some dynamic vector coupling of the one-dimensional basis fields associated with the domains oriented along x-, y- and z-axis. The dimensionality of the global boson field can be recognized from the number of in-equivalent domain orientations. Characteristic for two-dimensional thin ferromagnetic films is that there are two types of domains along x- and y-axis only [15]. At the surface of bulk magnets there can be also only two types of domains with spin orientations parallel to the surface [15]. Note that definition of the dynamic dimensionality by the number of the differently oriented domains is at variance with the atomistic concepts that do not consider magnetic domains [7]. The dimensionality of the spin is always three for the here considered magnets with pure spin magnetism. The dimensionality of the exchange interactions is unimportant when the bosons are the relevant excitations. Only for the cubic antiferromagnet RbMnF3 it is observed that the three domain types do not couple and that the bulk material exhibits the one-dimensional dynamic behavior pertinent to the isolated domain [8,16]. Interestingly, this applies also to the cubic ferromagnets EuS and EuO but only to the critical range near TC (see discussion of Fig. 16 below) [8]. The Goldstone boson fields and the magnons are not completely independent systems. Finite interactions between bosons and magnons modify the dispersion relations of both excitations characteristically. In other words, the magnon dispersions cannot be understood neglecting interactions with the Goldstone bosons and vice versa. It is observed that for all lattice structures the dispersion of the magnonic magnons is essentially as predicted by spin wave theory for the linear spin chain [17]. Linear chain dispersion can be understood as a consequence of the fact that magnon propagation is limited to the volume of the individual domain and that within each domain the boson field and the spin structure are exactly one-dimensional. In other words, the Goldstone boson fields do not only determine the temperature dependence of the spontaneous magnetization, they manipulate the wave-vector dependence of the magnon dispersions as well. Due to the coherence of the bosons all spins of the domain belong to the linear spin chain. Note that domains with collinear spin structure occur for all lattice symmetries. This is another manifestation of universality. Only the domain distribution can depend on the lattice structure. The one-dimensional boson field within each domain warrants the long-range magnetic order whereas magnons are non-relevant local excitations. Since the local interactions persist in the paramagnetic phase magnons persist also in the paramagnetic phase (see Figs. 8 and 13 below). Surprisingly, in the paramagnetic phase, magnon dispersions can be also as for linear spin chains (see discussion of Fig. 8 below). The coupling between the boson fields of the differently oriented domains determines the dimensionality of the global boson field and the universality class of the spontaneous magnetization. Since there are three dimensionalities of the global boson field and two types of spins (integer and half-integer), all magnets can be classified by six universality classes (Table 1). In other words, we have to distinguish between six values for the exponent ε of the Tε functions in the thermal decrease of the spontaneous magnetization and the corresponding six exponents x in the qx dispersion relations of the associated bosons [1,2]. We have every reason to identify the qx power functions of wave vector observed in the low q-range of the magnon dispersions with the dispersion of the global Goldstone boson field at the critical point T = 0. This is a very advantageous circumstance since it enables the evaluation of the otherwise difficult to obtain dispersions of the mass3

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dispersions of antiferromagnets contain a sine function of wave-vector but magnon dispersions of ferromagnets a sine function squared. In other words, the sine functions occur for all dimensionalities of the global boson field. Hence, the sine function must be characteristic of the individual domain. Note that domains with collinear spin structures occur for all lattice structures. In contrast to the qx functions at small qvalues, the sine functions do not depend on the spin quantum number, in agreement with spin-wave theory [20]. Linear chain dispersion in all magnets proves that in each domain a finite intra-domain interaction between the one-dimensional boson field and the spins is generally active. This interaction assures a perfect collinear spin structure, quite independent of the local exchange anisotropies. It is therefore not possible to verify the magnon dispersions of three-dimensional magnets as they can be calculated, in principle at least, using spin wave theory [20]. Only the absolute magnon energies are material specific, and are given by the near-neighbor exchange interactions. The wave-vector dependence of the magnons is boson defined. In other words, the Goldstone bosons manipulate the magnon dispersions. On the other hand, it is very comfortable that only sine functions (or sine functions squared) occur for all magnetic structures. In principle, spin wave theory expects as many types of magnon dispersions as there are magnetic space groups [31]. Moreover, spin wave theory has to consider the exchange interactions to the more remote spin neighbors as well. For KMn0.97Ni0.03F3 rather precise experimental magnon dispersion data are available to illustrate the sine function in the magnon dispersion relation of antiferromagnets with rather weak inter-domain interaction (compare also Figs. 7 and 17 below) [32]. The magnon dispersions of KMn0.97Ni0.03F3 agree quantitatively with the less precise data of pure KMnF3 [33]. In pure KMnF3 there is some weak indication of a magnon gap of the order of ~0.2 THz. This detail is not resolved at all in Fig. 1. It can be assumed that KMn0.97Ni0.03F3 has at room temperature the same cubic perovskite structure as pure KMnF3 but undergoes, as KMnF3, a structural phase transition at T*~186 K into a tetragonal phase [34,35]. At the magnetic ordering temperature of TN = 88 K, KMnF3 transforms into a monoclinic phase [34,35]. In our context it is important to note that these lattice distortions may change the domain configuration but they have no consequence on the property of the individual domain and, therefore, on the sine function of wave vector. On the other hand, a modified domain configuration, induced by lattice distortions, can become relevant for the dynamics and can change the universality class of the global boson field [6,18]. This is through a modified inter-domain interaction. Note that the universality classes either at T = 0 or at T = Tc (i.e. the critical exponents ε and β) depend on the domain distribution and on the dynamic interactions between the boson fields of the different domains. This interaction is always such that a definite universality class results for the global boson field. If there are significantly more or larger- domains along one crystallographic direction, the dynamic symmetry can be one-dimensional as for the individual domain. In fact, in spite of the pure sine-function in the magnon dispersion relation (Fig. 1), lattice distortions seem to be relevant for the dynamics of pure KMnF3, as it is evidenced by the observed critical exponent of the staggered paramagnetic susceptibility of γ = 4/3 [36]. This exponent is indicative of a one-dimensional boson field (assuming S = 5/2), and therefore is consistent with the

no domains. As a summary, one has to distinguish between the bosonic part of the magnon dispersion relation, at small q-values, and the magnonic part, at large q-values. The magnons of the two wave-vector regions have rather weak temperature dependence and persist into the paramagnetic phase. Only the energy gap at q = 0 exhibits a pronounced anomaly at Tc and tends to zero for T → Tc. As is well-known, the spontaneous magnetization tends also to zero at the ordering temperature [22,23]. A vanishing gap means a destabilized spin structure. It is the bosonic character of the magnons at small q-values that allows for electromagnetic radiation transitions at q = 0, using either infrared spectroscopy [24], AFMR [25,26] or Raman scattering [27]. Excitation of pure magnons is not possible by irradiation of electromagnetic waves. For instance, in the ferromagnetic resonance (FMR) absorption experiments [11] the irradiated rf-field accelerates the rotating transverse spin component and thereby creates Goldstone bosons. As is well-known, magnetic dipole radiation is generated by the rotating transverse spin components. Only by virtue of boson-magnon interactions, indirect excitation of magnons appears possible by FMR. On the other hand, direct excitation of magnons is possible for all wavevector values by irradiation of massive neutrons. The same applies to the excitation of phonons that requires also irradiation of massive neutrons. The mass-less Debye bosons (sound waves) cannot be excited by irradiating neutrons. Due to the bosonic character of the magnons in the low q-vector range, it is possible that all excitations within the bosonic q-range can collect in the gap-state at q = 0 before relaxation into the ground state occurs [28]. Here we present quantitative but rather phenomenological analyses of published magnon dispersion data. The conclusions drawn from these analyses should help in the development of future, more realistic field theories of magnetism then are presently available [29]. Realistic field theories of magnetism have to distinguish between the different fields generated in magnets with integer and with half-integer spin. Upon precession, the two spin species emit different types of field quanta. In other words, for the generation process of the field quanta (magnetic dipole radiation) the vector character of the spin is essential. Note that the emission process of magnetic dipole radiation by the precessing spins is not yet understood theoretically. This is in contrast to the emission of electric dipole radiation by the charge of the electron that is a well understood process of quantum-electrodynamics (QED) [30]. The charge of the electron is a scalar quantity. In addition, future field theories of magnetism have to consider that the two types of boson fields interact differently with the magnetic background of the spins. Only the isotropic boson field in magnets with half-integer spin does virtually not interact with the magnetic background. The dispersion relation of this boson field at the critical point T = 0 is linear [11], and the critical behavior is of mean field type (Table 2) [6]. The energy density of this boson field is ~T3 [1,2]. Note that the energy density of the electromagnetic radiation field is ~T4 (Stefan-Boltzmann law). For all other boson fields, magnon-Goldstone boson interactions are not negligible, and the energy densities of these fields are correspondingly different (Table 1).

2. Analysis of experimental magnon dispersion data As we have already mentioned, for all lattice structures magnon

Table 2 Proposed rational values of the critical exponents for the boson-driven magnetic ordering transition in dependence of the dimensionality of the boson field and of the spin quantum number. Dimensionality of the boson field

Integer spin

Half-integer spin

d=3 d=2 d=1

γ = 4/3, β = 1/3, ν = 2/3, 2‧β = d‧ν-γ γ = 5/3, β = 1/6, ν = 1, 2‧β = d‧ν-γ ?

γ = 1, β = 1/2, ν = 1/2, 2‧β ≠ d‧ν-γ ? γ = 4/3, β = 1/3, ν = 2/3, 2‧β ≠ d‧ν-γ

4

Journal of Magnetism and Magnetic Materials 502 (2020) 166533

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exponent, z, that describes the relaxation (line width) of the spontaneous magnetization at the critical temperature Tc [42]. It is not clear whether the dynamical critical exponent z can be correlated via scaling relations with the static critical exponents. For Fe, Ni, Co and EuO z ~ 5/2 has been observed in spite of integer spins for Fe, Ni, and Co but half-integer spin for EuO (S = 7/2) [43]. This could mean that in agreement with the universality classes of Table 2, the critical exponents z are identical for the two universality classes with d = 3, S = integer and with d = 1, S = half-integer. Another famous example of a relevant crystal field interaction is MnO. The saturation magnetic moment of MnO of 4.58 ± 0.03 μB/Mn is consistent with an effective spin moment of the Mn2+ ion of Seff = 2, instead of S = 5/2, and therefore provides clear evidence of a relevant crystal field interaction [44]. Spontaneous lattice distortions are additionally relevant in MnO. Integer effective spin of Seff = 2 and a relevant lattice distortion is revealed by observation of a T3 function in the thermal decrease of the spontaneous magnetization of MnO [18]. For orthorhombic LaMnO3 the effective spin of the Mn3+ ion is Seff = 3/2 instead of S = 2. Consistent with Seff = half-integer is the T5/ 2 function in the thermal decrease of the spontaneous magnetization [18,57]. In the case of KMnF3 crystal field interaction is not relevant as it reveals from the saturation magnetic moment of 5.06 μB/Mn2+ that is quite consistent with the full spin moment of S = 5/2 [45]. As a consequence, γ = 4/3 proves relevance of the tetragonal lattice distortion for T ≥ TN in KMnF3 (Table 2) [36]. As will be illustrated below, the necessary information on the dimensionality of the relevant global boson field is contained in the bosonic part of the magnon dispersion at small q-values (see Fig. 6). This detail is not resolved in Fig. 1. Additionally, the dimensionality of the global boson field results from the exponent ε in the Tε function of the thermal decrease of the spontaneous magnetization. Table 1 summarizes our earlier results for the exponents x and ε in dependence of the spin quantum number and the dimensionality of the global boson field [1,2]. As an example of a ferromagnet with weak inter-domain interaction Fig. 2 shows the magnon dispersion along [0, −ζ, 2] direction of YTiO3 measured at T = 5 (TC = 27 K) K [46]. Magnetism of YTiO3 is due to the Ti3+ ion with a half-integer spin of S = 1/2. The lattice structure of YTiO3 is orthorhombic [46]. As can be seen, along [0, −ζ, 2] direction

Fig. 1. For antiferromagnets with weak inter-domain interaction the magnon dispersion is given by a sine function of wave vector, independent of lattice structure [32]. This is the dispersion of the antiferromagnetic linear spin chain. Linear chain dispersion is a consequence of the fact that magnon propagation is restricted to the volume of the individual domain where spins are aligned perfectly parallel by the action of a one-dimensional boson field.

tetragonal lattice distortion for T > TN [8]. Note that for cubic lattice symmetry the exponent would be γ = 1 (Table 2). However in the compounds of the Mn2+ or Mn3+ ions one must be aware of a relevant crystal field interaction [19]. For instance, the crystal field is relevant in LaMnO3 (see Fig. 20 below). In the boson-controlled temperature range, the effect of a relevant crystal field is to reduce the spin quantum number of the Mn2+ ion from S = 5/2 to Seff = 2. In other words, one spin state is excluded from thermodynamics. This decreases the saturation magnetic moment and changes the universality classes at T = 0 and at T = Tc, that is, the critical exponents ε and β. As can be seen in Table 2, for cubic magnets (d = 3), change of the spin quantum number by ΔS = 1/2, i.e. from integer to half-integer or vice versa, changes the critical universality class [6]. For half-integer spin the critical behavior is of mean-field type while for integer spin the critical exponents are β = 1/3, γ = 4/3 and ν = 2/3 [8,18]. Table 2 presents a compilation of idealized rational critical exponent values for the boson driven magnetic ordering transition in dependence of the dimensionality of the boson field and of the spin quantum number. Minor deviations from rational values cannot be excluded but they should never be larger than the critical exponent η that, for instance, has been evaluated as η = 0.055 ± 0.010 for RbMnF3 [37]. Data for the universality class d = 3, S = integer are based mainly on results for iron, nickel and cobalt [38,39]. Note that Fe, Ni and Co are cubic in the critical range and have integer effective spins with Seff = 2 for Fe and Co but Seff = 1 for Ni [18,39]. Famous examples of the mean field critical behavior of the universality class d = 3, S = half-integer are cubic GdMg and GdZn [18]. Data for the universality class d = 2, S = integer refer to K2NiF4 [40]. The most prominent example of the universality class d = 1 and S = half-integer is tetragonal MnF2 [41]. Surprisingly, the critical exponents of the cubic materials RbMnF3 [37], EuS and EuO [6,8,18] agree quantitatively with those of one-dimensional MnF2. This is because in the critical range of these materials the boson fields of the three domain types along x-, yand z-axis do not couple dynamically such that the observed critical behavior is that of the isolated domain, that is, one-dimensional. It can be seen in Table 2 that the universality class for d = 3, S = integer agrees with the universality class for d = 1, S = half-integer. Moreover, the classical scaling relation 2‧β = d‧ν-γ appears to hold for the two universality classes with integer spin only. For completeness we should mention also the dynamical critical

Fig. 2. Magnon dispersion of ferromagnetic YTiO3 with TC = 27 K measured at T = 5 K [46]. For ferromagnets with weak inter-domain interactions the magnon dispersion is given by a pure sine-function squared of wave vector. This is the dispersion of the linear ferromagnetic spin chain. Linear chain dispersion is a consequence of the one-dimensional boson field within each magnetic domain. 5

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the magnon dispersion is excellently described by a sine-function squared of wave vector. As we have mentioned, the spin quantum number and the lattice structure are of no importance on the sine functions. The two critical universality classes of YTiO3 at TC = 27 K and at T = 0 are consistent with the low lattice symmetry and with the half-integer spin of the Ti3+ ion (see Fig. 4 below). The reported critical exponents are reasonably consistent with β = 1/3 and γ = 4/3 [47,48]. According to Table 2, β = 1/3, γ = 4/3 and ν = 2/3 apply to the critical universality class of the one-dimensional boson field at T = Tc [6,8]. This is consistent with the orthorhombic lattice structure of YTiO3. In the low wave-vector range, the data of Fig. 2 are not detailed enough to resolve the analytical crossover to the bosonic part of the magnon dispersion (~qx). The universal exponent x would provide information on the dimensionality of the global boson field and can be expected to be x = 3/2 (Table 1). In addition to the main ferromagnetic spin structure there is a weaker antiferromagnetic structure observed in YTiO3. YTiO3 therefore can be characterized as a weak antiferromagnet [49]. For this type of magnets with two spin structures it could be evidenced that ferromagnetic and antiferromagnetic spin components are perpendicular to each other [18,50]. Better known are the weak ferromagnets such as MnCO3 [51] or CoCO3 [52] for which the main spin order is antiferromagnetic and the second spin order type is ferromagnetic. It could be rationalized that occurrence of two ordering structures with all combinations of ferromagnetic and antiferromagnetic for the main spin order and antiferromagnetic and ferromagnetic for the second ordering structure [49] can be explained by higher order exchange interactions [53,54]. Occurrence of a second spin structure has no influence on the dynamic universality class [18]. Quite independent of the orthorhombic lattice structure and of the presence of a weak antiferromagnetic component, the magnon dispersion of the ferromagnetic component of YTiO3 along [0, −ζ, 2] direction is given perfectly by a sine function squared of wave vector (Fig. 2) [46]. For a stronger inter-domain coupling it turns out to be necessary to add a phase shift in the argument of the sine function (or sine function squared). At the same time the q-range of the bosonic part of the magnon dispersion relation (at small q-values) becomes larger (see Fig. 6 below). A phase shift can be identified in magnon dispersion measurements along c-axis of YTiO3 (Fig. 3). Comparison with Fig. 2 shows that in non-cubic magnets the inter-domain interaction strength can be different along different crystallographic directions (compare

Fig. 4. Spontaneous ferromagnetic moment of an YTiO3 powder sample (S = 1/2) measured in a field of 500 Oe as a function of temperature [56]. The field of H = 500 Oe is sufficiently larger than the demagnetization field of the powder material. The two critical exponents ε = 5/2 and β = 1/3 prove that the relevant boson field is one-dimensional in this orthorhombic material (Table 2) [8].

Figs. 18–20 below). It is evident that due to the phase shift, the sinefunction squared in Fig. 3 cannot continue until q → 0. Instead, an analytical crossover to a power function of wave vector with a finite gap at q = 0 can be anticipated (see Fig. 6 below). This detail is not resolved in Fig. 3. Measurements of the temperature dependence of the macroscopic spontaneous magnetic moment of YTiO3 [56] are consistent with the orthorhombic lattice structure [46,55] and with the half-integer spin of S = 1/2 of the Ti3+ ion. Fig. 4 shows the ordered magnetic moment of an YTiO3 powder sample measured in a field of 500 Oe as a function of temperature [56]. The applied field of H = 500 Oe is sufficiently larger than the demagnetization field that can be estimated to be ~450 Oe for the powder sample, assuming a demagnetization factor of 1/3 for the powder grains. As is frequently observed, the Curie temperature of the powder sample (TC = 30 K) is slightly larger than for the single crystal (TC = 27 K) [18]. This can be understood assuming that the linear dimensions of the powder grains are smaller than the mean free path of the Goldstone bosons. This means that the powder grains are smaller than the domains in the bulk material. The boson field then is compressed to the volume of the individual powder grain and its density is enhanced. This, possibly, increases the ordering temperature. In agreement with [47,48] the critical magnetization data at TC in Fig. 4 are well described by β = 1/3. Below critical range, thermal decrease of the spontaneous magnetization with respect to saturation at T = 0 is according to T5/2 function. The T5/2 function is indicative of a one-dimensional boson field and of a half-integer spin (S = 1/2) [1,2,18]. Note that for integer spin the exponent would be ε = 3 (Table 1). From the observed one-dimensional dynamic symmetry (ε = 5/2, β = 1/3) it follows that there can be only one domain type in orthorhombic YTiO3 with all spins pointing in the direction of the domain axis. This is as for orthorhombic LaMnO3 where all Mn3+ moments are aligned along the crystallographic b-axis (see discussion of Fig. 20) [57]. Along this direction the susceptibility tends to zero for T → 0. Normally, however, a one-dimensional dynamic symmetry requires an axial lattice structure. The best investigated one-dimensional material is the tetragonal antiferromagnet MnF2 [41]. Bulk MnF2 can be considered as one large magnetic domain. MnF2 therefore provides the opportunity to the study the properties of the isolated domain. The one-

Fig. 3. Magnon dispersion of ferromagnetic YTiO3 along crystallographic c-axis as a function of the reduced wave-vector, measured at T = 5 K (TC = 27 K) [46]. For this direction it is necessary to add a phase shift in the argument of the sine function squared (compare Figs. 16, 20, and 21 below). 6

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dimensional boson field is along c-axis and rigidly aligns all moments parallel to the c-axis such that the susceptibility along c-axis tends to zero for T → 0 [58,59]. A measure of how strong the spins are coupled to the c-axis is provided by the energy of the magnon gap (~1.1 meV) [41,69] and by the spin-flop field that is as large as 120 kOe for MnF2 [61]. Note that according to spin wave theory, there should be no significant anisotropy and virtually no magnon gap in the magnets with a pure spin moment such as MnF2. The isotropic paramagnetic susceptibility of MnF2 proves pure spin magnetism [58]. Typical for the one-dimensional dynamic symmetry is that the transverse correlation length (perpendicular to c-axis) and the transverse staggered susceptibility do not diverge at TN = 67.5 K [41]. In connection with the magnon gap of MnF2 we should mention the Haldane conjecture that in one-dimensional systems there should be a magnon gap only for an integer spin but not for a half-integer spin [60]. The rather large magnon gap of MnF2 with S = 5/2 appears to contradict Haldane’s conjecture. Note, however, that the prediction of Haldane’s theory is model-specific. In particular, Haldane’s model is essentially atomistic and is based on a Hamiltonian. As a consequence, the Haldane theory uses a different definition of the dimensionality. In MnF2 the dimensionality is given by the one-dimensional boson field. The spontaneous magnetization of antiferromagnetic MnF2 (S = 5/ 2) is excellently described by the same two critical exponents of ε = 5/ 2 and β = 1/3 as for ferromagnetic YTiO3 (S = 1/2) in Fig. 4 [8,18]. This proves that the boson-controlled spin dynamics is universal, i.e. independent of the spin structure. KCoF3 is another antiferromagnet to demonstrate different magnonboson interaction strengths along in-equivalent crystallographic directions. Magnetism of KCoF3 is due to the Co2+ ion with S = 3/2. In spite of a finite orbital contribution in the magnetic moment of the Co2+ ion [19], evidenced by a g-factor of larger than g = 2 (g = 2.22), crystal field interaction is not relevant in KCoF3 as reveals from observation of the full saturation magnetic moment of 3.33 μB/Co [45]. At room temperature KCoF3 has the cubic perovskite structure [34]. At TN = 114 K a spontaneous tetragonal lattice deformation sets in [34,45]. The tetragonal lattice distortion might modify the domain configuration but has no effect on the magnonic magnons within the individual domain that are always of the linear chain type. As can be seen in Fig. 5, magnon dispersion along [0 0 ζ] direction is essentially as for the linear antiferromagnetic spin chain [62]. However, in order to

Fig. 6. Along [ζ ζ ζ] direction of antiferromagnetic KCoF3 magnon-boson interaction is quite different compared to [0 0 ζ] direction (Fig. 5) [62]. Due to the negative phase shift in the argument of the sine-function the q-range of the q3/2 function is rather large. The two functions of wave-vector do not cross but approach each other tangentially only (compare discussion of Fig. 17).

obtain good agreement with the experimental data it is necessary to add a phenomenological phase shift in the argument of the sine-function [9,17,18]. The phase shift can be taken as a measure of the magnonboson interaction strength. Another measure of the magnon-boson interaction strength is the magnon gap at q = 0 that is ~1.1 THz for KCoF3 (~4.55 meV). Phase shift and magnon gap are unusually large for a material with a half-integer spin. The magnon gap energy is excellently confirmed using Raman scattering [63]. The observed Raman line at 38.5 cm−1 (at T = 2 K) corresponds to a frequency of 1.155 THz. In antiferromagnetic KCoF3 (TN = 114 K) the inter-domain interaction along [0 0 ζ] direction (Fig. 5) is such as to shift the sine function to the left-hand side (positive phase shift). The sine function therefore appears to be attracted by the dispersion of the Goldstone bosons, that is, however, not resolved in Fig. 5. Along [ζ ζ ζ] direction of KCoF3 (Fig. 6) the effect of the inter-domain interaction is to shift the sine function to larger q-values (negative phase shift). This increases the q-range of the bosonic qx function that is clearly resolved in Fig. 6. The qx function with x = 3/2 pertains to the 1d boson field and is consistent with the tetragonal lattice distortion of the perovskite lattice of KCoF3 [34]. The magnon dispersions along different crystallographic directions (compare Figs. 5 and 6) are surprisingly different in view of the rather weak tetragonal lattice distortion of KCoF3. Note that in KCoF3 lattice distortion sets in just at the Néel temperature of T = 114 K [34]. K2NiF4 is a well investigated antiferromagnet with layered tetragonal lattice structure and perfect two-dimensional magnon dispersions (Fig. 7) [22,40,64–68]. The lattice parameter along the c-axis (c = 13.04 Å) is much larger compared to the a-axis (a = 3.99 Å) [65]. As a consequence, the exchange interactions along the c-axis, that is, between the Ni2+ planes, can be expected to be weak but they are not likely to be perfectly zero. Nevertheless, within experimental error limits, there is no magnon dispersion at all along tetragonal c-axis (dashed horizontal line in Fig. 7). The magnon dispersion of K2NiF4 therefore is perfect two-dimensional. Nevertheless, along a- and b-axis the magnonic magnons are one-dimensional and follow precisely a sinefunction of wave-vector (Fig. 7), except for very low wave-vector values where the analytical crossover to ~qx function with a magnon gap at q = 0 of Egap = 2.43 meV occurs [22,68]. This gap-value is excellently confirmed using infrared spectroscopy [67]. A gap value of this order is typical for magnets with integer spin [18]. As a conclusion, in the

Fig. 5. In antiferromagnetic KCoF3, magnon-Goldstone boson interaction is stronger than in ferromagnetic YTiO3 (Figs. 1 and 2) [62]. It then proves necessary to add a phase shift in the argument of the sine-function. The phase shift and the excitation gap at q = 0 are two measures of the magnon-boson interaction strength. 7

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[65], the Ni2+ moments cannot be along the c-axis but must be in the ab plane. Consistent with domain orientations only along a- and b-axis is that the susceptibility along tetragonal c-axis (transverse to the domains) is finite for T → 0 [70]. For spin orientations along c-axis, the dynamic symmetry of K2NiF4 would be 1d and the susceptibility along c-axis would be zero for T → 0 [59]. At this point we have to mention that in low-dimensional or axial magnets the domain configuration is not very stable and can be manipulated by weak external perturbations such as the application of pressure [71]. Note that for ferromagnets the domain configuration can be changed from 3d to 1d by application of the weak demagnetization field. Unstable domain configurations mean unstable dynamic universality classes. This makes the exponents ε, x and β meta-stable. In large single crystals the domain configuration can depend on the strain in the sample, induced by the grows process of the single crystal. The observed dynamic symmetry class then can be sample dependent. Very surprising is that also in inhomogeneous or distorted magnets no other than the six known universality classes occur (Table 1). Since spin orientations are along domain axes, the spin orientations also depend on the domain configuration and therefore are not absolutely stable either. A clear example to illustrate the meta-stability of the domain structure and therefore of the dynamic universality class is K2MnF4. The reported magnonic magnon dispersions of K2MnF4 are completely identical to those of K2NiF4 [1,18,72]. However, in contrast to K2NiF4 [70], for the K2MnF4 sample investigated in [59] the susceptibility along tetragonal c-axis tends to zero for T → 0. This shows that in this K2MnF4 sample all Mn2+ moments are aligned parallel to the c-axis. As a consequence, there is only one domain type along c-axis. The dynamic symmetry therefore is 1d. In principle, the dimensionality of the global boson field reveals from the qx function in the bosonic part of the magnon dispersion. Assuming 2d symmetry we expect q5/4 dispersion in the bosonic magnon dispersion of K2NiF4 with S = 1, but q3/2 function for K2MnF4 with S = 5/2, assuming 1d symmetry [1]. Unfortunately, these fine but important details in the low q-range of the magnon dispersions of both materials are not sufficiently resolved. However, the same information is provided by the temperature dependence of the spontaneous magnetization. For the K2MnF4 sample investigated in [59] the spontaneous magnetization should decrease according to a T5/2 function [18]. However, data of [72] exhibit T3/2 function [18]. This is the universality class of the 2d-boson field in magnets with half-integer spin (S = 5/2). For this universality class the susceptibility along tetragonal c-axis must be finite for T → 0 [70]. On the other hand, neutron scattering data of the spontaneous magnetization of the K2MnF4 sample investigated in [73] exhibit T5/2 function, as it is consistent with the susceptibility data of [59]. As a conclusion, we have to assume that the K2MnF4 samples investigated in [59] and in [73] showing one-dimensional dynamic symmetry were axially distorted. Relevant lattice distortions are well-known to occur in the analogous Rb compounds Rb2NiF4 [18,67] and Rb2FeF4 [74]. At low temperatures Rb2FeF4 is no longer tetragonal but orthorhombic [74]. The spontaneous magnetization of the two compounds with integer spin and 1d symmetry, decreases as T3 instead as T2 function for 2d symmetry [18]. That magnons are not indicative of a long-range magnetic order, reveals clearly from observation of magnons far above the Néel temperature of TN = 2.7 K of the hexagonal antiferromagnet CsNiF3 [75–81]. Another typical example of well developed magnons in a material with no long-range magnetic order at a finite temperature is (CH3)4NMnCl3 [82]. In this one-dimensional material the correlation length diverges for T → 0 only but magnons with linear chain dispersion can be observed up to T ~ 40 K. (CH3)4NMnCl3 therefore proves that the classical linear spin chain is not ordered [21]. The perfect sine function in the magnon dispersion of (CH3)4NMnCl3 is indicative of a one-dimensional boson field (see discussion of Fig. 8). In the paramagnetic phase the Goldstone boson field is not ordered, there are no magnetic domains (see also Fig. 13 below). Nevertheless, the disordered boson field is able to provide a perfect dimensionality to

Fig. 7. Magnon dispersion along x-axis in the tetragonal layered antiferromagnet K2NiF4 measured at T = 5 K [64,65]. No dispersion is observed along tetragonal c-axis (dashed horizontal line). Sine-functions of wave vector along x- and y-axis prove that there are two types of domains within the a-b plane. By chance, the perfect two-dimensional magnon dispersion agrees with the dimensionality of the ordered boson field. Note that magnon energy at zone boundary is much larger than conforms to ordering temperature (TN·kB).

typical two-dimensional antiferromagnet K2NiF4 the observed dispersion of the magnonic magnons is also as for the linear antiferromagnetic spin chain, that is, one-dimensional. This proves again that the linear chain dispersion is characteristic of the individual domain, quite independent of the lattice structure and of the dimensionality of the global boson field. Typical for a two-dimensional global boson field is that there are domains along x- and y-axis only. These domains must be coupled. We must, however, recall two important points: first, observation of magnons is not indicative at all of a long-range magnetic order. Note that according to spin wave theory, the linear spin chain is not ordered at any finite temperature [21]. Only if the linear spin chain is the intrinsic behavior of a magnetic domain it belongs to a long-range ordered system. Second, the dimensionality of the observed magnons needs not to agree with the dimensionality of the ordered boson field (domain configuration). Since the boson field is the relevant excitation system, the magnetic material has to be characterized by the dimensionality of the global boson field. The dimensionality of the ordered boson field follows from the number of the differently oriented magnetic domains. Unfortunately, there is no direct experimental evidence of the domain structure of K2NiF4. As we have already mentioned, information on the dynamic dimensionality is provided by the exponent x of the qx function in the bosonic part of the magnon dispersion relation (at small q-values) or, equivalently, by the critical exponent ε of the Tε function in the thermal decrease of the spontaneous magnetization (Table 1). For the 2d-antiferromagnet K2NiF4 with integer spin of S = 1 we expect ε = 2 and a dispersion as q5/4 (see Figs. 13 and 19 below) [1,2,17,18]. Unfortunately, magnon dispersions of the low q-range are not detailed enough in Fig. 7 to allow for the evaluation of the exponent x. The twodimensional dynamic symmetry of K2NiF4 results, however, clearly from the exponent of ε = 2 in the thermal decrease of the spontaneous magnetization and of the magnon gap as well [1,2,18,66,68]. Additionally, the critical exponent of β = 1/6 proves a 2d-boson field [6]. In other words, by chance, the dimensionality of the magnons and the dimensionality of the relevant boson field are both two-dimensional in K2NiF4. There is long-range magnetic order along a- and b-axis only. This implies that there are two types of domains in the a-b plane. Long range magnetic order reveals clearly by the finite magnon gap that goes to zero for T → TN [68]. In contrast to the claimed spin orientation in 8

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Fig. 8. High-energy magnon dispersion along hexagonal c-axis of CsNiF3 measured at 4.9 K (TN = 2.63 K) [75]. A sine function squared of wave-vector (including a small phase shift in the argument) means a dispersion relation as for the ferromagnetic spin chain. The bosonic part of the magnon dispersion, at small q-values (see Fig. 10) is not resolved. For energies of the order of the Néel temperature (dashed horizontal line) no magnon data are available either.

Fig. 9. Low-temperature heat capacity of CsNiF3 as a function of absolute temperature [83]. The exponent of ε = 3/2 is indicative of a two-dimensional boson field and of a half-integer spin. It is reasonable to assume Seff = 3/2 (see text). From the fact that the fit curve does not pass into the origin it follows that the T3/2 function is not the asymptotic behavior for T → 0 and that a crossover to another power function of temperature will occur at a lower temperature.

the magnon dispersions. There seems to be no fundamental difference in the magnon dispersions above and below the magnetic ordering temperature. Evidently the dimensionality observed with the microscopic method of inelastic neutron scattering applies to the short atomistic length scale only. Note that above and below Tc, only the wavevector dependence but not the absolute magnon energies are determined by the bosons. Fig. 8 shows the magnon dispersion of CsNiF3 along hexagonal caxis measured at T = 4.9 K [75]. This temperature is much above the Néel temperature of TN = 2.63 K but still in the critical paramagnetic range. Most surprising for a material that orders antiferromagnetic at TN = 2.63 K are magnon dispersions as for the linear ferromagnetic spin chain for T > TN. Note that the magnon energies in Fig. 8 are much higher than conforms to the Néel temperature of TN = 2.63 K. A small positive phase shift proves weak magnon-boson interactions. The observed paramagnetic magnons in Fig. 8 agree with those observed in the long-range ordered state of YTiO3 in Figs. 2 and 3. This proves, once more, that there is no evidence of a long-range magnetic order on the short length scale of the magnons. In the basal plane, transverse to caxis no magnon dispersion is observed for CsNiF3 [75]. Observation of sine functions in the paramagnetic range indicates that the bosons are active also for T > Tc. This seems to be condition for a definite symmetry classification of the boson driven ordering transition. In contrast to the one-dimensional ferromagnetic magnons in the paramagnetic phase, in the antiferromagnetically ordered state of CsNiF3 the spins are oriented within the hexagonal basal plane [78–80]. The dynamic symmetry therefore is two-dimensional. As a consequence, for the assumed integer spin quantum number of S = 1 of the Ni2+ ion, the spontaneous magnetization should decrease, as for K2NiF4 [6,8], according to a T2 function (Table 1) [1,2,18]. Unfortunately, there are no sufficiently detailed elastic neutron scattering data available to clearly confirm the anticipated T2 function [79]. However, lowtemperature heat capacity data provide the same information (Fig. 9) [83]. As we know, thermal decrease of the spontaneous magnetization is controlled by the heat capacity of the boson field at T = 0 [18]. Fortunately, for an ordering temperature of as low as TN = 2.7 K the contribution of the Debye boson field to the observed heat capacity is negligible [83]. In other words, the observed heat capacity is exclusively due to the Goldstone boson field. As Fig. 9 unambiguously shows, instead of the expected T2 function, the heat capacity follows a

T3/2 function. This proves that the relevant boson field is, in fact, twodimensional but that the effective spin is half-integer (Table 1) [1,2,18]. A famous example of the T3/2 universality class is K2CuF4 with S = 1/2 [84]. The fact that the fitted T3/2 function in Fig. 9 does not pass into the origin indicates that the T3/2 function is not the asymptotic behavior for T → 0, and that a crossover to another power function of temperature will occur at a lower temperature [8,19]. An analytical change in the temperature dependence of an integral quantity such as the heat capacity does not necessarily mean a symmetry change [5,19]. This is because heat capacity contributions of subsystems that interact with each other (i.e. Debye bosons, Goldstone bosons) do not superimpose but appear alternately in the temperature dependence of the heat capacity [5,6]. It is reasonable to assume that the effective spin of the Ni2+ ion is Seff = 3/2 in the whole boson-controlled temperature range. On the other hand, from the high-temperature Curie-Weiss susceptibility an effective magnetic moment of μeff = 3.6 μB/Ni results for CsNiF3 [80,81]. Assuming S = 1 for the Ni2+ ion a reasonable g-factor of g = 2.55 is calculated from μeff = 3.6 μB/Ni. The larger g-factor than g = 2 is indicative of implicit orbital moment contributions in the magnetic moment of the Ni2+ ion. At about 86 K a crossover to smaller χ−1 values with respect to the Curie-Weiss law can be identified in the χ−1(T) data of CsNiF3 [76,81]. This crossover event has to be identified as a quantum state crossover from S = 1 at high-temperatures to an effective spin of Seff = 3/2 in the paramagnetic critical range and for all lower temperatures as well [18,19]. With the observed saturation magnetic moment of 2.25 μB/Ni a g-factor of g = 1.5 follows from Seff = 3/2 [79]. A g-factor of smaller than g = 2 is typical for orbital moment contributions. In other words, the orbital moment can become explicitly apparent by an increased effective spin quantum number and by a g-factor that is decreased below g = 2 [19]. For most of the compounds of the 3d-elements, the effect of a relevant crystal field is to reduce the spin quantum number by ΔS = −1/ 2 only [19]. This means that one of the 2S + 1 spin states gets eliminated from dynamics. The saturation magnetic moment then gets reduced accordingly [19]. Note that for the compounds of the Rare Earth elements with an explicit orbital moment, crystal field interaction can be stronger such that the number of states eliminated from the 9

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dynamics can be much larger [18,19]. The saturation moment then can be strongly reduced compared to the free-ion value [18]. It is important to note that the number of states remains an integer, also under a relevant crystal field interaction. This is condition for the observation of the same universality classes as for the magnets with a non-relevant crystal field interaction. The action of the crystal field is, so to say, quantized. A change of the effective spin is a threshold induced crossover event. The probability for the crystal field to become relevant depends on the ratio between the crystal field interaction and the magnetic interactions and therefore is the larger the lower the magnetic ordering temperature is [18]. There is, however, a competition between a relevant crystal field interaction that tends to decrease the spin quantum number and relevant orbital degrees of freedom that can increase the spin quantum number. For instance, in RbNiCl3 the saturation magnetic moment is 1.3 ± 0.1 μB/Ni [85], in CsNiCl3 the saturation magnetic moment is 1.05 ± 0.1 μB/Ni [86]. For both materials it can be evidenced that the effective spin is integer (see Figs. 14 and 15 below). According to the low saturation magnetic moments the effective spin can be Seff = 1 only for both materials. As a consequence, the relevant crystal field has reduced the number of states by ΔN = −1, i.e. the spin by ΔS = −1/2, but at the same time the relevant orbital moment has increased the number of states by ΔN = +1, i.e. the spin by ΔS = +1/2. In this way the effective spin quantum number remains unchanged, but the g-factor and the saturation magnetic moment get reduced due to the orbital degrees of freedom. For CsNiF3 the crystal field interaction seems not to be relevant but the orbital degrees of freedom seem to be relevant. Obviously, the spinorbit coupling is such as to align spin and orbital moments parallel. This increases the effective spin from S = 1 to Seff = 3/2. The g-factor thereby gets reduced from g = 2.55 to g = 1.5. Consistent with a halfinteger spin of Seff = 3/2 and with an assumed 2d boson field is observation of a q2 function in the bosonic part of the magnon dispersions (Fig. 10) [11]. Note that for integer spin, the dispersion of the 2d boson field would be ~q5/4 (see Figs. 13 and 19 below) [1,2,18]. It is interesting to see that the q2 function is observed at T = 1.9 K, i.e. below TN = 2.63 K, and at T = 12 K, i.e. above TN as well. In other words, there is no qualitative change of the magnon dispersions due to the ordering process. Moreover, characteristic for a half-integer effective spin is the relatively small magnon gap (Fig. 10). For CsNiF3 the

Fig. 11. Critical field Bc(T) of antiferromagnetic CsNiF3 in the critical range just below the Néel temperature of TN = 2.757 K [87]. These data have been obtained from anomalies in the elastic constants. Two sections with exponent β = 1/4 but with different pre-factors (amplitude crossover) are identified [6]. If a non-relevant parameter, for instance magnetostriction, changes strongly with temperature, this can induce a crossover in the pre-factor of the universal power function [6].

magnon gap can be estimated to be ~0.15 meV for an applied magnetic field of zero. A typical gap value for a magnet with 2d boson field and integer spin is ~2…3 meV (see Fig. 7) [18,22,64,67,68]. Note that the magnon energy at zone boundary of CsNiF3 (~2 THz) corresponds to a temperature of 96 K that is dramatically higher than the ordering temperature of TN = 2.61 K (Fig. 8). This also shows that the magnons are local, high-energy excitations that are not responsible for the longrange magnetic order. As a conclusion, there is ample experimental evidence that the effective spin of CsNiF3 is half-integer for temperatures of T < 86 K. Seff = 1/2 can be ruled out in view of an ordered saturation magnetic moment of 2.25 μB/Ni [79]. The only reasonable assumption is Seff = 3/2. Further evidence of the two-dimensional dynamic symmetry of CsNiF3 can be obtained from the critical exponent β of the spontaneous magnetization. For antiferromagnets it is possible to obtain the critical exponent β alternatively from the critical field curve in the limit Hc (T) → 0 [18]. Fig. 11 shows the critical field of CsNiF3 in the vicinity of TN, evaluated from anomalies observed in the temperature dependence of the elastic constants as a function of temperature [87]. These data display the interesting phenomenon that we have called amplitude crossover [6]. In the temperature range of Fig. 11 two sections with critical exponent values of β = 1/4 but with different pre-factors can be identified. An amplitude crossover means no change of the critical universality class, that is, no change of the exponent β. However, if a non-relevant parameter changes appreciably as a function of temperature, this can induce a crossover in the pre-factor of the universal power function. It is characteristic of non-relevant degrees of freedom that they leave the universality class unchanged but can change the prefactor of the critical power function, if they have exceeded a threshold value. This proves the stability of the universality classes. The intrinsic temperature dependence of the non-relevant parameter, therefore, becomes not apparent. As a non-relevant parameter, we can think of the spontaneous magnetostriction that commonly exhibits the strongest variation with temperature just below the ordering temperature [6,18]. The critical exponent of β = 1/4 pertains to planar systems that are not perfectly two dimensional. Possibly this is because there can be three types of domains in the hexagonal basal plane transverse to c-axis

Fig. 10. Bosonic part of the magnon dispersions of CsNiF3 along hexagonal caxis for an applied magnetic field of H = 10 kOe and for two temperatures [80]. Assuming a two-dimensional dynamic symmetry, the q2 function is indicative of a half-integer spin (Seff = 3/2) (see text). Note that 0.25 meV corresponds to 0.06 THz (compare Fig. 8). The rather low gap energy conforms to the halfinteger spin [18]. 10

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Fig. 13. Magnon dispersions of CsNiCl3 measured along crystallographic c-axis at T = 6.2 K (TN = 4.74 K) [90]. For the integer spin of Seff = 1 of the Ni2+ ion (see text) a rather large gap of ~1.5 meV occurs and the q-range of the qx function is correspondingly large. This allows rather precise fits of the exponents x for the two crystallographic directions.

Fig. 12. Critical field of CsNiF3 to a power of four as a function of temperature proving β = 1/4 [78]. These data have been obtained from neutron scattering experiments as a function of an applied magnetic field.

instead of two for tetragonal crystals [78,79]. For the 2d systems with tetragonal lattice symmetry the critical exponent is β = 1/6 [6,18]. As a summary, well established β-values are β = 1/2, 1/3, 1/4 and 1/6 (Table 2). Further clear examples of the critical exponent of β = 1/4 are CsFeF4 [6,88] and RbFeF4 [6,89]. The critical exponent of β = 1/4 is excellently confirmed from evaluations of the temperature dependence of the critical field using elastic neutron scattering measurements as a function of a magnetic field (Fig. 12) [78]. Since we are rather sure that the critical exponents are rational numbers it is appropriate to plot the critical field data to a power of 1/β = 4 as a function of temperature (Fig. 12). In this way, the critical temperature is not a fit parameter but comes out automatically by the extrapolation Bc(T) → 0. However, in the case of CsNiF3 the situation is more complicated due to the amplitude crossover [6]. Comparison with Fig. 11 shows that the data of Fig. 12 cover the temperature range more away from TN only. The amplitude crossover therefore is not resolved and the extrapolated critical temperature is somewhat lower compared to Fig. 11. CsNiCl3 is another, well investigated material [77,86] for which detailed magnon dispersion measurements have been performed in the paramagnetic phase [90]. In contrast to CsNiF3 (ms = 2.25 μB/Ni) [79], the saturation magnetic moment of CsNiCl3 is 1.5 ± 0.2 μB/Ni only [86]. Since there is sufficient experimental evidence that the spin of the Ni2+ ion is an integer (see Figs. 13–15), the only reasonable assumption is Seff = 1. In other words, a relevant crystal field interaction has decreased the number of states from N = 3, for S = 1 to N = 2 for Seff = 1/2 but the orbital moment has added one state such that the number of states remains N = 3 and the effective spin is Seff = S = 1. Fig. 13 shows the magnon dispersions along hexagonal c-axis measured at T = 6.2 K which is clearly above the Néel temperature of TN = 4.74 K [90]. Since the magnon gap in this material with an integer spin of Seff = 1 is fairly large (~1.5 meV), the q-range of the qx function of the bosonic magnons is also large. The large q-range of the qx function allows for a rather precise evaluation of the exponent x. Integer spin is confirmed by the dispersion as q5/4 (Table 1) which is the dispersion of the two-dimensional boson field in magnets with an integer spin (see Fig. 19 below) [1,2,18]. On the other hand, the q1.5 function seen for q > 1 in Fig. 13 is the dispersion of the one-dimensional boson field (Table 1). Observation of different qx functions along different crystallographic directions indicates some tendency for instability of the lattice- and/or spin structure [2]. In fact, in CsNiCl3 a phase instability is indicated by a second phase transition observed at

Fig. 14. Ordered magnetic moment per Ni2+ ion of antiferromagnetic CsNiCl3 as a function of absolute temperature to a power of three, evaluated from elastic neutron scattering measurements [92]. The T3 universality class is typical for the 1d boson field in magnets with integer spin (Seff = 1). The dispersion of the 1d boson field is q1.5, as it occurs in Fig. 13. The critical exponent of this universality class of β = 1/3 is reasonably confirmed (Table 2).

T = 4.34 K, i.e. just below the Néel temperature of TN = 4.74 K [91,94]. The relevant crystal field interaction that decreases the spin quantum number by ΔS = −1/2, and the explicit orbital moment that increases the effective spin by ΔS = +1/2 is in keeping with an assumed unstable lattice structure. Decision of about which of the two qx dispersion relations in Fig. 13 is the dispersion of the relevant global boson field that controls the spin dynamics is possible on account of the exponent ε of the Tε function in the thermal decrease of the spontaneous magnetization. Data for the temperature dependence of the spontaneous magnetization in Fig. 14 [92] are consistent with ε = 3. The T3 function is known for the heat capacity of the one-dimensional boson field in magnets with integer spin, and therefore conforms to the q1.5 dispersion in Fig. 13 [1,2,17]. In fact, the T3 function is confirmed by 11

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dynamic universality classes (Table 1) [1,2,18]. Additionally, crystalline imperfections also generate no new universality classes. This constitutes an enormous convenience in the dynamic classification of all magnets, quite independent of sizable differences on the microscopic length scale such as the spin- and the lattice structure. Although the data of Figs. 14 and 15 have not the highest precision, they confirm this phenomenon. For all magnets, the global Goldstone boson field has a clearly defined dimensionality. Very particular materials are the cubic ferromagnets EuS and EuO [6,8]. Judged from the atomistic point of view, the two materials fulfill all conditions to be ideal realizations of the Heisenberg model [96]. However, as we now know, the spin dynamics is not controlled by the exchange interactions between neighboring spins but by the relevant global Goldstone boson field [3,4]. In particular, the dimensionality of the global boson field depends essentially on the coupling between the one-dimensional boson fields associated with the differently oriented domains. This inter-domain coupling seems to be a matter of the ratio between the mean free path of the Goldstone bosons and the linear dimension of the domains. Quite generally, it can be assumed that the mean free path of the bosons decreases as a function of an increasing temperature. It therefore can happen that at a sufficiently high temperature the mean free path of the bosons has fallen below the linear dimension of the domains. The boson fields of neighboring domains then are no longer in dense contact and are no longer coupled. The global dynamics of the bulk sample then is determined by the boson field of the isolated domain and is one-dimensional. This seems to happen in the critical range of EuO and EuS [6,8]. Note that a lower dimensionality at a higher temperature is a puzzle for the atomistic models. Conventionally it is assumed that anisotropies generally decrease as a function of an increasing temperature. The critical dynamics of EuO and of EuS agrees with the critical dynamics of the best investigated one-dimensional antiferromagnet MnF2 [8,18,41]. Within the statistical and the systematic experimental error limits the reported critical exponents of EuO and EuS converge reasonably towards the rational numbers β = 1/3, γ = 4/3 and ν = 2/3 [8,96,97]. This is the critical universality class of the one-dimensional boson field, and holds independent of the spin quantum number [41]. Note that for a threedimensional coupling of all domains, the “normal” critical behavior of the cubic ferromagnets EuO and EuS with half-integer spin would be of mean field type (Table 2) [6]. This, for instance, is realized for cubic GdMg and GdZn with half-integer spin of S = 7/2 [6,18]. Note that the universal exponents of Table 1 and of Table 2 hold also for metallic magnets. Further metallic systems are discussed in [19] and in [39]. Rather detailed experimental data of the magnon dispersion relations of EuS are available [14,96,98]. Fig. 16 shows magnon dispersions of EuS measured at T = 1.3 K along the face diagonal of the cubic unit cell [14]. The magnonic part of the magnon dispersion, at large q-values, is well described by a sine-function squared of wave vector, including a small phase shift in the argument. The sine-function squared conforms to the ferromagnetic spin order. Surprisingly, the bosonic part of the magnon dispersion follows q1.5 function plus a very small gap. As we know, the q1.5 function is the dispersion of the one-dimensional boson field. If this boson field would be relevant, the spontaneous magnetization would decrease according to a T5/2 function [8]. This disagrees with the T2 function observed in zero field measurements [6,8,18]. This shows again very clearly that one has to distinguish between relevant and non-relevant boson fields and their respective dispersion relations. Evidently, the q1.5 dispersion in the magnon dispersion of EuS (Fig. 16) indicates the tendency for a one-dimensional dynamic behavior. This holds in the critical range at TC only. As we know, any change of the dynamic symmetry occurs as a crossover event. In EuS and in EuO the crossover from 3d to 1d dynamic symmetry coincides with the crossover from T2 function to the critical power function with exponent β = 1/3 (at TCO) [8]. Typical for the coincidence of two crossover events at TCO is an unusually large (or narrow) width of the critical

Fig. 15. Normalized spontaneous magnetization of a CsNiCl3 single crystal different from that one of Fig. 14 as a function of absolute temperature to a power of 9/2. These data are obtained from elastic neutron scattering measurements (compare Fig. 14) [86]. The T9/2 universality class is typical for the 3d boson field in magnets with integer spin (Seff = 1). The dispersion of this field is q2. The critical exponent corresponding to the T9/2 universality class of β = 1/3 is reasonably confirmed.

direct heat capacity measurements [94]. The accuracy of the spontaneous magnetization data in the critical temperature range of Fig. 14 is rather limited but is consistent with β = 1/3. This exponent conforms to the 1d dynamic symmetry (T3 function) in magnets with integer spin (Seff = 1) [18]. In the spontaneous magnetization data of Fig. 14 there is no obvious indication of the second phase transition at T = 4.34 K that is clearly visible in the heat capacity [91,94]. This is typical for the boson controlled spontaneous magnetization that is independent of the spin structure and largely also of the lattice structure. However, data for the temperature dependence of the spontaneous magnetization of another CsNiCl3 single crystal investigated in [86] reveal T9/2 universality class within experimental error limits (Fig. 15). The T9/2 universality class pertains to the isotropic boson field in magnets with an integer spin. This boson field has q2 dispersion (Table 1) [1,2,17]. The q2 dispersion does not occur in Fig. 13, and, possibly, can be observed in the basal plane only. Note, however, that the exponents x of the qx functions can dependent on sample preparation [93]. This is in contrast to the sine-functions that are independent of lattice structure. The different Tε functions in Figs. 14 and 15 can be due to the preparation process of the single crystals [93]. Note that the Tε universality classes depend on the domain configuration and on the interactions of the boson fields of the different domains but not on the spin structure. For no interaction between the domains the global dynamic symmetry is that of the isolated domain and is one-dimensional as in Fig. 14 [8]. No interaction is possible for weak magneto-elastic interactions. The domains then are large. This is realized in cubic RbMnF3 [8,16]. Comparison of Figs. 14 and 15 indicates that for magnets with integer spin, the critical exponent is β = 1/3 for the isotropic boson field (T9/2 universality class) and for the 1d boson field (T3 universality class) as well (Table 2). Additionally, β = 1/3 is specific for the 1d boson field in magnets with half-integer spin (T5/2 universality class). Observation of β = 1/3 therefore does not enable an unambiguous symmetry classification of the critical behavior of the investigated material [95]. It is much surprising that in spite of very different and occasionally complicated spin- and lattice structures, all magnets fit one of the six 12

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propagating mass-less bosons. Moreover, the critical behavior of this universality class is of mean field type (Table 2) [6]. As a consequence, in plots of the reciprocal susceptibility as a function of temperature, critical susceptibility and Curie-Weiss susceptibility are linear and intersect, in spite of the different translational symmetries of the associated excitations [6]. By the same argument, it is necessary that for all other universality classes with finite boson-magnon interactions, i.e. with a non-linear dispersion of the bosons, the critical exponent γ must be larger than unity in order to avoid a crossing of the critical susceptibility and of the Curie-Weiss susceptibility. Non-crossing is commonly observed [18]. Experimentally well established γ-values in addition to γ = 3/3 [18] are γ = 4/3 [97] and γ = 5/3 [40]. As a consequence, for all universality classes with γ > 1 the critical behavior is strongly modified by boson-magnon interactions and cannot be understood neglecting these interactions. In a similar way crossing of the bosonic magnon dispersion relation and the magnonic magnon dispersion relation seems to be allowed only if there is virtually no interaction between bosons and magnons. This is a rare exception. Indication of crossing of the two dispersion lines is obtained in the mixed magnetic material Rb2Mn0.5Ni0.5F4 [99]. In solid solutions of two structurally similar magnetic materials but with chemically different magnetic atoms, two magnon branches are commonly observed [101,102]. In the case of Rb2Mn0.5Ni0.5F4 the high-energy magnon branch can be attributed to the Rb2NiF4 sub-system and the low-energy branch, displayed by Fig. 17 (filled circles), to the Rb2MnF4 sub-system [101]. The fact that the magnon dispersions of the pure materials persist in the mixed material can be explained by the very weak Heisenberg interactions between chemically in-equivalent magnetic atoms [102]. Note that a free exchange of electrons between the interacting magnetic atoms requires a mirror plane between the atoms. This holds for chemically identical atoms only. However, in the mixed magnetic system the magnon dispersions of the pure materials are considerably modified. In particular, the magnon gaps of both constituents are much larger than in the pure materials [101–103]. Since for magnets with an integer spin the gap is generally larger compared to magnets with a half-integer spin, it becomes evident that the high-energy branch in the Rb2Mn0.5Ni0.5F4 mixed system belongs to the Rb2NiF4 sub-system with S = 1 and the low-energy branch to the Rb2MnF4 sub-system with S = 5/2 [99].

Fig. 16. In the cubic ferromagnet EuS the q-range of the bosonic magnons with q1.5 dispersion is surprisingly large [14]. The exponent of x = 1.5 is characteristic for a one-dimensional boson field (Table 1) [1,2,18]. Since the lattice structure of EuS is cubic for all temperatures, the q1.5 function has to be attributed to the one-dimensional boson field of the isolated domain. This field becomes relevant in the critical range only (see text).

range. For EuS with TCO = 12.2 K the width of the critical range is (TC − TCO)/TC = 0.26. A normal value would be ~0.15. At this point we have to mention another very intriguing complication when the universality class is determined from the thermal decrease of the spontaneous magnetization [39,66]. In macroscopic measurements of the spontaneous magnetization of ferromagnets, the sample is in the mono-domain state. The dynamic symmetry of the saturated ferromagnet therefore should be 1d. This means, the exponent ε of the Tε function in the thermal decrease of the spontaneous magnetization can be different when evaluated from macroscopic magnetization measurements compared to zero-field methods such as NMR [19,39] or neutron scattering [96,98]. In the zero-field methods cubic magnets are in the multi-domain state and are 3d. However, the universality classes observed in both methods can depend additionally on the size and shape of the sample [39]. For a mean free path of the bosons of larger than the diameter of the (spherical) sample, it is possible that a considerable fraction of the bosons gets scattered at the inner surface of the sample. The boson field then can become isotropic in spite of the single-domain state prepared by the applied demagnetization field. This has been observed for cubic magnets with a halfinteger spin [39]. Note that for this class of magnets the boson-magnon interaction is virtually zero and the dynamics can be 3d even for the mono-domain state forced by ferromagnetic saturation. The parallel orientation of the spins then is exclusively due to the applied demagnetization field whereas the dynamics is determined by the reflected bosons and is 3d. In general, the qx function and the sine function of wave vector avoid each other and approach tangentially only. A non-crossing behavior can be understood since the excitations of the two functions of wave vector have different (translational) symmetry [6]. For a finite interaction between the two excitations the corresponding dispersion relations attract each other but crossing is forbidden. We discuss here one observed exceptional case where crossing is observed [99,100]. This is indicative of a vanishing interaction between the two excitations [6]. The exceptional case of virtually no interactions between bosons and magnons is the symmetry class with isotropic boson field and halfinteger spin. No interaction between both systems reveals from the fact that for this universality class the dispersion of the bosons at T = 0 is a linear function of wave-vector [11]. Linear dispersion means freely

Fig. 17. For the low-energy Rb2MnF4 branch in the magnon dispersions of Rb2Mn0.5Ni0.5F4 (filled circles) the qx function and the sine-function of wavevector cross [99,100]. This is indicative of very weak interactions between bosons and magnons. The particular exponent of x = 1.625 is typical for mixed systems with different spin quantum numbers of the pure materials (see text) [18]. For pure Rb2MnF4 (circles) magnon dispersion is given by a sine-function of wave vector except for the crossover to a small energy gap at q = 0 (not resolved) [100]. 13

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Fig. 17 shows the low-energy Rb2MnF4 magnon branch in the magnon dispersions of Rb2Mn0.5Ni0.5F4 measured along the a-axis (filled circles) [99] together with the magnon dispersion of pure Rb2MnF4 (circles) [100]. Magnon dispersion of pure Rb2MnF4 follows a sine function of wave vector rather perfectly except for the not resolved crossover to a small gap at q = 0 (Egap = 0.62 ± 0.02 meV) [100]. For this bosonic part of the magnon dispersion, a q2 function can be anticipated for Rb2MnF4 with 2d dynamic symmetry and half-integer spin of S = 5/2 [1,2,18]. In Rb2Mn0.5Ni0.5F4, the two universality classes for integer spin of S = 1 due to the Ni2+ ion and for half-integer spin of S = 5/2 due to the Mn2+ ion coexist. Under this condition, the universality class of the global boson field, and therefore the exponent x in the qx dispersion of the bosonic magnons is not clear a priory. It appears that in the case of the coexistence of two universality classes, mixed exponents are possible, in particular for compositions near to 1:1 [18,102]. For the pure material Rb2NiF4 with a 2d-global boson field and integer spin the exponent is x = 5/4 (compare Figs. 13 and 19). For Rb2MnF4, with S = 5/2, the corresponding exponent is x = 2. In fact, the fitted value for the exponent x in both magnon branches of Rb2Mn0.5Ni0.5F4 is consistent with x = 1.625 (Fig. 17). This value can be understood as the average of x = 5/4 and of x = 2 of the pure materials. Crossing of the q1.625 dispersion curve and the sine-function of wave vector indicates weak interactions between the mixed boson dispersion and magnons, in spite of a considerable phase shift in the argument of the sine function. Crossing seems to be allowed due to the special character of the q1.625 dispersion. It is natural to assume that mixed exponents are possible also for the Tε function in the thermal decrease of the spontaneous magnetization. For instance, in the ordered mixture Fe3O4 (magnetite) the universality class for integer spin of S = 2 due to the Fe2+ ion (FeO-component) and the universality class for half-integer spin of S = 5/2 due to the Fe3+ ion (Fe2O3-component) coexist. In fact, in neutron scattering measurements of the spontaneous sublattice magnetization the exponent ε = 3.5 is obtained from [h,k,l] scattering intensities to which both subsystems contribute. This value can be understood as ε = 2 + 3/2 assuming 3d-anisotropic dynamic symmetry for both sub-systems [18]. In other words, in Fe3O4 the mixed exponent ε is the sum of the exponents of the pure materials. Binary mixed magnetic systems have the great advantage that the energy gaps of the two observed magnon branches are larger than in the pure materials. This implies a wide q-range of the qx function of the bosonic magnon dispersion section, and therefore allows for a rather precise evaluation of the exponent x. It is observed that the gap of each constituent increases with increasing concentration of the second constituent. For instance, in the MnxCo1−xF2 composition series [101], the energy gap of the CoF2 magnon branch increases with increasing MnF2 concentration. For pure CoF2 the magnon gap is ~1.15 THz [103] but for Mn0.95Co0.05F2 the gap energy of the CoF2 magnon branch is ~3.6 THz [101]. Figs. 18 and 19 demonstrate the large q-ranges of the qx magnon dispersion section of Mn0.3Co0.7F2 [104,105] and of Mn0.05Co0.95F2 [101]. Shown is the high-energy magnon branch pertinent to CoF2 subsystem with integer spin and therefore with the larger magnon gap. One complication with the interpretation of these data is that in pure CoF2, crystal field interaction is relevant [19]. This means that in CoF2 the Co2+ ion has an effective spin of Seff = 1 instead of S = 3/2 [19]. The large magnon gap in CoF2 of 1.15 THz (~4.75 meV) conforms to the integer effective spin [103]. Figs. 18 and 19 show again that the exponent x can be different along different crystallographic axes. The q1.5 function is the dispersion of the 1d boson field and the q2 function is the dispersion of the 3d-boson field in magnets with integer spin (Fig. 18). However, in view of the low (tetragonal) lattice symmetry, observation of the q2 dispersion of the 3d-boson field is surprising. Perhaps, the spin of the Co2+ ion in Mn0.3Co0.7F2 is not unambiguously clear and could depend on sample preparation. For the full, half-integer

Fig. 18. Magnon dispersion of the high-energy CoF2-branch in the magnon excitation spectrum of Mn0.3Co0.7F2 measured along a-axis and c-axis. Filled points are from [104], circles are from [105]. The q1.5 dispersion is typical for a 1d boson field, the q2 function for a 3d boson field in magnets with an integer spin. In CoF2 the spin of the Co2+ ion is Seff = 1 (see text). Note that 1.9 THz correspond to 91 K or 7.86 meV.

Fig. 19. Magnon dispersions of the high-energy CoF2 branch in the mixed magnetic compound Mn0.05Co0.95F2 measured along two crystallographic directions [101]. The q1.25 dispersion belongs to the 2d-boson field in magnets with integer spin (Seff = 1). The q3/2 dispersion belongs to the 1d-boson field. In pure CoF2 the magnon gap energy is 1.15 THz [103].

spin of S = 3/2 of the Co2+ ion the q2 dispersion would mean a 2dboson field, which appears to be the more plausible explanation. In comparison to pure CoF2 with a gap energy of ~1.15 THz the gap energy of the CoF2 magnon branch in Mn0.3Co0.7F2 is ~1.9 THz. The gap of the CoF2 magnon branch in MnxCo1−xF2 seems to increase linearly with increasing MnF2 concentration. Magnon data of Fig. 19 are consistent with the assumption of an integer spin of Seff = 1 of the Co2+ ion [19]. The dispersion as q1.25 means a 2d boson field and the dispersion as q3/2 means a 1d boson field (Table 1). Which boson field is the relevant field to determine the dynamics could be found out by the Tε function in the thermal decrease of the spontaneous magnetization. If the boson field with q1.25 dispersion is the relevant field ε = 2, if the boson field with q3/2 dispersion is relevant ε = 3. Very interesting magnon dispersions occur for antiferromagnets 14

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with ferromagnetically ordered crystallographic planes and antiferromagnetic spin orientations from plane to plane. Examples for this type of spin structure are MnO [106], LaMnO3 [57], CoCl2 [107], Fe2O3 [108], FeCl2 [109,110], EuTe [111] and K2FeF4 [112]. For these layered spin structures two magnon branches are predicted [113] and are, in fact, observed experimentally [18] for MnO [114], EuTe [115], CoCl2 [116,117], LaMnO3 [57] and in K2FeF4 [112]. The high-energy magnon branch has to be characterized as antiferromagnetic because its energy decreases as a function of an applied magnetic field [115,116]. By the same argument, the low-energy magnon branch has to be characterized as ferromagnetic because its energy increases as a function of an applied magnetic field [115,116]. In orthorhombic LaMnO3 (TN = 139.5 K) the spins in the a-b plane are ferromagnetically ordered with alternating spin orientations from plane to plane along c-axis [57]. Orientation of all spins along the b-axis proves that there is only one domain type. Along this axis the macroscopic susceptibility can be expected to tend to zero for T → 0. The spin dynamics therefore is one-dimensional [18]. However, according to the low saturation magnetic moment of 3.87 ± 0.03 μB/Mn, the spin of the Mn3+ ion has to be assumed to be Seff = 3/2 instead of S = 2 [57]. In other words, the crystal electric field is relevant and has eliminated one of the 5 states of the total spin of S = 2 from dynamics [19]. Halfinteger spin and 1d dynamic symmetry is excellently confirmed by observation of T5/2 function in the thermal decrease of the spontaneous magnetization [6,18]. This is as for the tetragonal one-dimensional antiferromagnet MnF2 with S = 5/2 and spin orientation along c-axis [8]. The interesting point with Fig. 20 is the crossover from q2 function of wave-vector to a sine-function squared of wave-vector. It is evident that the sine-function squared is due to the ferromagnetic spin order in the a-b planes. The large energy range of the sine-function squared indicates that the ferromagnetic interactions between the spins in the ab planes are larger than the antiferromagnetic coupling of the spins from plane to plane. This seems to be common to all antiferromagnets with ferromagnetically ordered planes. Note that in spite of a planar ferromagnetic spin order, the observed dispersion is as for the linear ferromagnetic spin chain. In other words, all ferromagnetically ordered moments within the a-b plane belong to the spin chain. The fitted q1.5 and q2 functions in Fig. 20 pertain to a 1d-boson field and to a 2d-boson field, respectively. The low dimensionalities of the two boson fields conform to the low lattice symmetry. According to the T5/2 function in

Fig. 21. Magnon dispersion in the hexagonal a-b plane of antiferromagnetic FeCl2 measured at T = 5 K (TN = 23.5 K) showing crossover from q2 function to sine-function squared of wave vector [109]. The sine function squared is indicative of the ferromagnetic spin order in the hexagonal a-b planes. For integer spin of S = 2 the q2 dispersion means a 3d boson field (see text). No dispersion is observed along hexagonal c-axis (dashed horizontal line).

the thermal decrease of the spontaneous magnetization [18] the relevant boson field is that with q1.5 dispersion. Very similar magnon dispersions as for LaMnO3 are observed for hexagonal FeCl2 (Fig. 21) [109]. A low-energy ferromagnetic magnon branch is, however, not resolved by the inelastic neutron scattering measurements of [109]. In FeCl2 the Fe2+ moments (S = 2) are ferromagnetically ordered in the hexagonal a-b planes with alternating spin orientations from plane to plane [109]. The Fe2+ moments are claimed to be oriented parallel to the hexagonal c-axis. For all spins oriented along c-axis only one domain is given, and the dynamic symmetry should be 1d. The spontaneous magnetization therefore should decrease according to T3 function. However, in [118] it was shown that the susceptibility along c-axis is finite for T → 0. This is not consistent with a single domain along c-axis. As we will see, the larger experimental evidence is in favor of a 3d dynamic symmetry [18,110]. Connected with the unclear spin and domain structure of FeCl2 is the strange peculiarity that different Tε functions result for the thermal decrease of the spontaneous magnetization when evaluated from different magnetic [h,k,l] Bragg reflection intensities [110]. When evaluated from the rather strong [1,0,−1] magnetic scattering intensity, the spontaneous magnetization decreases according to a T9/2 function [18]. The T9/2 function means an isotropic 3d boson field. When evaluated from the weaker [2,0,1] intensity, the spontaneous magnetization decreases according to T3 function [18]. Both exponents are consistent with an integer spin of S = 2. For both Bragg reflections the

Fig. 20. Magnon dispersion along orthorhombic a-direction of LaMnO3, measured at T = 20 K (TN = 139.5 K) [57]. The low saturation magnetic moment of 3.87 μB/Mn, indicates that crystal field interaction is relevant and that the spin of the Mn3+ ion is Seff = 3/2 instead of S = 2 [19]. The sine-function squared in the magnonic q-range is indicative of the ferromagnetically ordered spins in the a-b planes. 15

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critical exponent β is, as for hexagonal CsNiF3, (Figs. 11 and 12) β = 1/ 4 [6,18]. That the same critical exponent of β = 1/4 is observed for both scattering intensities is another puzzle [18]. For both symmetry classes (T9/2, T3) the critical exponent should be β = 1/3. The exponent of β = 1/4 applies to 2d systems [18]. For 2d systems with integer spin, thermal decrease of the spontaneous magnetization should be by T2 function, which disagrees with the observed T9/2 and T3 functions. The exponent fitted to the bosonic part of the magnon dispersion in Fig. 21 of x = 2.005 ± 0.061 is perfectly consistent with x = 2. The q2 function is the dispersion of the 3d boson field in magnets with an integer spin and therefore conforms to the T9/2 function observed in the thermal decrease of the spontaneous magnetization when evaluated from [1,0,−1] scattering intensity. For the 1d dynamic symmetry class (T3 function), q1.5 should hold in the bosonic q-range of the magnon dispersion. As a conclusion, the spin structure of FeCl2 is not completely clear. A final solution of this problem requires a careful investigation of the domain structure. It is, at least, satisfying that even for complicated spin structures no new critical exponents appear. A relatively consistent picture is obtained considering only the strong [1,0,−1] reflection intensities. The observed T9/2 function is consistent with the q2 dispersion in the bosonic q-range of the magnon dispersions (Fig. 21). Consistent with the isotropic T9/2 universality class is that the longitudinal susceptibility is finite for T → 0 [118]. However, for the 3d boson field a critical exponent of β = 1/3 should hold. The observed value of β = 1/ 4 means a symmetry reduction to 2d behavior which could be explained by a dynamic decoupling of certain types of domains due to a deceasing mean free path of the bosons as a function of increasing temperature [8].

of the bosons which corresponds approximately to the linear dimension of the domains. As we have shown here, the dynamics of the long-range ordered magnetic system is exclusively due to the Goldstone bosons and exhibits universality, that is, spin-structure independence. Universal power functions of temperature observed for the magnetic heat capacity, spontaneous magnetization, susceptibility etc. are the typical signature of the bosons of the continuous magnetic solid. Classical description by spin wave theory is restricted to genuine Ising magnets. This is because Ising spins do not precess and therefore are unable to generate field quanta (magnetic dipole radiation). The boson field of the Ising magnets therefore remains empty, and the dynamics is determined, in fact, by the exchange interactions. However, the only two clearly identified Ising magnets are the tetragonal layered antiferromagnets K2CoF4 [124] and Rb2CoF4 [125] with planar (2d) exchange interactions. Fortunately, for the atomistic 2d-Ising model there is a closed theoretical expression for the whole temperature dependence of the spontaneous magnetization available [126]. Note that for the boson-controlled dynamics the spontaneous magnetization consists of two universal power functions of temperature pertinent to the critical temperatures T = 0 and T = Tc. Onsager’s exact solution of the 2d Ising model allows a rigorous quantitative comparison between theory and experiment. In fact, the temperature dependence of the spontaneous magnetization of K2CoF4 [124] and of Rb2CoF4 [125] agrees perfectly with Onsager’s solution. The quantitative agreement proves that the dynamics can be either atomistic or continuous only. As a consequence of the atomistic dynamics, there are no domains in Ising magnets [128]. No domains, means no axial anisotropy that would align all spins rigidly parallel to the domain axes. The susceptibility therefore exhibits no sharp anomaly at TN [127]. Moreover, in contrast to MnF2 [58] and K2MnF4 [59], the susceptibility along the axis of the Ising spins remains finite for T → 0 [127]. This indicates that local exchange anisotropies are important, and that the parallel spin orientation is not well stabilized. The magnon dispersions of the 2d-Ising antiferromagnets therefore do not exhibit the typical anisotropy gap of a few meV. Magnons of Rb2CoF4 show very little temperature- and wave-vector dependence [128]. For all magnets with a three-dimensional spin the dynamics is universal, that is boson defined. This means, the boson field is the relevant excitation system. There is no thermal energy in the system of the interacting spins. The passive spins are only indicators of the dynamics of the boson field. Due to their different (translational) symmetries the two excitation systems cannot become relevant at the same time. This finding entails two severe implications: first, the only useful information provided by the magnons is whether the spin structure is ferromagnetic or antiferromagnetic. These microscopic details are of no importance on the universal, boson defined dynamics. Second, using inelastic neutron scattering, no (direct) information on the relevant mass-less bosons can be obtained. However, owing to boson-magnon interactions the dispersion relations of both sub-systems get significantly modified. Magnons then assume a wave-vector dependence that is inexplicable by spin wave theory. This concerns the large magnon gaps and the analytical crossover from ~qx power functions of wave-vector to sine functions of wave-vector. Note that a magnon gap of a few meV in magnets with a pure spin moment [41,69] cannot conclusively be explained by spin wave theory. On the other hand, boson-magnon interactions make the dispersion of the bosons a nonlinear function of wave-vector (Table 1). However, the exponent x in the dispersion relation ~qx remains always a rational number that is independent of the spin structure. The boson-magnon interaction appears to be quantized. This is condition for the observed universal power functions of temperature in the thermal decrease of the spontaneous magnetization. Except for the gap that tends to zero for T → Tc, there is no principal change of the magnon dispersion relations on passing through the magnetic ordering transition [120]. This shows that the exchange interactions are not involved in the magnetic ordering process. Instead,

3. Conclusions Symmetry considerations are extremely important basic concepts in all parts of physics. As is well-known from high-energy physics, particles can be classified by symmetries [119]. This applies also to solid state physics. The two particle generating symmetries that we treated on here are the continuous translational symmetry of the macroscopic solid and the discrete and periodic translation symmetry of the atomic solid. Surprisingly, in addition to their discrete atomic structure, all solids exhibit properties of a continuous medium such as if the atoms would not exist [3,4]. This aspect is well-known from practical experience. The excitations of the continuous solid are mass-less bosons that cannot be observed using inelastic neutron scattering. For instance, the particles of the continuous elastic solid are the mass-less sound waves (Debye bosons) [9]. The bosons of the continuous magnetic solid are essentially magnetic dipole radiation (Goldstone bosons) while the bosons of the electronic continuum of the metals are virtually not explored [10]. Phonons and magnons are the excitations of the discrete and periodic translational symmetry of the atomic structure. These localized atomistic excitations can be excited by the irradiation of massive neutrons. According to their atomistic symmetry, magnon dispersions are different for ferromagnets and for antiferromagnets. The absolute dispersion energies of the magnons and phonons are determined essentially by the inter-atomic near-neighbor interactions. This makes these excitations material specific. However, in spite of the atomistic origin of the magnons, the observed wave-vector dependence of the magnonic magnons does not depend on the lattice structure and is always as for the linear spin chain, i.e. one-dimensional [18]. This is because magnon propagation is restricted to the volume of the individual domain. Within each magnetic domain the Goldstone boson field is perfectly one-dimensional and manipulates the wave-vector dependence of the magnon dispersions. According to spin wave theory, the linear spin chain is not ordered at any finite temperature [21]. Quite generally, magnons provide no evidence of a long-range magnetic order. The long-range ordered system is the boson field. The typical length scale of the ordered boson field is given by the coherence length 16

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Only the magnon gap shows a pronounced anomaly at the critical temperature [22,23,68]. For the here considered magnets with a pure spin moment the gap is a measure of the stability of the collinear spin structure. As far as experiments show, the gap tends to zero on approaching the magnetic ordering temperature and exhibits the same critical exponent β as the spontaneous magnetization [18,22,23,68]. In other words, the collinear spin structure gets unstable just at Tc. Reported examples of a vanishing gap for T → Tc are the typical twodimensional magnets K2NiF4 [22,68], K2CuF4 [120] and also one-dimensional MnF2 [23,41]. For CsNiCl3 it is observed that the magnon gap rises again in the paramagnetic phase as a function of increasing temperature [123]. It appears that a boson controlled paramagnetic critical dynamics requires a finite magnon-boson interaction and therefore a finite gap. Only the bosonic part of the magnon dispersion relation (at small qvalues) provides information on the dimensionality of the global boson field that is responsible for the dynamic universality class. This part of the magnon dispersion relation is given by a gap plus a power function of wave vector (~qx). The exponent x is characteristic of the dimensionality of the global boson field (Table 1). However, the qx function gives also no evidence of a true long-range magnetic order (see Fig. 10). The ~qx function with the same exponent x can be observed above and below the ordering temperature [41,80]. This shows that the dimensionality of the boson field stays constant on passing through the ordering transition. This is condition for a homogeneous critical behavior and allows for a clear symmetry classification of the magnetic ordering transition. The critical exponents observed for the thermodynamic observables above and below the ordering temperature therefore are correlated and occasionally obey one of the known classical scaling relations (Table 2) [7]. Concluding we have to admit that we are far away from a theoretical understanding of the dynamics in ordered magnets. Some of the conclusions drawn from the here performed analyses of published experimental magnon dispersion data might appear very suggestive but they need further experimental as well as theoretical support for a final approval. Since the advent of the RG theory [3] it is clear that the spin dynamics of the ordered magnets is controlled by boson fields, and not by the exchange interactions [3]. Hamiltonians therefore are inadequate. As a consequence, only field theories can give a correct description of the magnetic dynamics. The nature of the field bosons could, however, not be clarified by the RG theory. Moreover, in the available field theories, a strict distinction between the two translational symmetries is not made [4]. In particular, it is not considered that integer and half-integer spins generate different types of boson fields (magnetic dipole radiation). The dynamics of the boson field therefore is different for magnets with an integer or a half-integer spin. One obstacle in the development of exact field theories of magnetism is that, with only one exception (the 3d boson field in magnets with half-integer spin), the boson fields are not free fields but interact significantly with the magnetic background. Possibly, this interaction can be accounted for by an anisotropic index of refraction for the bosons. Finite interactions between the differently oriented domains change the dispersion relations of the global boson field. This inter-domain coupling appears to be quantized and results into the different universality classes.

there is every reason to assume that the magnetic ordering transition is driven by the Goldstone boson field. At the critical temperature the boson field orders. Ordering of boson fields always seems to be associated with the formation of domains [13,15]. In the ordered state, the propagation of the bosons is restricted to the few domain axes. Domains are, however, difficult to observe using neutron scattering [121]. Magnetic domains with a perfect collinear spin structure occur for all lattice- and spin structures. This can be viewed as another variant of universality. The axial spin structure within each domain is a direct consequence of the generation process of the bosons (magnetic dipole radiation) by stimulated emission. Stimulated emission provides one explanation for the phenomenon of broken symmetry. Emergence of the magnetic domains at Tc is noticed by a sudden stop of the thermal increase of the macroscopic susceptibility as a function of a decreasing temperature. This is a consequence of the strong axial anisotropy generated by the ordered one-dimensional boson fields within each magnetic domain. Detecting magnetic order means detecting magnetic domains. Verification of a long-range magnetic order requires measurements of the two-spin correlation length. In non-cubic magnets the behavior of the correlation length can be different along different crystallographic directions. When the correlation length diverges along all main symmetry directions, the dynamic symmetry is three-dimensional and isotropic. This means that there are domains oriented along these crystallographic directions. In MnF2, for instance, there is only one domain type along tetragonal c-axis [41]. All Mn2+ moments are rigidly aligned along the c-axis. The longitudinal susceptibility tends to zero for T → 0 [58]. Typical for the one-dimensional dynamic symmetry of MnF2 is that the transverse correlation length does not diverge [41,122]. Nevertheless, magnons are observed transverse to the tetragonal c-axis [69]. This shows, once more, that magnons are not indicative of a longrange magnetic order. The nearly isotropic magnon dispersions of MnF2 [69] give a misleading information on the boson defined one-dimensional dynamic symmetry. Surprisingly, the bosons are the relevant excitations also in the critical paramagnetic range. The critical paramagnetic dynamics shows the typical boson determined universality: the critical range is finite, and the critical exponents are rational numbers to a good approximation (γ = 3/3, 4/3, 5/3, ν = 3/6, 4/6, 6/6) (Table 2) [6]. Furthermore, boson dynamics reveals from the dispersion relations of the paramagnetic magnons that are also of the linear chain type (Fig. 8) [75,76]. Since there are no domains in the paramagnetic phase, we have to assume that the observed perfect one-dimensional symmetry of the paramagnetic magnons [76] applies to the atomistic length-scale only. A similar phenomenon is observed for the acoustic phonons [9]. As the Goldstone boson field in the paramagnetic phase of magnets, the Debye boson field is not ordered at any finite temperature. The critical temperature of the Debye boson field is T = 0. Nevertheless, the Debye bosons can manipulate the wave-vector dependence of the acoustic phonons. The standard dispersion of the acoustic phonons is as for the linear atomic chain [9,10,17]. This suggests that the generation process of the Debye bosons is also by stimulated emission. Only the absolute phonon energies are determined by the inter-atomic interactions. From the identical dimensionality of the boson dispersions above and below the ordering temperature (Fig. 10) it follows that the paramagnetic dynamics is determined by the same boson type as in the critical range below ordering temperature. These bosons drive the magnetic ordering transition and determine its dimensionality. Observation of magnons above the ordering temperature conforms to the fact that thermal equivalent of the nearest-neighbor exchange interactions is generally larger than conforms to the boson driven ordering temperature. In other words, at the crossover from atomistic dynamics (Curie-Weiss susceptibility) to boson dynamics (critical susceptibility) the ordering temperature gets shifted to a lower value [6]. The Goldstone bosons have an anti-binding functionality for the magnetic system.

CRediT authorship contribution statement U. Köbler: Conceptualization, Methodology, Formal analysis, Writing - original draft. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 17

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Acknowledgment

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