WAVE-VECTOR DEPENDENT MAGNON RENORMALIZATION ; A NEW STATISTICAL APPROACH TO THE HEISENBERG FERROMAGNET

HAL Id: jpa-00213978

https://hal.archives-ouvertes.fr/jpa-00213978

Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

WAVE-VECTOR DEPENDENT MAGNON RENORMALIZATION ; A NEW STATISTICAL

APPROACH TO THE HEISENBERG FERROMAGNET

M. Lines

To cite this version:

M. Lines. WAVE-VECTOR DEPENDENT MAGNON RENORMALIZATION ; A NEW STATIS- TICAL APPROACH TO THE HEISENBERG FERROMAGNET. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-477-C1-481. �10.1051/jphyscol:19711158�. �jpa-00213978�

ANTIFERRO ET FERROMA GNETISME

WAVE-VECTOR DEPENDENT MAGNON RENORMALIZATION ; A NEW STATISTICAL APPROACH

TO THE HEISENBERG FERROMAGNET

by M. E. LINES

Bell Telephone Laboratories Incorporated, Murray Hill, New Jersey, U. S. A.

RbumB. - On dkrit une mkthode permettant une renormalisation du spectre de magnons dkpendant du vecteur d'onde. Ce calcul est effectue sur la base du dkouplage au premier ordre des fonctions de Green dans un modkle d'Hei- senberg. On explique ainsi I'existence d'ondes de spin de faible longueur d'onde au-dessus de la tempkrature de Curie.

Des resultats numkriques sont prksentks pour des rkseaux magnktiques B deux et trois dimensions, comportant des calculs d'aimantation, de susceptibilite, de chaleur spkifique, de corrklation et d'exposants critiques.

Abstract. - A method is described for introducing wave-vector dependent magnon temperature-renormalization into simple first-order decoupling Green's function theories of Heisenberg ferromagnetism, thereby allowing for the existence of short wavelength propagating magnons above the Curie temperature. Numerical results are presented for both two and three dimensional magnetic lattices and include calculations of magnetization, susceptibility, specific heat, correla- tions, and critical exponents.

1. Introduction.

-

A great deal of progress, however, has been made in developing low temperature approximations and excellent reviews are available by Mattis [l] and Kef- fer [2]. The simplest approximation is to neglect spin- wave interactions entirely (simple spin-wave theory) in which case magnon energy ET(K) is temperature independent and equal to its value at absolute zero E,@). A simple improvement is accomplished by including magnon interactions in a Hartree (or ran- dom phase) approximation. This has the effect of producing a temperature dependent magnon disper- sion ET(K) = S(T) E,(K) where <(T) is a function of temperature but not of K-magnitude. We referto a temperature renormalization (or simply renormaliza- tion) of magnon dispersion governed by the function S(T) which, in the random phase approximation (RPA) cited above, is proportional to the square root of a near neighbor spin-pair correlation and has been given a physical interpretation by Keffer and Loudon [3].

Spin-wave interactions, in general, lead to two impor- tant modifications of the simple spin-wave picture.

Firstly they provide a relaxation mechanism giving the magnon excitations finite lifetime, and secondly they give rise to magnon energy renormalization. Static properties of the thermodynamic spin system are thought to be primarily dependent on the latter effect

and, in view of the enormous complexity of the detailed problem, efforts to extend theoretical calculations beyond low temperature limits and still retaining a tractable formalism have led to the development of many first order w approximations which neglect magnon lifetime effects and focus attention on energy renormalization. One of the more significant develop- ments along this line has been the study of lowest order decoupling approximations for thermodyna- mic Green's functions. Useful references are Zuba- rev [4], Bonch-Bruevich and Tyablikov [5], Tahir- Kheli and ter Haar [6], Callen [7], and Haas and Jarrett 181.

In the most simplistic terminology, the Green's function method in magnetism consists of writing down the equation of motion for a spin correlation function. This equation, in general, involves additio- nal correlations more complex than the original.

One can then proceed either by writing further equa- tions of motion for the ever more complicated correla- tions generated (thereby giving rise to an infinite hie- rarchy of coupled equations), or by << decoupling >>

at some stage to close the set of equations by relating a more complex correlation to a simpler one using some plausible, and hopefully physically transparent, statistical approximation. The only reason for cou- ching all of this in terms of Green's functions rather than correlation functions (to which they are directly and simply related) is that the former evolves as the more natural function for the solution of the problem.

First order decoupling takes place at the earliest possible stage and in general gives rise to elementary excitations of infinite lifetime.

Over the years, many varied decoupling schemes have been devised for the Heisenberg ferromagnet.

Almost all of them, especially those aiming for vali- dity over a wide temperature range, have used first order decoupling, and this has been done almost from necessity rather than choice because of the enor- mous increase in mathematical complexity which results from delaying decoupling until a later stage (unless further approximations restricting the tempe- rature range to low values are also introduced). A

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711158

C 1 - 478 M. E. LINES

representative list of publications, in addition to Ref. [4] to [8], which illustrate the developing sophis- tication of first order Green's function decoupling might include Bogolyubov and Tyablikov [9], Oguchi and Honma [lO], Tahir-Kheli [Ill, [12], [13], Cope- land and Gersch [14], Dembinsky [15], Katsura and Horiguchi [l61 and Mubayi and Lange 1171.

Regardless of their degree of sophistication, all these first-order decoupling approximations have produced excitation energies for which the scaling with temverature is comvletelv indevendent of wave- vector.

hat

It is now very well established experimentally that real magnetic second-order phase transitions affect, in a fundamental way, only long wavelength excita- t i o n ~ . Both neutron diffraction [18], [l91 and Raman [20], [21] experiments indicate that short wavelength magnon energies undergo no obvious anomaly at Tc but that such magnons continue to exist in a more or less wzll defined form into the paramagnetic phase. The phenomenon is most marked for systems of low dimensionality where narrow linewidth magnon excitations, with almost completely unrenorrnalized energies, persist well into the para- magnetic phase for all except the very longest wave- lengths. Pertinent neutron diffraction and Raman experimental results have been reported by Birgeneau and coworkers [22], [23], [24], and by Fleury [25].

Even for linear chains, where ideally, T, = 0, propa- gating magnons are observed over nearly all the Bril- louin zone to quite elevated temperatures (Birge- neau [26]). On the other hand, Birgeneau et al. [27]

have established that, even for magnetic systems of low dimensionality, the limiting zone center (infinite wavelength) magnons do have energies which renor- malize to zero at T,.

It seems clear that any theoretical approximation hoping to describe this situation realistically must, at the very least, introduce some wave-vector dependence into magnon energy renormalization.

We shall demonstrate the manner in which such an approximation can be constructed within the Green's function formalism by extending the physical picture

.

of energy renormalization first given by Keffer and Loudon [3].

Using this new (closed form) statistical approxima- tion to calculate static ensemble averages, we have computed detailed numerical results for the Heisen- berg ferromagnet in both two and three dimensions.

The main accomplishments of the new approximation, apart from those for which it was specifically desi- gned (e. g. the description of propagating <t short wavelength paramagnetic magnons, etc.) are the prediction of a phase transition to a zero magnetiza- tion, infinite susceptibility, low temperature phase for the isotropic two dimensional system, and the

computation of critical indices for paramagnetic susceptibility and correlation length essentially in agreement with high temperature series estimates for the isotropic three dimensional system.

The two dimensional result supports the series expansion findings of Stanley and Kaplan [28], [29]

and is not open to the same criticism [30] levelled at the earlier decoupling scheme used by Mubayi and Lange [l71 and Tahir-Kheli [13]. The three dimensio- nal result is, to the authors knowledge, the first of its kind claimed for a closed form avvroximation.

2. The basic approximation. - The Green's func- tion method in statistical magnetism is by now so well documented in the literature that the present argu- ments can be outlined briefly with emphasis on the physical aspects of the approximation. The retarded double-time temperature-dependent Green's function for Heisenberg operators A and B is defined by in which square brackets denote a commutator, angular brackets an ensemble average, and 6(t) is the step function 8 = 0, t < 0 ; 8 = 1, t > 0. In units for which Planck constant h = 2 ^R, it has the equation of motion

where Je is the system Hamiltonian. For the Heisen- berg ferromagnet we have

in an obvious notation, and it proves convenient to study the operators A(t) E s:(t) and B(0) E (S:)" S;

(see, for example Lines [31]), where n is an integer, and subscripts g and h refer to sites in the spin lattice.

In a translationally invariant lattice we can Fourier transform with respect both to time t and lattice g-h to discuss the equation of motion for the Green's function transform GK(E) of

(<

>),

where K denotes a reciprocal lattice wave-vector. The equation of motion takes the form

where F =

<

>

Instead of writing higher order equations of motion for the higher order Green's functions, first order decoupling approximations (as exemplified by refe- rences [4] to [17]) close the chain of equations at the earliest opportunity by linearizing eq. (4). These many and varied decoupling schemes are normally deduced from Green's function approximations made in real space and thereafter transformed to K representation.

They all reduce essentially to writing

= t(T) E,@) , (5)

where E,(K) is the low temperature spin-wave disper- sion and ((T) is a real wave-vector independent tempe- rature scaling function which differs from one approxi-

WAVE-VECTOR DEPENDENT MAGNON RENORMALIZ ATION, A NEW STATISTICAL APPROACH C 1 - 479

mation to another. Substituting (5) into (4) allows solution for the original Green's function as

= (F12 X) [E - t(T) E,(@] - l 7 (6) with poles (i. e. elementary excitations) at the real frequencies t(T) Eo(K).

These solutions involve three basic assumptions : (i) that AK(E) can be related to GIL(E) through a ^(< mass operator equation

AK(E) = M,@) G,@) 7 (7) where MK(E) does not involve terms proportional to the inverse of GK(E) (the restrictions inherent in this assumption have been discussed by Tahir-Kheli [l11 and Wortis [32]) ; (ii) that [E

-

neglect of magnon lifetime effects) and (iii) that MK can be written as ((T) E,(K), where t(T) is indepen- dent of K.

It is only restriction (iii), implying that all magnons renormalize in the same way with temperature, which will be lifted here. We shall thus attempt to define a temperature renormalization c,(T) which is K depen- dent. Consider first a paramagnetic phase. Spin corre- lations at infinity, for zero applied field, vanish.

Nevertheless, we can in general imagine a tempera- ture dependent volume V or length L(V -- L ~ , where d is dimensionality) over which spins are closely correlated in some sense. We suppose that spin-waves with wavelength A & L can be well described as exci- tations from the thermally averaged equilibrium state (i. e. first random-phase approximation with compo- nents of each spin perpendicular to total spin averaged to zero) giving t,(T) =

<

Clearly, however, such as approximation is quite inappropriate for spin-waves with A & L. For these, additional excitations are (using Keffer and Loudon's analogy in Ref. [3]) like ripples superimposed onto the instantaneous nonequilibrium position of the exis- ting waves. In considering nearest neighbor exchange effects in this limit the second random phase approxi- mation is more appropriate, averaging to zero those components of nearest neighbor spins So and S6 which are perpendicular to S,

+

v,,

< s,.s,

<

Clearly for intermediate A the function tK(T) should vary smoothly between these two extremes.

For the simplest tractable formalism, however, it is convenient to think of magnons (in the first Bril- louin zone) as belonging only to one or the other of two classes ; << long wavelength ⁾⁾ (with 1 K I < n/L) for which we take t,(T) =

<

>

tK(T) =

<

We expect L to define some sort of coherence length for which magnons with A < 2 L experience much less damping and renormalization than do magnons with A > 2 L. Clearly, only long wavelength D

magnon energies renormalize to zero at T,.

Substituting these tK(7') expressions into (6) allows solution for GK(E) and therefore enables us to close

the equations and generate results for thermodyaa- mic properties of interest in the usual way [4]. Details will be published elsewhere [33]. One difficulty remains.

The solutions are given in terms of L, so that the pro- blem is not completely determined in closed form until a precise physical meaning is attached to L.

We have chosen to associate coherence length L with the spin separation R for which

<

>

falls to some fixed fraction of its near neighbor value.

But this introduces an additional problem. We require

the <t cut-off D wave-vector (with magnitude n/L)

to be a continuous variable (approaching the farthest reaches of the first Brillouin zone at extremes of high temperature) whereas correlation

<

>

as evaluated in a self-consistent fashion from the Green's function theory, only has physical significance at actual spin sites. We wish t o extend the concept of spin correlation to that of a continuous function which interpolates smoothly between the physically signi- ficant values.

A function w h i ~ h is fairly adequate in this sense (though not ideal because of a developing oscillatory behavior at high temperatures) is the self consistent Green's function result

<

>

Quantitatively we define coherence length L by the equation

where @ < 1 is a numerical donstant to be determined.

The form of the left hand side denominator (where A is a distance parameter) is dictated by the require- ment that the theory go over smoothly to RPA Green's function theory at high temperatures i. e.

n/L ⁴ !K,,,,, I, a zone corner wave-vector, in this limit. We shall find that critical behavior, for example, is in no way dependent on A. In one dimension A = 8, the nearest neighbor distance, and is indepen- dent of CD. In two and three dimensions A < 6 and is

@ dependent.

The definition serves for both the ferromagnetic and paramagnetic phases. In general, L becomes infinite below a temperature T' in the ordered phase.

Thus, for T < T', all magnons are << short wavelength n and we recover the essence of interacting spin-wave theory. At high temperatures, with properly chosen parameters, the theory goes over to RPA Green's theory, all magnons becoming c< long wavelength ^)>

and renormalizing as

<

All detailed calculations will be published elsewhere, Lines [33]. The more significant findings we shall summarize below.

3. Results. - 3.1 THE SIMPLE CUBIC HEISENBERG FERROMAGNET. -Using the Hamiltonian of eq. 3, but restricting exchange to nearest neighbors only, we have examined first the CD dependence of the A = 0 approximation [eq. 81. The value CD = 1 corres- ponds exactly t o RPA Green's function theory with the familiar results for critical exponents y = 2, v = 1, y = 0 ; and for Curie temperature

C 1 - 4 8 0 M. E. LINES which are to be compared with the best known series

extrapolation findings (taken from Bowers and Woolf [34]) y = 1.375, v w 0.70, V 0, and

The last of these results is valid only for the high spin (classical) limit which is also the limit, in view of eq. (7), for which we expect the Green's function appro- ximation to be best (see for example Tahir-Kheli [ll]).

As @ is decreased, RPA critical exponents (for the paramagnetic phase) are maintained but the tempera- ture range above Tc over which the simple power laws hold is gradually reduced. Eventually, at a value

@ = 0.568, this ⁽⁽ RPA ^)> range goes to zero and we find new critical exponents y = 1.36

+

v = 0.68 f 0.01, V = 0 ,

essentially in agreement with the series values. The error limits are controlled by the numerical accuracy of computer evaluated Brillouin zone integrals.

Even the amplitude of the susceptibility divergence is in agreement with the Pad6 estimate for this @ value.

Writing, in an obvious notation,

we find a Green's function result B = 2.14

+

kTc/JS(S

+

For 0 c 0.568 the susceptibility X becomes double valued, thereby making @ = 0.568 a sort of singular point and fixing @ unambiguously. With this CD value, we find that the coherence length L for the simple cubic lattice approaches a finite value L(Tc) = 0.806 6 as T + T,f according to the power law

[6/L(T)] - [G/L(T,)I = 1.61 [T - T a r ; V = 0.68 (1 0) where o is a nearest neighbor distance. Thus, for T

> Tc, only magnons out near the zone boundary

(particularly the zone corners) are well defined.

Numerical work in the ferromagnetic phase has not yet been performed.

Now it turns out, for the simple cubic lattice at least, that the A = 0 approximation above is very close to the solution which joins smoothly onto RPA theory as T -, CO, so that it seems to be valid to an excellent approximation throughout the paramagnetic phase. It is nevertheless interesting to examine the A dependence of the more general theory.

We find that as A is allowed to vary from zero, a singular value can be located (as described above) for each case. The critical amplitude B and exponents y, v, y, are all seemingly independent of A. Curie temperature Tc is, however, A dependent and, to a very good approximation, susceptibility curves are just shifted parallel to the T axis by this parameter.

We have also evaluated paramagnetic specific heat and it is here that we find the first discrepancies with the Pad6 results. Critical exponent a (again indepen- dent of A) takes the value 0.32 whereas the most likely cc from series extrapolation is thought to be close to zero.

3.2 THE QUADRATIC LAYER HEISENBERG FERRO-

MAGNET. - For the quadratic layer lattice the para-

meter A, which adjusts the theory to join smoothly to RPA theory at high temperatures, is only weakly

@ dependent and approximately equal t o a/&. ^The

first striking result of the new theory is that, whereas the familiar RPA Green's function approximation

@ = 1 gives T, = 0 for the isotropic system, we now find (for any @ less than unity) a finite temperature T?) for which paramagnetic susceptibility diverges.

There is, however, no obvious discontinuity in cri- tical behavior as a function of CD to enable us to locate the most appropriate CD value in the manner used for the three dimensional case.

We have therefore plotted a family of susceptibility curves for various @ values, which we show in figure 1.

FIG. 1. -Temperature dependence of magnetic susceptibility for the isotropic quadratic layer Heisenberg ferromagnet as calculated for different values of the parameter @ (I = 1 cor- responding to the RPA Green's function approximation). The open circles are the high temperature series result for classical

spins. Also shown is the molecular-field result.

Also plotted is the classical spin high temperature series estimate of susceptibility [35]. We see, on the scale of figure 1, a virtual identity between the @ = 0.74 results of Green's function theory and the series sus- ceptibility. The temperature T?) (which is @ depen- dent, but not A dependent) is given for the best fit Green's function theory by k ~ : ~ ) = 1.19 JS(S

+

which is perhaps a little larger than the most likely value suggested by series extrapolation techniques [35].

For critical exponents we find y w 2.0, v w 1.0, y = 0. There are no published Pad6 series estimates for this case but, if the general feeling that critical exponents reflect primarily dimensionality is true, it might be pertinent to compare them with the two dimensional Ising results y = 1.75, v = 1, y = 0.25, For the isotropic system, with @ = 0.74, we find that coherence length L diverges at T:~) as

SIL = 8 . 9 5 [ ( ~ / ~ , ' ~ ' ) - l]", V = 1 .O , (1 1) and that L is directly proportional to the correlation length in this limit, but is about a factor of ten smaller.

Also, in this limit, correlations

<

>

+

The theory is just as easy to compute for the weakly anisotropic Hamiltonian and a weak dependence of

WAVE-VECTOR DEPENDENT MAGNON RENORMALIZATION, A NEW STATISTICAL APPROACH C 1

-

T;" with anisotropy is found. Also, for the weakly anisotropic system, we have computed the magnetiza- tion below the Curie temperature and studied the approach to the isotropic limit. The results, shown in figure 2, indicate the general situation. The tempera- ture T,(') for which magnetization goes to zero is iden- tical with the temperature TZ2) for which paramagne- tic susceptibility diverges, except in the isotropic limit itself. As the limit is approached,

<

>

t o zero as l/ln(l/A), where A is the ratio of anisotropy field to exchange field, over the entire range 0 c T <

T:". Finally, for the isotropic limit, the specific heat has been computed and is zero for T < T;'), rises sharply (but linearly) above T:') to a rounded maxi- mum at NN 1.05 T:) before tailing off t o high tempera- tures.

The most significant difference between the theore- tical two and three dimensional situations is, perhaps, the divergence of coherence length at Ti2) in the former.

Thus, just above a phase transition in two dimensions, well defined propagating magnons are expected over virtually the entire Brillouin zone. I n three dimensions, on the other hand, they will exist (if well defined at all) only out near the zone edges and corners. Although the work for the linear chain is only just getting under-

References MATTIS (D. C.), The Theory of Magnetism (Harper

and Row, New York, 1965).

KEFFER (F.). Handbuch der Phvsik Vol. XVIII12 (springer-~erlag, New ~ o r k , . 1966), 1.

KEFFER (F.) and LOUDON (J.), J. Appl. Phys., 1961, 32, 2 S.

ZUBAREV (D. N.), Usp. Fiz. Nauk., 1960, 71, 71.

BONCH-BRUEVICH (V. L.) and TYABLIKOV (S. V.), The Green Function Method in Statistical Mecha- mcs, edited by D ter Haar (North Holland, Publi Co. Amsterdam. 1962).

TAHIR-KHELI (R. A.) and ^TER HAAR (D.), Phys.

1962, 127, 88.

CALLEN (H.), Phys. Rev., 1963, 130, 890.

HMS (C. W.) and JARRETT (H. S.), Phys. Rev., 135, A 1089.

BOGOLYUBOV (N. N.) and TYABLIKOV (S. V.), Acad. Nauk. SSSR, 1959, 126, 53.

OGUCHI (T.) and HONMA (A.), J. Appl. Phys., 34. 1 153.

FIG. 2. - A qualitative sketch of the temperature dependence of magnetization as we approach the isotropic limit for the weakly anisotropic quadratic layer ferromagnet. Parameter A is the ratio of anisotropy field to exchange field and curves A S and A6 are quantitative theoretical calculations (@ = 0.74)

for spin ^7/2 and A = 10-4 and 10-2 respectively.

way, it seems clear that paramagnetic magnons will exist for this situation also over virtually the entire Brillouin zone a t low enough temperatures, even though T:') = 0 in one dimension.

Rev.,

1964, Dokl.

1963,

TAHIR-K&- (R. A.), Phys. Rev., 1963, 132, 689.

TAHIR-KHELI (R. A.), Phys. Rev., 1967, 159, 439.

TAHIR-KHELI (R. A.), Phys. Rev., 1970, 1, 3163.

COPELAND (J. A.) and GERSCH (H. A.), Phys. Rev., 1966, 143, 236.

DEMBINSKY (S. T.), Can. J. Phys., 1968, 46, 1021.

KATSURA (S.) and HORIGUCHI (T.), J. Phys. Soc.

Japan, 1968, 25, 60.

MUBAYI (V.) and LANGE (R. V.), Phys. Rev., 1969, 178, 882.

[l81 COWLEY (R. A.) and STEVENSON (R. W. H.), J. Appl.

Phys., 1968, 39, 1116.

1191 NATHANS (R.), MENZINGER (F.), and PICKART (S. J.), J. Appl. Phys., 1968, 39, 1237.

C201 FLEURY (P. A.), Phys. Rev., 1969, 180, 591.

[21] FLEURY (P. A.), J. Appl. Phys., 1970, 41, 886.

[22] SKALYO (J.), SHIRANE (G.), BIRGENEAU (R. J.) and GUGGENHEIM (H. J.), Phys. Rev. Letters, 1969, 23, 1394.

[23] BIRGENEAU (R. J.), GUGGENHEIM (H. J.) and SHIRANE (G.), Phys. Rev. Letters, 1969, 22, 720.

1241 B~RGENEAU (R. J.), SKALYO (J.) and SHIRANE (G.), J. Appl. Phys., 1970, 41, 1303.

1251 FLEURY (P. A.), Phys. Rev. Letters, 1970, 24, 1346.

1261 BIRGENEAU (R. J.), private communication.

[27] BIRGENEAU (R. J.), DE ROSA (F.) and GUGGENHEIM (H. J.), Solid State Commm., 1970, 8, 13.

1281 STANLEY (H. E.) and KAPLAN (T. A.), Phys. Rev.

Letters, 1966, 17, 913.

1291 STANLEY (H. E.) and KAPLAN (T. A.), ^J. Appl. Phys., 1967, 38, 975.

[30] KENAN (R. P.), Phys. Rev., 1970, 1, 3205.

1311 LINES (M. E.), Phys. Rev., 1964, 135, A 1336.

r321 WORTIS IM.1. Phvs. Rev.. 1964. 138. A 1126.

[33j LINES (M. E:), phYs. R&., to'be pbblished.

1341 BOWERS (R. G.) and WOOLF (M. E.), Phys. Rev., 1969. 177. 917.

[35] STANLEY '(H. E.), Phys. Rev., 1967, 158, 546 ; ibid., 1967, 164, 709.