anderson localization of spin waves in random heisenberg antiferromagnets

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PHYSICAL REVIEW 8 VOLUME 35, NUMBER 10 1 APRIL 1987 Anderson localization of spin waves in random Heisenberg antiferromagnets Jun-ichi Igarashi* Department of Physics, University of California, Los Angeles, California 90024 (Received 6 October 1986) Anderson localization of spin waves is studied for Heisenberg antiferromagnets with random uni- axial anisotropy energy and random exchange interaction. We show that spin waves are localized in the long-wavelength limit in the presence of the uniaxial anisotropy energy. The effect of quantum fluctuations which do not conserve the number of spin waves is estimated to be small. The longitu- dinal spin-correlation function which is sensitive to the spin-wave diffusion constant is calculated by use of a scaling argument near the mobility edge. Our most important result is that localization should also, against expectation, strongly affect the transverse spin-correlation function. It leads to a pronounced, temperature-dependent shift in the spin-wave energy near the mobility edge as a re- sult of the enhanced effect of spin-wave interaction for nearly localized spin waves. This could be observed in neutron scattering experiments. I. INTRODUCTION Recently, there has been much progress in understand- ing Anderson localization of electrons in a random poten- tial. ' It is known that all electron states are localized in one and two dimensions, and that there is a "mobility edge" in three dimensions (the energy at which the states change their character from localized to extended). It is also known that the density of states of single-particle states in a random potential is profoundly affected by electron-electron interaction. ' In addition to the electron case, localization effects on other quasiparticles (phonons ' and photons ) have been studied for systems in which the mass density or dielectric constant takes a random value. There is an important difference in these cases: Since phonons and photons are Cxoldstone bosons even in the presence of this type of ran- domness, the elastic mean free path depends strongly on the energy. This symmetry requirement prevents phonons and photons from being localized in the long-wavelength limit. In contrast to these cases, little work has been done on the localization of spin waves, except the recent work of Bruinsma and Coppersmith, who studied random Heisenberg ferromagnets (FM) within the harmonic ap- proximation. They showed that the localization problem of the system with random uniaxial anisotropy was sim- ply mapped onto the Anderson model for electron locali- zation. They discussed localization effects on the longitu- dinal spin-correlation function on the basis of this equivalence. Experiments are easier to perform on antiferromagnets (AFM), because there is no nuclear scattering to obscure central peaks and more suitable materials are available. Localization theory of AFM, however, is more complicat- ed than that of FM because of quantum fluctuations which do not conserve the number of spin waves and two kinds of spin wave modes which are degenerate without external magnetic field. Spin waves of AFM cannot be simply mapped onto the Anderson model of electron lo- calization. The purpose of the present paper is to con- struct a theory of localization of spin waves in Heisenberg AFM's with random uniaxial anisotropy energy and ran- dom exchange interaction, by focusing on the correlation functions which are determined by neutron-scattering ex- periments. We first study the longitudinal spin-correlation func- tion and later discuss the transverse spin-correlation func- tion. Because the longitudinal spin-correlation function is governed by diffusive motion at small momenta and low frequencies, it is directly related to the localization prob- lem. Quantum fluctuations are unimportant for diffusive motion, because they only give rise to a small renorlriali- zation of impurity potential, as estimated in Appendix B. Neglecting quantum-fluctuation terms, we find that the remaining diagrams are the same as those of the self- consistent theory of Vollhardt and Wolfle for electronic systems. By applying their theory to our system, we find that spin waves are localized in the long-wavelength limit, as in the electronic case, in the presence of anisotropy en- ergy, while random exchange interaction does not give rise to the localization of spin waves in the long-wavelength limit in the absence of anisotropy energy (just as for pho- nons and photons). Since the mobility edge is located near the bottom of the energy band of spin waves in the pres- ence of anisotropy energy, the "strong localization" re- gime is also important for calculating the longitudinal spin-correlation function. Making use of the scaling argu- ment of Imry, Gefen, and Bergmann, ' we can calculate the longitudinal spin-correlation function not too far from the mobility edge. The transverse spin-correlation function is dominated by the spin-wave peak. Within the harmonic theory, ran- domness merely gives rise to a spectral width because of elastic scattering. There is no anomaly near the mobility edge. When we take account of spin-wave interaction, the peak shifts toward higher energies. Using perturbation theory, we discuss the effect of randomness on spin-wave interaction. We show that the shift is strongly enhanced by the diffusive motion of spin waves in the disordered 35 5151 1987 The American Physical Society

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Page 1: Anderson localization of spin waves in random Heisenberg antiferromagnets

PHYSICAL REVIEW 8 VOLUME 35, NUMBER 10 1 APRIL 1987

Anderson localization of spin waves in random Heisenberg antiferromagnets

Jun-ichi Igarashi*Department of Physics, University of California, Los Angeles, California 90024

(Received 6 October 1986)

Anderson localization of spin waves is studied for Heisenberg antiferromagnets with random uni-

axial anisotropy energy and random exchange interaction. We show that spin waves are localized inthe long-wavelength limit in the presence of the uniaxial anisotropy energy. The effect of quantumfluctuations which do not conserve the number of spin waves is estimated to be small. The longitu-dinal spin-correlation function which is sensitive to the spin-wave diffusion constant is calculated byuse of a scaling argument near the mobility edge. Our most important result is that localizationshould also, against expectation, strongly affect the transverse spin-correlation function. It leads toa pronounced, temperature-dependent shift in the spin-wave energy near the mobility edge as a re-sult of the enhanced effect of spin-wave interaction for nearly localized spin waves. This could beobserved in neutron scattering experiments.

I. INTRODUCTION

Recently, there has been much progress in understand-ing Anderson localization of electrons in a random poten-tial. ' It is known that all electron states are localized inone and two dimensions, and that there is a "mobilityedge" in three dimensions (the energy at which the stateschange their character from localized to extended). It isalso known that the density of states of single-particlestates in a random potential is profoundly affected byelectron-electron interaction. '

In addition to the electron case, localization effects onother quasiparticles (phonons ' and photons ) have beenstudied for systems in which the mass density or dielectricconstant takes a random value. There is an importantdifference in these cases: Since phonons and photons areCxoldstone bosons even in the presence of this type of ran-domness, the elastic mean free path depends strongly onthe energy. This symmetry requirement prevents phononsand photons from being localized in the long-wavelengthlimit.

In contrast to these cases, little work has been done onthe localization of spin waves, except the recent work ofBruinsma and Coppersmith, who studied randomHeisenberg ferromagnets (FM) within the harmonic ap-proximation. They showed that the localization problemof the system with random uniaxial anisotropy was sim-ply mapped onto the Anderson model for electron locali-zation. They discussed localization effects on the longitu-dinal spin-correlation function on the basis of thisequivalence.

Experiments are easier to perform on antiferromagnets(AFM), because there is no nuclear scattering to obscurecentral peaks and more suitable materials are available.Localization theory of AFM, however, is more complicat-ed than that of FM because of quantum fluctuationswhich do not conserve the number of spin waves and twokinds of spin wave modes which are degenerate withoutexternal magnetic field. Spin waves of AFM cannot besimply mapped onto the Anderson model of electron lo-

calization. The purpose of the present paper is to con-struct a theory of localization of spin waves in HeisenbergAFM's with random uniaxial anisotropy energy and ran-dom exchange interaction, by focusing on the correlationfunctions which are determined by neutron-scattering ex-periments.

We first study the longitudinal spin-correlation func-tion and later discuss the transverse spin-correlation func-tion. Because the longitudinal spin-correlation function isgoverned by diffusive motion at small momenta and lowfrequencies, it is directly related to the localization prob-lem. Quantum fluctuations are unimportant for diffusivemotion, because they only give rise to a small renorlriali-zation of impurity potential, as estimated in Appendix B.Neglecting quantum-fluctuation terms, we find that theremaining diagrams are the same as those of the self-consistent theory of Vollhardt and Wolfle for electronicsystems. By applying their theory to our system, we findthat spin waves are localized in the long-wavelength limit,as in the electronic case, in the presence of anisotropy en-

ergy, while random exchange interaction does not give riseto the localization of spin waves in the long-wavelengthlimit in the absence of anisotropy energy (just as for pho-nons and photons). Since the mobility edge is located nearthe bottom of the energy band of spin waves in the pres-ence of anisotropy energy, the "strong localization" re-gime is also important for calculating the longitudinalspin-correlation function. Making use of the scaling argu-ment of Imry, Gefen, and Bergmann, ' we can calculatethe longitudinal spin-correlation function not too far fromthe mobility edge.

The transverse spin-correlation function is dominatedby the spin-wave peak. Within the harmonic theory, ran-domness merely gives rise to a spectral width because ofelastic scattering. There is no anomaly near the mobilityedge. When we take account of spin-wave interaction, thepeak shifts toward higher energies. Using perturbationtheory, we discuss the effect of randomness on spin-waveinteraction. We show that the shift is strongly enhancedby the diffusive motion of spin waves in the disordered

35 5151 1987 The American Physical Society

Page 2: Anderson localization of spin waves in random Heisenberg antiferromagnets

5152 JUN-ICHI ICxARASHI 35

system, because the spin waves spend a comparativelylarger time in a given region in space. Since the trans-verse spin-correlation function is more easily measurablethan the longitudinal spin-correlation function, this pro-vides us with an alternative means of looking for localiza-tion effects in neutron-scattering experiments.

The paper is structured as follows. In Sec. II, wedescribe our model Hamiltonian. In Sec. III, we calculatethe longitudinal spin-correlation function and discuss theAnderson localization. In Sec. IV, we discuss the effectsof spin-wave interaction. Section V is devoted to conclud-ing remarks.

II. HAMILTONIAN

We employ the Heisenberg model with random ex-change interaction and anisotropy energy:

a;=X ' gagek

b =% ' gbkek

(2.4)

In order to diagonalize the quadratic part of the bosonoperators with the average anisotropy energy and ex-change interaction, we introduce the Bogoliubov transfor-mation:

The Fourier transforms of boson operators a; and bj aredefined by

H = g 2J;)S;S) —g D; (S,') —g D/(S~')(i j& i

(2.1) k kak vkP kb ——k vkak +11kP (2.5)

where (i,j ) indicates a sum over pairs of nearest neigh-bors and indexes i and j refer to site a (up) and b (down)sublattices, respectively. The JJ ( =Jp+5J 1 ) and D;(=Dp+5D; ) are random variables, with configurationalaverages

where

1/23 +ukUk=

28k

1/23 —Ek(2.6)

(D;); p—Dp, (5D; ); p

——b„/(2S)

( J,) ); p——Jp, (5J~J ) =4,„/(2S)

(2.2)

where S denotes the magnitude of spin. D;, Dj, and J;jare assumed to be positive in order to guarantee antifer-romagnetic order at low temperatures. In our units,A= k~ ——1.

We use the Holstein-Primakov transformation as fol-10 s:11 12

with

ek ——(3 gk)', A—=2JpSz+2DpS,

11k 2JpS——~yk, yk =~ ' g exp(ikp) .P

(2.7)

Here z and p denote the number of nearest-neighbor sitesand the vector to the nearest neighbors, respectively. Wepick a body-centered-cubic lattice for concreteness. Apartfrom a constant term, we get the following expression forthe HamiltonianS+ =V2S 1— a; a; a;, Sl+ 3/2Sbj 1 ——— bj b&4S ' ' " ' ' 4S

=v'2S a; 1 — a; a;, Sl ——3/2S 1—4S

bb- b4S ' '

J

H =HP+~lnt+~imP (2.8)

(2.3) where

Hp =g Ek(akak +PkPk ) (2.9)

kl, k2, k3, k4

(ck k g g ak ak ak ak +ck g g, g Pk, Pk, Pk, Pg +ck k1k3k4ak~Pk1ak3Pk4)5(k, +k2 —k3 4)a-p

+terms with nonconserving spin-wave number, (2.10)

imp g 1 kk akak +'gk Pk'Pk'+ vk'k akP k'+'( vkk' ) akP k' ~— —a-a 8-p a-p a-p

k, k' (2.1 1)

with

Ck k k k = (JpZ/2%)(11k Uk Vk Vk Yk + 1lk 1lk 1lk Uk Yk + 11k Uk 11k k Yk +Uk, Vk 11k Vk ) k—11k Uk 11g Vk 'Yk k

—(Dp/%)(ak ak Qk ak +Uk Uk Uk Uk ) ~ (2.12)

Page 3: Anderson localization of spin waves in random Heisenberg antiferromagnets

35 ANDERSON LOCALIZATION OF SPIN WAVES IN RANDOM . . ~ 5153

Ck k k k (Joz/N)( uk uk uk vk Yk +uk vk vk vk rk +uk uk vk uk rka-P

+Uk Qk Uk Uk pk +Uk Uk Uk Qk ~k2+ Ukl ~k2+k3~ k4~k3

+Uk Uk ~k Uk yk +~k Uk ~k +k V k 2~k ~k ~k ~k Fk —k

2uk vk vk uk rk —k 2vk uk uk vk rk +k 2vk vk vk vk rk —k, )

—4(DO/N)(uk vk uk vk, +vk uk, vk uk, ) ~

Vkk =2S[ 5D'(k —k')ukuk +5D (k —k')vkvk +5J(k —k', 0)ukuk

+5J(O,k' —k)vkvk 5J(k—, k')ukvk —5J( —k', —k)vkuk ],Vkk

——2S [ 5D'(—k —k')ukvk —5D "(k k')vk—uk 6J(k——k', 0)ukvk

—5J(O, k' k)vk—uk +5J(k, k')ukuk +5J( k', ——k)vkvk ],5D' ' '(k) =(1/N) g 5D; (i)exp( ikR;—(~)),

i (j)

5J(k, k') =(1/N) g exp( ikR;)5—J~ exp(ik'RJ ) .

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

—J()z/(4N)+ (2.18)

2Jpz 2 2 2 8DpC~ = (u, +v, )(u, —v, ) — u,v,N

The H;„, describes spin-wave interaction, whereKronecker's 5 represents the conservation of momentum.The coefficient Cp k k k is given by Eq. (2.12) inter-

changing u and U. In the long-wavelength limit, we find

Jpz 4 4Coooo= uovo(uo —vo) — (uo+vo)

N N

Note that the contribution of random anisotropy energy isenhanced by a factor up+Up from the bare value b.„andthat the contribution of random exchange interaction is afactor 2/z smaller than that of random anisotropy energy.In the absence of anisotropy energy, the scattering due torandom exchange interaction vanishes with k, k'~0, as itshould be by rotational invariance. In random FM, evenin the presence of anisotropy energy, the scattering due torandom exchange interaction vanishes in the long-wavelength limit.

——Jpz/N+ . -- (2.19)

III. LONG ITUDINAL SPIN-CORRELATIONFUNCTION

The H; ~ describes the scattering of spin waves due torandom anisotropy energy and random exchange interac-tion. The Vg~ is given by Eq. (2.14) interchanging u andv The last t. wo terms in Eq. (2.11) represent quantumfluctuations which do not conserve the number of spinwaves. In the long-wavelength limit, we get

In this section, we study the longitudinal correlationfunction defined by

Fs(„)(k,co)= f e' '

)& « 5Ss („)(k,t)5Sg („)(—k, O)) ); p, (3.1)

&l

vik'l

&)mp- (ukuk +vkvk )''N

where the suffixes g and u stand for the "total" spin andthe "staggered" spin, respectively:

2 2+ z (uk —vk ) (uk —vk )N

(2.20)5Ss(k) =5S,'(k)+5Sb(k), 5S„'(k)=5S,'(k) —5Sb(k),

(3.2)

Jpz Dp—(1/N)4D '" 2J

(2.21)with

5S,'(b)(k)=N '~ g (S,'(J) —&S,'(J) ) )

i (j)

The rms deviations of 6D and 6J are at most the order ofthe average values of Dp and Jp,

X exp( i kR; (~) ) .— (3.3)

( 1/N)( J0Sz)Jpz

21+—z

6, -(2D()S), b,,„-(2JoS)So &

lVoo

l ); pisoforder

) stands for a thermal average. The total magneti-zation is a conserved quantity, so the correlation functionsatisfies the diffusion equation for small momenta andfrequencies. The staggered magnetization is not a con-served quantity and is reduced by zero-point quantumfluctuations. Nevertheless, the corresponding correlation

Page 4: Anderson localization of spin waves in random Heisenberg antiferromagnets

5154 JUN-ICHI IGARASHI 35

function also satisfies the diffusion equation for smallmomenta and frequencies, as shown below. Since dif-fusive motion is strongly affected by backscattering due toself-interference of wave packets, consideration of thesefunctions should reveal the nature of the Anderson locali-zation. We first consider the positive anisotropy energyand later discuss the random exchange interaction.

(p) happ (k tvt qn):

a I a

aen+ &)9+k

a&n9

a I"'a

Spp +

a~n+~ i

p+k

a&n9

I a

a&n9'

A. D;,D, )0First we introduce the single-particle G-reen s function:

G (k, iE„)=—j dre

(c)

(~)a I a a a

+

a a

X/t

a

a a

X

1

'L /

aX

a

/X

a

//

/

X

X ( ( T,[ak(r)ak(0)] ) );—ic r (3.4)

G~(k, ie„)= — dr e0

X ( ( T,[lgk(r)pk(0)] » ; p,

where T stands for the temperature, ~ the imaginary time,c,„=2vrnT with n being an integer, and T, the tirne-ordering operator. G and G~ are equivalent withoutexternal magnetic field. The lowest-order diagram of thecontribution of impurity scattering to the self-energy isshown in Fig. 1(a). In the long-wavelength limit, the ma-trix element is replaced by its zero-momentum value:

p(E) = E(e —Ep)' /(JpSz)2

(3.7)

The real part of the self-energy is neglected for simplicity.Note that while the elastic lifetime, r( s ), diverges as(E —Ep) ', the mean free path, l(e)=c(e)r(s), remainsconstant near c.—c0.

FIG. 2. (a) Diagram for P~~(k, cp&, e„). (b) Diagrammaticequation for the vertex function [I ~~(k, rp&, e„)] . . (c) The ir-reducible vertex function [I ~~ (k, rp„e„))

77 (JpSz)l(e)- a0[ba Jpz/(4Dp)+ b,,„Dp/(2Jp)]

(3.8)

[6 (up+vp)+5 z(up —vp) ]( I/N)

Xl Cn —E,k

(3.5)

The imaginary part of the self-energy of the retardedCareen's function is given by

X (k, e+i6) — is[a, (u p—+ p)v+6 „z(up —vp) ]P(e)

Jpzl(e)-(vr/2)

2Dp

1

(1+2/z)(3.9)

The Careen's functions for the total spin and the stag-gered spin are reduced to

because the spin-wave velocity c(E) vanishes at Ep. Herea0 is the lattice constant. In the typical situation given byEq. (2.22), l (c.) is quite long,

2

l

2r(c. )

where p(E) stands for the spin-wave density of states,

(3.6)

(a)a a a

(b)a p a

(c)a a a a a

&n

k

&nk'

//

X

&n

k

&n -&n &n

k -k' k

//

//

X

&n &n &n &n &n

k

X X

Path 1

a p a p a/ /

//

X

a p a p p a+

X

'x

X

Path 2FICx. 1. Diagrams for the self-energy of elastic scattering of

6 (k, ic,„). The dashed lines represent the impurity potential.(a) and (b) represent the second-order perturbation of the impur-ity potential. (b) comes from quantum fluctuations. (c) and (d)are examples of higher order perturbation. In order to get thediffusion pole in Eq. (B5), the contribution of (d) has to be addedto that of (a) ~ FICx. 3. Sketch of the contour, C, composed of paths 1 and 2.

Page 5: Anderson localization of spin waves in random Heisenberg antiferromagnets

35 ANDERSON LOCALIZATION OF SPIN WAVES IN RANDOM. . . 5155

Gg (k, cubi)

= —T g (1/N) g (up up—

)Ppp (k,cu„E„)(up —u~ ),n P~P

(3.10)

Gg (k, cubi)

where Pzz(k, cubi, e„) is defined by the diagrams in Fig.2(a). The other type of diagrams generated by the totalspin and the staggered spin gives rise to only a smallcorrection to Eqs. (3.10) and (3.11) as far as the singularbehavior for small momenta and frequencies is concerned,as shown in Appendix A.

We take the sum over c.„by using the replacement,

= —T g (1/N) g (uz+v~)P~~ (k,co„E„)(uz +u~ ),1 p dc

2ni "&n

With / E,n ~F (3.12)

P~P

(3.1 1)where the contour C is given by the solid line in Fig. 3.By replacing ice] by co+i 5, we get

G' (k, co)

G„' (kco) f ds[nr(e) —nT(E+co)](1/N) g (u~+u~)P~~ (k, co, E)(u~ +u~ ),2&l

P'~P

(3.13)

where Ppp (k, co, E) is defined by Fig. 2(a) with the replacement of G (p +k, i

E„+icosi�

) and G (p, i e„) by the retarded func-tion, G (p+k, E+~), and the advanced function, G (p, E), respectively. Pz~ (k, co, E) is a strongly peaked function atEz ——Ez

——s. Since we are interested in the singular behavior with co, we have omitted the G (p+k, E+co)G (p, E) term.In the first factor, nT(E) comes from path 1 and nT(a+co) from path 2, where nr(co) is the Bose distribution function,1/(e "/ —1). The correlation function is found by applying the fluctuation-dissipation theorem to Eq. (3.13):

Fg (k, co)1

ImGg (k, co)e co/T —1—

F„(k,co) ~ ImG„" (k, co)

[u (s)—u (E)]=f dEnT(E)[1+nT(E+co)]X '2 2 q

X'u (e)+u (c, ) 2mi

1 1Im —g P~~ (k, co, s) .

N(3.14)

The vertex function approximately satisfies the integral equation shown in Fig. 2(b):

[r„.(k,~„s„)]....=[r,',".(k,~„s„)]....+g [rI,",.(k,~„s„)]....q, (k,~„E„)[r,-,.(k,~„e„)]....,P

(3.15)

with I "being the irreducible vertex function, and

(k, co&-, e„)=G (p"+k,iso&+iE )G „(p iE "), . „ (3.16)

Quantum fluctuations make the vertex function couple to other channels. However, such corrections to the vertex func-tion only give rise to a small renormalization of impurity potential, as estimated in Appendix B. We neglect quantumfluctuations in the following analysis.

To lowest order in the impurity concentration, the irreducible vertex function is found by keeping the first diagram inFig. 2(c). In the long-wavelength limit, it becomes

(&) ~a 4 4 ~ex 4[r~~ (k,co„E„)] . =I p= (up+up)+ z(up —up)pp & n aa aa— N

By summing the ladder diagrams, we get

(3.17)

(1/N) g Ppp (k, co,e ) = (1/N) g fq (k, co,c).P~P

1 —r, g q, (k,~,.)— 2mp(c)

ico+Dp(e)k—(3.18)

where the diffusion constant is given by

(JpSz) (2 2)1/2

Dp(E) = —,' c'(s)r(E) =- ao

3 [A, (up+ut)+b, ,„z(up —up) ] s(3.19)

Now we consider higher-order corrections. If we neglect quantum fluctuation terms, then the diagrams to be con-sidered are the same as electron's, and the localization theory of electron system of Volhardt and Wolfle can be applied

Page 6: Anderson localization of spin waves in random Heisenberg antiferromagnets

5156 JUN-ICHI ICzARASHI 35

to our system. (See Appendix C.) The diffusive form of Eq. (3.18) still holds with a renormalization of the diffusionconstant:

D '(co, s) =D '(e) 1+ g (p-k)&G (p)[l "(k,co, c)] . &G (p')(p '-k)2n.p(c, )

(3.20)

with

b, G (p) =G (p +k, s+ co) —G (p, E), (3.21)

where I z~ (k, co, c, ) is defined by the diagrams of I "with the same replacement as mentioned before in the definition ofp and k denote the direction of the vector p and k.

The most important diagrams for an analysis of localization are the "maximally crossed diagrams, " shown in Fig. 4. '

By summing the ladder diagrams, we get

[I q~ (k, co, c)] . =[5,,(uo+Uo)+b. ,„z(uo —Uo) ] r(s) ice+—Do(c)~ p +. p'~

(3.22)

for ez ——Ez——e, small

~ p +p ~, k, co. By inserting this into Eq. (3.20), we get the diffusion constant,

D '(co, e) =Do '(c, ) 1+~p(E) g —i co+Do(E)Q

(3.23)

where the sum over Q is restricted within Q &1(s) '. We take account of inelastic scattering effects phenomenologicallyby introducing the lifetime r;„, and replace —ice by —in+ 1/r;„ in Eq. (3.23). The result is

Do 1+ 1 — tan2m' Dppl

for co~;„&&1

D '(co, E)- '

Dp 1+ 1 11 — v'6cor —for cur;„» 1,

2~ Dopl 4

(3.24)

where 1;„=(Doe,„)'/ in Eq. (3.24), the backscatteringterm due to self-interference of wave packets diverges, be-cause Dppl ~0 with c.~cp, the perturbation breaks down.

In the self-consistent theory of localization at zero tem-perature, Do(s) in the denominator of Eq. (3.23) should bereplaced by D(co, s}:

with4

E, —so ——(3/m. )2 2

(JOSz}

X [h, (u,"+Uo }+b,,„z(uo —Uo)']' . (3.27)

D (~ s)/Do(E) = 1— 1

2m. Do(s)p(E)

x f,' Q'dQ

i co/D (co, E ) +—Q(3.25)

So c, —cp is of order

(192/~ )(J0Sz)JpZ

4t . 221+—z

2 2cc —c,oD(~~O, s) =Do(s) 1 —

2—cofor e & E, , (3.26)

n+(d~ 6n+~)P+ )t P'+)t

In the static limit, this equation becomes for the condition given by Eq. (2.22). For E & s„ there isno solution of D(co= , 0)&s0: spin waves are localized.The localization length, g(s ), which is defined bylim 0[ ice/D(co, k)]=/ —(E), is given by

g(c) — E [6 (uo+Uo)3 1 4 4

4m (JOSz) 2

I

X +II

//

x 3c/

I /

XX X X

+ ~ ~ ~

+b,,„z(uo —Uo) ] ~

E —s,~

ao,

a&n

P

a&n

P

FICx. 4. Maximally crossed diagrams.

(3.28)

where v= 1 within this theory. [For c, &c,„the charac-teristic length g is also given by Eq. (3.28) in analogy to

Page 7: Anderson localization of spin waves in random Heisenberg antiferromagnets

35 ANDERSON LOCALIZATION OF SPIN WAVES IN RANDOM. . . 5157

the localization length. ]Even near the mobility edge, (1/N) g, P~~ (k, co, E)

can be written as

where the diffusion constant becomes k and co dependent.A scaling argument is given by Imry, Gefen, and Berg-mann. By considering the static limit,

(1/N) g P~~ ( k, co, E ) = 2m.p(c. )

i co—+D (k, co, e)k(3.29)

D(k, co, e)k &co,

we find the diffusion constant,

(3.30)

(I „) /r;„ for l „=(D*gr;„)' «g and kl „«1D(k, co=0,e) — g /r;„ for I „=(D*r;„)'~&&g and kg&&1

D*gk for kg, kl „»1,

(3.31)

(3.32)

(3.33)

where D' is the macroscopic diffusion constant,

D' -Do(E)l (e)/g(e) . (3.34)

The situation is illustrated in Fig. 5. By substituting Eqs. (3.31)—(3.33) into Eqs. (3.14) and (3.29), we find the correla-tion function for (e, —Ep) « T as

F„(k,co~0) 1 nT(E)p(E)Fg (k,co~0) = dc

(up+Up)' nk' D(k, ~~O, E)

g JOSz 2 2 ]/2 3nT(e, ) h& 4 4 4 (E,, —ep)' (e, ep) r;„—

e(kao) [b (uo+vp)+b, ,„z(uo —Uo) ]

9 [b,, (up+Up)+b, ,„z(up —Up) ]+h2 7

7T' (JpSz)'(3.35)

with

1 3 [A~(up+Up)+5~„z(up —Up) ]Io= nT(e&)T E) '1+m.(kao) 2m-3 (JoSz) 2m'Dp(ei)p(E])1(e])

(3.36)

where h&

and h2 are numerical constants of order unity.The first term proportional to v;„ in Eq. (3.35) is the con-tribution of region I and III, and the second term is thatof region II. The Io represents the contribution of the re-gion where the perturbation theory is valid. The lowestenergy of this region is denoted by E.&, which is of orderEp+ h (E, —eo) with h —3—4. Note that if the exponent ofthe correlation length, v, is not equal to 1, the dependenceon k and ~;„of the contribution from region II is not kbut

B. D; =DJ ——0

Now we consider random exchange interaction in theabsence of anisotropy energy. As already mentioned inSec. II, the scattering due to random exchange interactionvanishes in the long-wavelength limit. So we expand thescattering potential with respect to small momentum.The result is

(3.39)

and

k —2 (1—1/v)/3+in or in «

k-"-"] for k-'«I, „.

(3.37)

(3.38)

z ~- c1/r(E) =167T JoSz JoSz

(3.40)

The elastic lifetime of spin waves is estimated from thesecond-order diagram of the self-energy:

'4

The power law with an exponent 3 —1/v is the same asdiscussed in Ref. 7 for FM.

Therefore the mean free path l(E) diverges for the low en-

ergy of spin waves. With increasing spin-wave energy,

Page 8: Anderson localization of spin waves in random Heisenberg antiferromagnets

5158 JUN-ICHI IGARASHI 35

1(e) decreases and the backscattering term due to self-interference of wave packets becomes important. Usingthe self-consistent theory described in the preceding sub-section, we find for the diffusion constant

D(to~0, E)=Dp(e)[1 —(s/e, ) ) for E &e, ,

where

(JpSz) JpSz4

Dp(E) =- Opex E,

(3.41)

(3.42)

e, /(JpSz) =1/3

1617 ~ (JOSz)

Z ex(3.43)

(nThis result is the same as the case of localization of pho-nons. In a typical situation, the mobility edge c, can beestimated as

e, /(JpSz)-4. 7 for b, ,„-(2JpS), z =8 . (3.44)

At these energies we are not anymore in the long-wavelength regime and the inelastic lifetime, as estimatedfrom nonrandom systems, ' is then of the same order asthe elastic lifetime. Thus we should expect that in thiscase localization effects are obscured by inelastic scatter-ing.

IV. TRANSVERSE SPIN-CORRELATION FUNCTION

In this section we study the transverse spin-correlationfunction defined by

EO EC

FIG. 5. Sketch of the various length scales near the mobilityedge. For I „«k ', the boundary between regions I, II, andIII is given by dashed lines. In region II, the relevant lengthscale is I „. For k '« l „, the boundary between regions I, II,and III is given by dotted lines. In region II, the relevant lengthscale is k

F („)(k,p))= I e' '((5$g+(„)(k,t)5' („)(—k, O)) ); ), ,

(4.1)

with

5Sg (k) =5S, (k)+5$b (k),5$„—(k) =5S,—(k) —5'—(k),

(4.2)

and discuss how randomness modifies the effect of spin-wave interaction. We restrict ourselves to the case thatpositive anisotropy energy is present in the following dis-cussion.

Within the harmonic approximation, the transversespin-correlation function is given by applying thefluctuation-dissipation theorem on Eqs. (3.4) and (3.5):

ek ek y (Ckq +Ckq )nT(Eq )

q

(4.4)

with

The spin-wave peak is broadened by the elastic scattering.There is no critical behavior with passing through the mo-bility edge.

Spin-wave interaction gives rise to a temperature-dependent shift in the spin-wave energy, which is ob-served by measuring the peak positions of the transversespin-correlation function. The shift can be estimated bytaking account of the Hartree term. In pure systems, it isgiven by

Fg (k, tp)

2S(uk —Uk )

F„(k,tp)

2S(Ok +Uk )

=1 [1+nT(cp)](2r(cp) )

(tp Ek)'+ [2r(tp)]—

kqqk+ k k + qkP +~qkqk

Ckq ——Ckqgq .

(4.5)

In the long-wavelength limit, we find from Eqs. (2.18) and(2.19) that

[2r(p))]+nT co(p)+Ek) +[2r(p))]

Cpp ~Jp+ + ' ' ' Cpp Jpg + o ~ ~aa aP (4.6)

(4.3)The leading terms are canceled in Eq. (4.4).

The diffusive motion of spin waves in random systems

Page 9: Anderson localization of spin waves in random Heisenberg antiferromagnets

35 ANDERSON LOCALIZATION OF SPIN WAVES IN RANDOM. . . 5159

enhances the interaction between the same kind of modes(a-a and P-P). The above-mentioned cancellation of theleading terms of interaction is destroyed. This is differentfrom the case of FM, where the spin-wave interaction due

to exchange interaction always vanishes in the long-wavelength limit. The self-energy enhanced by random-ness is found by taking account of the diffusion propaga-tor (Fig. 6):

X(kit )=(—T) pe " g . Cppten ek+Q Xk+g(lEn )

X 1++ (i'k" (Q ~] E E )[rk"k(Q ~1k"

(4.7)

The vertex function I kk to the lowest-order approximation is calculated by summing the ladder diagrams as discussed inthe preceding section. The sum over e„ is replaced by the integration given by Eq. (3.12). The result is

X (k,at)=— 1 ~ de ~ —1

2~i 0 e c fT 1 &P ck+g —i/27

1 2 1 1

&(E) i (at——E)+Dp(e)Q r(E) [ i (co ——e)+Dp(E)Q ](4.8)

where the sum over Q is restricted within Q &l (co). The Ek+~ is approximated by ek. Summing over Q, we get thereal part of the self-energy,

Re[X (k co)]= — C Nae' —1 & (e —Ek) +[2r(E)] 4vr Dp(E)1(E)r(E) 16%2m' Dp(e) ~ r(e) V~

co —E~

(4.9)

1 1 1 1Jpzn7—(ek )ap +

4~ Do(Ek )l (ek )r(ek ) 8' [Dp(sk )r(Ek )]for

~

co —Ek~

r(Ek ) && 1, Tr(ek ) && 1 .

(4.10)

The imaginary part of the self-energy vanishes for Tr(Ett ) » 1. The diagram of the self-energy due to the Cooper propa-gator is given by reversing the direction of the loop in Fig. 6(b). The contribution should be the same as from the dif-fusion propagator, if the momentum dependence of the interaction is neglected. The renormalized spin wave energy c isgiven by

Ek ——ek+Re[X (k, ek)] . (4.11)

1, 040

(o)

(b)

a4J)k

a~ ~n~ k+Q

a(dik

a, e„, k+Q a, a„, k+Q

y a+

~V

1.035

twk

C

E'p

1.030-

a a aQJ t (d ik k

aQt ik

a au) i

k

a a a a(d 1 4) tk k

I

0.05koo

I

0, 1

I

0.15

FICs. 6. The self-energy due to spin wave interaction in theHartree approximation. The wavy lines represent the interac-tion. (a) For nonrandom system. (b) For random system. Theblocks with slanting lines represent the vertex function Ii.e., the diffusion.

FIG. 7. Renormalized spin-wave energy as a function ofmomentum in the condition given by Eq. (2.22) with D0 ——J0and S =2. The dashed line represents the critical momentumcorresponding to the mobility edge. Near this line our theory isinapplicable.

Page 10: Anderson localization of spin waves in random Heisenberg antiferromagnets

5160 JUN-ICHI IGARASHI 35

The bare value, Ek, in Eq. (4.3) should be replaced by this renormalized value.We take account of higher-order corrections due to randomness by replacing Do(s) in the above expressions with the

renormalized diffusion constant, D (co~0,s). By using Eq. (3.26), we find

[~a(uo+uo)+~exz("0 —uo) ] Ek 3 Ek —so 3V'3Sk Ek +JOZnT(ek ) 12 5 2 2+

(JQSz) 277 ck —Eq 4&

2 2 3/2Ek —~o

2 2Ek —&c

(4.12)

This relation is displayed in Fig. 7. The shift dependsstrongly on temperatures. This behavior is quite differentfrom the temperature-independent shift calculated by thecoherent potential approximation within the harmonictheory. The shift becomes very large when we approachthe mobility edge. The large shift however simply meansthe breakdown of the Hartree approximation. The imagi-nary part of the self-energy is zero within our theory, buthigher-order terms will give a finite contribution, whichcould be divergent at the mobility edge.

V. CONCLUDING REMARKS

We have developed a theory of localization of spinwaves in random AFM. AFM are important, because ex-periments are easier to perform on AFM than on FM.The theory of AFM looks more complicated than that ofFM, because quantum fluctuations and two kinds ofspin-wave mode are present. However, they are unimpor-tant for diffusive motion, because they only give rise tosmall corrections, as estimated in Appendices A and B.So the essential character of localization of AFM is quitesimilar to that of FM.

We studied the longitudinal spin-correlation function,which was sensitive to the spin-wave diffusion constant.Both the longitudinal correlation function for the totalspin and that for the staggered spin were governed by thesame diffusive mode. The intensity for the staggered spinwas larger than that for the total spin by a factor( u 0 + u 0 ) . We also found the following.

(1) In the presence of random anisotropy energy, spinwaves are localized in the long-wavelength limit, andthere is a mobility edge in the vicinity of the bottom ofthe spin-wave band. (2) In the absence of anisotropy ener-

gy, random exchange interaction does not give rise to thelocalization of spin waves in the long-wavelength limit,and localization occurs in the high-energy region just as

with phonons and photons. (3) The temperature-dependent part of the longitudinal spin-correlation func-tion is given by c(kao) nT(E, )r;„ in the low-frequencylimit, where c is a numerical constant. If we use the esti-mate of r;„ in nonrandom systems, ' r,„-T exp(EO/T)Measuring the temperature dependence on the longitudi-nal spin-correlation function in the low-frequency limit isuseful to check the theory.

In addition we studied the transverse spin-correlationfunction which was dominated by the spin-wave peak anddiscussed the effect of spin-wave interaction. By takingaccount of the diffusion propagator and the Cooper prop-agator coupled to the Hartree term of the interaction, wefound a pronounced, temperature-dependent shift inspin-wave energy near the mobility edge by the enhancedeffect of spin-wave interaction for nearly localized spinwaves. This behavior could be observed in the transversespin-correlation function by neutron-scattering experi-ments. Our theory is however not applicable in the vicini-ty of the mobility edge. In order to discuss the energyshift and the lifetime of spin waves in this "critical" re-gion, we have to consider higher-order effects of interac-tion terms.

In this paper, we confined ourselves to positive aniso-tropy energy. If we allow anisotropy energy to take ran-domly positive and negative values, the problem may bequite difficult, because the long-range order of AFM isdestroyed and the concept of spin wave loses its meaning.To study low-lying excitations in such a system may be aninteresting problem left for the future.

ACKNOWLEDGMENTS

The author expresses his sincere thanks to Professor R.Bruinsma for valuable discussions and for the criticalreading of the manuscript. He is also grateful to IBM andUCLA for financial support.

as

APPENDIX A: VALIDITY OF EQS. (3.10) and (3.11)

In this Appendix, we examine the validity of Eqs. (3.10) and (3.11). The total spin and the staggered spin are written

5S'(k)=N ' g(u —u )(a +ka —P kP ),P

5S„'(k)=N ' g [( u~+u)( a~+k~ap+~ kp ~) —2u~u~(a~+kp ~+p ~ ka~)] for small k .P

(A 1)

(A2)

The singular contribution to the longitudinal correlation functions in small k and co comes mainly from the coupling tothe vertex function [I zz (k, co, s)] . , the diagrams of which in the lowest-order perturbation are shown in Fig. 8. Thefollowing relations are satisfied in small k and m:

N 'gg~ (k, co, E)=N 'gG (p+k, s+co+i5)G (p, E i5) —2mp—(E)~(e),P

(A3)

Page 11: Anderson localization of spin waves in random Heisenberg antiferromagnets

35 ANDERSON LOCALIZATION OF SPIN WAVES IN RANDOM. . . 5161

(0)aE' +(zi + I Sp+k

(b)

P—6 -(z) —

I S-p-k

{c)

—E' —(z) —I S

-p-k

ae' +t() + I Sp+k

r aa; ii, v

aE + (z) + I Sp+k

raP; ii.v

06 -(zi —

I S-p-k

6 + (zi + I Sp+k

rPP; )uv

E' -(z) —IS-p-k

(2

t —ISP

P- e +ISP

(IE —ISP

P-f +ISP

(It-ISP

0-q+IS

P-6+ IS

FIG. 8. Diagrams coupled to I .„.These are created by5S~(k) and 6S„'(k). FIG. 9. Various channels of the vertex function.

g(t)p (k, (v, E)=N g Gp( —p —k, —e —(v i5—)G (p, E i5)———] ap —] —imp(E)

P P

g Pp (k, ro, e)=N g G (p+k, E+co+i5)Gp( —p, —e+i5)——1 pa —1 i rrp( e)

P P

—pp P(ep )dip ~0

N ggp (k, (Q, E)=N QGp( —p —k, E —co i—5)G—( —p, —c+i5)—( E+ ep ) 4~(JQSz)

(A4)

(A5)

(A6)

The diagrams shown in Figs. 8(c) and 8(d) are canceled due to Eqs. (A4) and (A5). The contribution of the diagramshown in Fig. 8(b) is a factor RQF smaller than that of the diagram shown in Fig. 8(a), due to Eqs. (A3) and (A6), whereR&F is given by

RQF 3 [5 (uo+vo}+5 z(uQ vQ) ]4 4 4

4m. (J()Sz)(A7)

So RQF is of order (W2/2')(DQ/Joz) r (1+2/z) for the condition given by Eq. (2.22). Equations (3.10) and (3.11) arederived from the diagram shown in Fig. 8(a), so the neglected terms are at most a factor RQF small.

APPENDIX 8: QUANTUM FLUCTUATION IN THE VERTEX FUNCTION

In this Appendix, we estimate perturbationally theshow that it is small.

Generally speaking, quantum fluctuations make thefined in Fig. 9. The equation for the vertex function is

(i) (i) aa (i) apraa;pv r aa;pv r aa;aa(It r aa;ap(ti(i) (i) aa (i) ap

rap;pv r ap;pv r ap;aalu r ap;ape{i) + (i) aa (i) ap

rpa;pv r pa;pv r pa;aalu r pa;apt(i) (i) aa (i) apr pp p r pp p r pp;aalu r pp;apl

contribution of quantum fluctuation to the vertex function and

(i) par aa, pa4

r (i)y pa

(i) par pa; pap

r (i)q pa

r(i)

happ

r pp pp~

I pq

I p.p„(B1)

vertex function a 4&&4 matrix, the components of which are de-given by

We abbreviated the notation by omitting the explicit dependence on k, (v, and c.. 1 ". g 1 .p should be read as(i)$p" ( rpp" )aa;aalu p" ( rp",p')aa;pv.

To lowest order in the impurity concentration, the irreducible-vertex function I "is given by

r.".' ..=r pp) pp= [a.(u,'. +v,')+ a,„z(u, —v, )']/N —=r, ,

(i) (i) (i) (i) (i) (i)~ 'p. p=~ p -p ——~ 'p. p =~p . P=~pp. aa=~aa. pp

=[5.,2u v +A,„z(u —v ) ]/N—=I, ,

(i) (i) (i) (i) (i) (i) (i) (i)I aaap= I aapa= I paa= I pa; = I ppap= I pppa= I appp= I papp

=[—z5 uovo(uo+vo)+mezz(uo —vo) ]/N .

(B2)

(B3)

(B4)

Because N ' g gpP(k, Qi, c)= N' g (tip (k, co, E—)

[Eqs. (A4) and (A5)], the contribution of the secondcolumn and that of the third column are canceled in Eq.(Bl). Then I .p„ is given by with

r.".' „.+ r,y«(1 r,qP.P) 'r pp—-1 [r,+r', q pp(1 rg—pp) I]q-—-

Page 12: Anderson localization of spin waves in random Heisenberg antiferromagnets

5162 JUN-ICHI IGARASHI 35

P""=gg~p"(k, a), e) .P

(B6)

The last term in the denominator represents the renormal-ization of the scattering potential due to quantum fluctua-tions. By noting that I of —1, and I o and I

~are the

same order of magnitude, we find that the last term in thedenominator is an order of R&F smaller than the barevalue I 0. The second term in the numerator of Eq. (B5) isalso smaller than the first term in the same order. Itshould be noted here that the contribution of the second-order diagram shown in Fig. 1(a) to the imaginary part of

I

the retarded function of the self-energy is not sufficient,and the contribution of the diagrams shown in Fig. 1(d)has to be added to it, in order to get the diffusion singu-larity from the denominator of Eq. (BS).

APPENDIX C: DERIVATION OF EQ. (3.20)

Since diagrams are the same for electrons with neglect-ing quantum fiuctuation terms, the theory of Volhardtand Wolfle for electrons is applied to the present prob-lem. We review this theory in this appendix.

We consider the equation for the response function P~~'

(k, cu, e)=P (k, co, E)5 ~ +g P (k, co, E)[I " (k, co, E)] . P " (k, co, E) .P

Defining b, G(p)=G (p+k, c, +co) —G"(p, c, ), we obtain

(Cl)

~—k. ' —r,' „(E+~)+r,"(E) y„.(k,~,s)= —aG(p) s„.+g[r,'," (k, ~-, e)]....y, , (k,~, E)ap '+

II(C2)

By keeping the correspondence between the self-energy and the irreducible-vertex function just as Fig. 1(c) and Fig. 2(c),we can prove the Ward identity,

k(e+ ) —&"(s)=y [I "(k, , &)] . &G (p") .P

By using this identity and summing over p and p, we get the following equation:

cog(k, co, E)—PJ(k, n), c, ) =2mip(e),

where

tI)(k, a), s) =N ' g Ppp (k, co,s),PP

(C3)

(C4)

(C5)

$~(k, co, E)=N ' g k.Ppp (k, co, E) .

PP (C6)

By expanding with respect to the angle of p on g, P~z (k, co, e), we get the relation

gP~~(k, co,s)= [—2~ip(E)—] 'bG(p) g [1+d(p k)(p" k)]P~-~(k, co, E), (C7)P PP

where p and k stand for the direction of the vector p and k. By multiplying k.BE~/Bp in Eq. (C2) and summing over pand p', we get

[co+M(k, co, e)]PJ (k, co, e) — P(k, co, c. ) =0,k c (E)—(C8)

with

M(k, co) = + g (p.k)b, G (p)[I " ] . b, G (p'),r(E) 2mp(E)

where d is the dimension of the system. By solving Eqs. (C4) and (C8), we get

~(k )2vri p(E)

cu+iD (co,E)k

with

(C9)

(C10)

D '(co, s) =D '(e) 1+ g (p k)b G(p)[I "(k,co, e)] . EG(p')(p'. k)2vrp(e)

(Cl 1)

Equation (Cl 1) is the Eq. (3.20) given in the text.

Page 13: Anderson localization of spin waves in random Heisenberg antiferromagnets

35 ANDERSON LOCALIZATION OF SPIN WAVES IN RANDOM . ~ . 5163

'Permanent address: Department of Physics, Osaka University,Toyonaka Osaka 560, Japan.

For a review, P. A. Lee and T. V. Ramakrishnan, Rev. Mod.Phys. 57, 287 (1985).

~B. L. Altshuler and A. G. Aronov, Zh. Eksp. Teor. Fiz. 77,2028 (1979) [Sov. Phys. —JETP 50, 968 (1979)).

P. A. Lee, Phys. Rev. B 26, 5882 (1982).4S. John, H. Sompolinsky, and M. J. Stephen, Phys. Rev. B 27,

5592 (1983).5E. Akkermans and R. Maynard, Phys. Rev. B 32, 7850 (1985).6A. A. Cxolubentsev, Zh. Eksp. Teor. Fiz. 86, 47 (1984) [Sov.

Phys. —JETP 59, 26 (1984)].~R. Bruinsma and S. N. Coppersmith, Phys. Rev. B 33, 6541

(1986).D. Volhardt and P. Wolfle, Phys. Rev. B 22, 4666 (1980); in

Anderson Localization, edited by H. Nagaoka and H. Fukuya-ma (Springer, Berlin, 1982), p. 26.

Y. Imry, J. Appl. Phys. 52, 1817 (1981); Y. Irnry, Y. Gefen,and D. J. Bergmann, Phys. Rev. B 26, 3436 (1982).The use of the scaling argument is the same as in Ref. 7. Inthis paper we discuss in more detail inelastic effects.T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).

~~R. M. White and R. Freedman, Phys. Rev. B 10, 1039 (1974).L. P. Gor'kov, A. I. Larkin, and D. E. Khmelnitzkii, Pis'maZh. Eksp. Teor. Fiz. 30, 248 (1979) [JETP Lett. 30, 228

(1979)].' A. B. Harris, D. Kumar, B. I. Halperin, and P. C. Hohenberg,

Phys. Rev. B 3, 961 (1971); in the presence of anisotropy ener-

gy, w;„ is estimated from the imaginary part of the self-energyof the second-order diagram of spin-wave interaction for thek =0 spin wave of nonrandom systems,

1/(2~;„)= 7T(1 —e /T)

X g nr( aE, )[1+ nr( ae, )]k (,k2, k3

x[1+nr(«)]6(k, —kp —kg)3

x lc l'@so+«, —«

3 21 ~p T

exp ( —cp / T)4~ g (Jp+z ) Jp

L

for T «cp,where C

lis the matrix element given by Eq. (2.10). The

small contribution of the order Dp/(Jpz) to the matrix ele-

rnent is neglected.