magnon localization in one-dimensional disordered xyz heisenberg ferromagnets

phys. stat. sol. (b) 205, 633 (1998)

Subject classification: 75.10.Dg; 75.30.Et

Magnon Localization in One-DimensionalDisordered XYZ Heisenberg Ferromagnets

E. P. Nakhmedov (a, b), M. Alp (a), �O. A. SaCË li, (c),and K. Al-Shibani (d)

(a) Department of Physics, Faculty of Sciences and Letters,Istanbul Technical University, Maslak, 80626 Istanbul, Turkey

(b) Institute of Physics, Azerbaijan Academy of Sciences, H. Cavid Street 33,370 143 Baku, Azerbaijan

(c) Department of Physics, Faculty of Sciences and Letters, Marmara University,G�oztepe, Istanbul, Turkey

(d) Department of Physics, College of Science, King Saud University,P.O. Box 2455, Riyadh 11451, Kingdom of Saudi Arabia

(Received June 23, 1997)

Disordered one-dimensional XYZ Heisenberg ferromagnets with substitutional impurities are studied.The substitutional spins give rise to a random distribution of the exchange couplings. By introducinga new transformation for spin operators, the XYZ Heisenberg Hamiltonian can be exactly diagonalizedand expressed by Bose operators. Being restricted by the spin wave approximation, the Hamiltonianobtained is shown to be similar to the Anderson Hamiltonian for electronic systems. We have shownthat the magnon localization should be destroyed at the middle of the band due to the effects of com-mensurability of the magnon wavelength and lattice constant. The magnon density of states and thelocalization length are shown to have singularities at the middle of the band irrespective of the strengthof disorder. The middle of the magnon band can be reached not only by changing the magnon energybut also by varying the external magnetic field which facilitates the observation of the phenomenon.

Eindimensionale XYZ Heisenberg-Ferromagnete mit substitutioneller Unordnung werden untersucht.Die substituierenden Spinmomente verursachen eine zuf�allige Verteilung der Austauschkopplungen.Mit Hilfe einer neuen Transformation der Spinoperatoren l�a�t sich der XYZ Heisenberg-Hamiltonianexakt diagonalisieren und durch Bose-Operatoren ausdr�ucken. Der resultierende Hamiltonian ist vomSpinwellentyp und ist dem Anderson-Hamiltonian �ahnlich. Wir beweisen, da� aufgrund derKompatibilit�at der Magnonenwellenl�ange und Gitterkonstante, die Magnonlokalisierung in der Mitteder Zone zerst�ort wird. Unabh�angig von dem Grad der Unordnung, sind Magnonenzustandsdichteund Lokalisierungsradius in der Mitte der Zone singularit�atsfrei. Die Mitte des Magnonbandes ist so-wohl durch Variation der Magnonenenergie, als auch durch Variation des �au�eren Magnetfeldes zuerreichen. Diese Tatsache soll die Beobachtung des beschriebenen Ph�anomens erleichtern.

1. Introduction

Effects of disorder on the kinetic and thermodynamic properties of quasi-particles insolids are intensively studied [1 to 4]. A wide knowledge has been reached in understand-ing of Anderson localization in low-dimensional electronic systems. The electron±elec-tron interactions in the presence of a random potential have been also taken into accountwhich reduce the density of single electronic states and change the thermodynamic prop-erties of low-dimensional metals [1, 2].

E. P. Nakhmedov et al.: Magnon Localization in 1D Heisenberg Ferromagnets 633

Relatively little attention has been paid to the magnon localization in magnetic sys-tems [5, 6]. Since magnons are Goldstone bosons, the character of the spin wave localiza-tion in a random potential changes from ªstrong localizationº to ªweak localizationºregime with increasing magnon energy. It is known from the localization theory for elec-tronic systems that in one-dimensional (1D) and two-dimensional (2D) disordered sys-tems all electronic states are localized irrespective of the degree of randomness [1, 2].Therefore, all magnon states in 1D and 2D magnetic systems are expected to be loca-lized for arbitrary magnon energies. However, if a 1D disordered system has a crystallinestructure with substitutional impurities, the strong Bragg reflections of electrons havebeen shown to destroy the localization at the center of the band [7, 13]. In this paper,we shall study the effects of substitutional impurities on the magnon localization in 1DXYZ Heisenberg ferromagnets in an external magnetic field. Generally speaking, disor-der in this system may be caused by the random distribution over sites of the magnitudeof spin, the exchange coupling constant I�n; m�, the anisotropy parameters g and D,and the magnetic moment m. Here we shall investigate particularly the randomnesswhich is connected to the exchange coupling constant I�n; m�. However, the generaliza-tion of the results found is easy for the cases when all the parameters presented aboveare random.

This paper is structured as follows. In Section 2 we introduce a new transformation tobosonize and diagonalize the XYZ Heisenberg Hamiltonian. Assuming that the numberof spin waves is conserved, we present the Hamiltonian written in the harmonic approx-imation. In Section 3 we study the magnon density of states r�e�, and show that r�e�has a Dyson singularity near the middle of the band for an arbitrary strength of impur-ity potential. In Section 4 the magnon localization length lloc is calculated. It is shownthat lloc can diverge at the center of the band. In this case the density±density correla-tor decreases as a power law. The center of the magnon energy band may be reached byvarying the external magnetic field.

2. The Hamiltonian

We consider the anisotropic XYZ Heisenberg chain of spins in an external magnetic fieldH, the Hamiltonian of which is given as

H � ÿ 12

Pm;n

I�n; m� f�1� g� SxnSxm� �1ÿ g� SynSym � DSznSzmg ÿ

Pn

m0BSzn ; �1�

where Sn � fSxn; Syn; Szng is an Heisenberg spin operator on the site n with magnitudeS0. The exchange coupling I�n; m� in eq. (1) is taken to be positive as to ensure thatthe system is ferromagnetic and the parameters of anisotropy vary in the intervals0 � D � 1 and ÿ1 � g � 1.

2.1 Diagonalization of the anisotropic Heisenberg Hamiltonian

To study the properties of the low-lying elementary excitations as spin waves we applythe following canonical transformation for the operators S�n � Sxn � Syn at site n :

Sÿn � un��2S0

pa�n

��1ÿ a

�n an2S0

sÿ vn

��2S0

p ��1ÿ a

�n an2S0

san ; �2�

634 E. P. Nakhmedov, M. Alp, �O. A. SaCË li, and K. Al-Shibani

S�n � un��2S0

p ��1ÿ a

�n an2S0

san ÿ vn

��2S0

pa�n

��1ÿ a

�n an2S0

s; �3�

Szn � S0 ÿ a�n an ; �4�where a�n and an are Bose creation and destruction operators, respectively. The coeffi-cients un and vn are defined from the following conditions: (i) the operators S�n and Sÿmmust satisfy the commutation relation �S�n Sÿm� � 2Szndn;m and (ii) the bilinear terms thatappear after applying the transformations (2) to (4) to the Hamiltonian (1) must van-ish. As a result we obtain two equations for the coefficients un and vm,

u2n ÿ v2

n � 1 ; �5�ÿunvm � 1

2 g�unum � vnvm� � 0 : �6�To solve the system of equations (5) and (6) we use the symmetry of the second equation,

unvm � 12 g�unum � vnvm� � umvn ;

which gives the condition

vmum� vnun� a

for the arbitrary site number n. This condition allows us to solve eqs. (5) and (6).The solution of eqs. (5) and (6) gives the following results for the coefficients un and vn :

un � g��2�g2 ÿ 1�

��1ÿ g2

p�

q ; �7�

vn ��1ÿ

��1ÿ g2

p2

��1ÿ g2

ps: �8�

The validity of the results obtained is easily verified. For g! 0 the Hamiltonian shoulddescribe the XXZ Heisenberg model. For the bosonization of this model the Holstein-Primakoff transformation is required. Indeed, the condition of g! 0 turns our transfor-mations (2) to (4) into the Holstein-Primakoff transformation ensuring vn ! 0 andun ! 1 with g! 0.

After these canonical transformations we get the following exact expression for theHamiltonian:

H � E0 �Pn�m0B� DL�n�� a�n an ÿ

1

2DPn;m

0 I�n; m� a�n ana�mam

ÿ S0

��1ÿ g2

p Pn;m

0 I�n; m��1ÿ a

�mam2S0

sa�n am

��1ÿ a

�n an2S0

s; �9�

where E0 is the ground state energy given as

E0 � ÿm0NS0Bÿ 12 NSs DL�0�

and is ignored in the following calculations. Also,

L�n� � S0

Pm 6�n

I�n; m� and L�0� � S0

Pn 6�m

I�n; m� : �10�

Magnon Localization in 1D Disordered XYZ Heisenberg Ferromagnets 635

In the linear spin wave approximation we also neglect two-magnon interactions andhigher terms in eq. (9). Assuming the number of spin waves are conserved, we restrictourselves to the following harmonic Hamiltonian:

HL �Pn�m0B� DL�n�� a�n an ÿ S0

��1ÿ g2

p Pn;m

0 I�n; m� a�mam : �11�

For g! 0 and D! 1 the isotropic results are realized from eqs. (9) and (11).

2.2 Impurity Hamiltonian

A disorder in the system is assumed to be realized due to the random substitution of theimpurities on the sites of the host spins. In this case the crystalline structure maintainsits regularity. We assume that the magnitude S0 of the spin is the same across thesystem. However, the exchange coupling varies randomly from site to site as

I�n; m� � I0�nÿm� � dI�n; m� : �12�Substituting eq. (12) in the linearized Hamiltonian (11), we obtain

HL � H0 � Himp : �13�Here, H0 is the magnon Hamiltonian for a pure system given as

H0 � �m0B� DL�0��Pna�n an ÿ S0

��1ÿ g2

p Pn;m

0I0�nÿm� a�n am ; �14�

where

L�0� � S0

Pn �6� 0�

I0�n� : �15�

The impurity Hamiltonian Himp in eq. (13) has the form

Himp � ÿS0

��1ÿ g2

p Pn;m

0 dI�n; m� a�n am : �16�

The spin wave is assumed to be scattered elastically by impurities.Performing the Fourier transformations given as

an � 1��Np P

k

ak eikna and a�n �1��Np P

k

a�k eÿikna �17�

with a being the lattice spacing, we get the following expressions for H0 and Himp :

H0 �Pk

e�k� a�k ak �18�and

Himp �Pk; k0

V �kÿ k0� a�k ak0 : �19�

The magnon energy spectrum e�k� in eq. (18) is

e�k� � m0B� DL�0� ÿ��1ÿ g2

pL�k� ; �20�

where

L�k� � S0

Pn �6�0�

I0�n� eikna : �21�


Also, the impurity potential V �kÿ k0� in eq. (19) has the following form:

V �kÿ k0� � ÿS0

��1ÿ g2

p 1

N

Pn;m

0 DI�n; m� eÿikn� ik0m : �22�

In what follows we shall study the nearest-neighbor spin interaction, i.e. the exchangecoupling

I�n; m� � I0 6� 0 for m � n� 1 and = 0 otherwise :

Then the magnon energy spectrum e�k� becomes

e�k� � m0B� 2S0I0�Dÿ��1ÿ g2

pcos ka� ; �23�

where I0 is the exchange coupling constant of the nearest neighbors.For the unfilled magnon band, i.e. if ka < 1, the cosine term in eq. (23) can be ex-

panded up to second-order terms in k as

e�k� � e0 � �h2k2

2m*; �24�

where

e0 � m0B� 2S0I0�Dÿ��1ÿ g2

p� �25�

and

m* � �h2

2S0a2I0

��1ÿ g2

p �26�

being the magnon effective mass.We write the impurity Hamiltonian for the nearest-neighbor spin interaction in real

space as

Himp � ÿS0

��1ÿ g2

p Pn�dI�n; n� 1� a�n an� 1 � dI�n; nÿ 1� a�n anÿ 1�

�PnV �n� �a�n an� 1 � a�n anÿ 1� ; �27�

where we propose

dI�n; n� 1� � dI�n; nÿ 1� � dI�n�and take

V �n� � ÿS0

��1ÿ g2

pdI�n� :

The correlators of the random potential V �n� in the Born approximation are

hV �n�iimp � 0 and U�nÿm� � hV �n�V �m�iimp 6� 0 ;

where h. . .iimp denotes averaging over the random potential. In this paper we considerthe case when the random potential V �n� of chaotically distributed substitutional spinsis arbitrary. So all the impurity correlation functions

Uk�n1; n2; . . . ; nk� � hV �n1�V �n2� . . . V �nk�iimp

are nonzero essential. The mean distance cÿ1 between the impurities of concentration cis assumed to be large with respect to the magnon wavelength l, i.e. c� lÿ1. This is


equivalent to the condition of P �e� l�e�=�h� 1. The expressions (13) to (16) for the one-particle Hamiltonian with off-diagonal random potential describe the motion of a quan-tum particle in a 1D regular lattice where one has a random field of substitutional impu-rities. Here the spin wave localization problem is mapped onto the Anderson electronlocalization. We shall study the magnon density of states as well as density±densitycorrelator using the model under consideration.

3. Magnon Density of States

The magnon density of states is calculated by using the known formula

r�e� � ÿ 1

pIm hG��n; n ; e�iimp ; �28�

where G��n1; n2 ; e� is the retarded Green function (GF) of a magnon with energy e,and n1; n2 are the numbers of sites. Since the linearized spin wave Hamiltonian (13) isequivalent to the Hamiltonian for 1D disordered electronic systems, we can use the re-sults obtained for the density of states of electronic systems.

The density of states for 1D disordered crystals was investigated by many authors [7to 14]. These studies reveal the so-called Dyson singularity [15] near the middle of theband. The singularity in the phonon density of states was shown first by Dyson [15], fora 1D disordered harmonic chain.

To study the magnon density of states we use the Berezinskii diagrammatic technique[16], which was applied to electronic systems for the summation of diagrams in realspace.

In the coordinate±energy representation the unperturbed magnon Green functionstake the following forms:

G�0 �n; n0; e� � � i

�hv�e� exp f�P �e� jnÿ n0j ag ; �29�

where G�0 and Gÿ0 represent the retarded and advanced Green functions, respectively.Also v�e� and P �e� are the magnon velocity and momentum given as

v�e� ��2�eÿ e0�m*

s; �30�

P �e� ��2m*

�h2�eÿ e0�

s: �31�

The magnon effective mass m* and the energy e0 are defined above by expressions (26)and (25), respectively.

When the magnon energy e approaches to e0, the spin wavelength increases rapidly asl � 2p=P �e0� ! 1. There is no magnon with energy less than e0, so the Green func-tions G�0 �n; n0; e < e0� decrease exponentially for e < e0. The character of localizationchanges from ªweak localizationº regime to ªstrong localizationº regime with increasingmagnetic field or magnetic anisotropy D in z-direction. This crossover is also possible ifthe modulus of the anisotropy parameter g decreases. The condition for ªweak localiza-tionº is P �e� l�e�=�h� 1 which is equivalent to �eÿ e0� t=�h� 1. Here l�e� is the magnonfree path and t � l�e�=v�e� is the magnon relaxation time. On the other hand, when the


Ioffe-Regel criterion of P �e� l�e�=�h � 1 is satisfied, the ªstrong localizationº regime takesover.

The usual cross technique [17] is used to average over the impurity positions. Thesummation of diagrams is performed in two steps. First of all, the diagrams correspond-ing to multiple scattering of a spin wave by a single impurity, are summarized. As aresult of such summation, the Born amplitude for a single scattering is renormalized togive the total amplitudes f� and fÿ for forward and backward scatterings. The ampli-tudes f� and fÿ are connected by the unitary relations [10, 13]

ÿ�f� � f*�� jf�j2 � jfÿj2

and

ÿ�fÿ � f*ÿ� � f�f*ÿ � fÿf*� :

The magnon mean free path l is determined by the reflection coefficients of h � jfÿj2 foran individual impurity as

l � 1

chwith 0 � h � 1 : �32�

Below we give the main characteristic pecularities of the Berezinskii diagrammatictechnique. As the unperturbed Green functions G�0 �n; n0; e� are factorable (see, eq.(29)), the coordinate dependence can be transferred from the lines to the scatteringvertices. The selection of essential diagrams is based on neglecting the contribution ofthose containing strongly oscillating factors of the type exp fiP �e� niag (We shall usehereafter the unit �h � 1.) Due to the condition of �P �e� l�e��ÿ1 < 1 the average of thisfactor over the impurity sites ni gives small contribution. When the magnon energy isnear the middle of the band, i.e. P �e� ' P0 � p=2a, a new class of essential diagramsappears due to the periodicity of the disordered 1D crystal. The contribution of thesediagrams leads to a singularity in the density of magnon states r�e� near the middle ofthe band.

According to eq. (28) the diagrams in the real space for the density of states start andend on the same point x � an. Therefore all diagrams can be cut in two parts along thevertical xx axis, each part of which is denoted as Rm�x� with x � an. The density ofstates r�e� is expressed in terms of Rm�x� as

r�e� � r0�e� 1� 2 ReP1m� 1

R2m

� �; �33�

where

r0�e� �1

pv�e� �1

p

��m*

2�eÿ e0�

s; �34�

Rm�x� is the sum of the contributions of all the individual right parts having 2m lines incross-section. By shifting the point x, the following recurrent equation for Rm�x� is ob-tained [10, 11]:

ÿ d

dxRm�x� � c

PsVmsRm� s e4is�P0 ÿP � x ÿ cRm � cR0f

2mÿ e4im�P ÿP0� x �35�


with

Vms �PkCk

2mCk� 2s2m� 2sÿ 1�fÿ�2k� 2s �1� f��4mÿ 2k : �36�

The solution of eq. (35) for Rm [10 to 12] makes it possible to study the magnon densityof states near the middle of the band. When the amplitude f� for forward scattering hasreal value, the density of magnon states turns out to have singularities in all rationalpoints of the band and the Dyson singularity at the center of the band is enhanced.From eq. (31), f� can be expressed as

1� f� � �1ÿ h�1=2 eij0 ;

where h � jfÿj2 is the reflection coefficient and j0 is a phase shift in the forward scatter-ing. For j0 � 0; f� takes real values.

Equation (35) for Rm can be solved for large m and for arbitrary values of h with theboundary conditions R�0� � 1 and R�1� � 0. Near the middle of the band, whenk � ÿi4�P �e� ÿ P0� lm� 1 is satisfied, the following solution for R�k� is obtained [12]:

R�k� ' A ln k ; �37a�where

A � 1= ln �ÿi4�P �e� ÿ P0� l� : �37b�Using this result in the expression (33) for the magnon density of states, a singularity ofthe Dyson type is obtained near the middle of the band,

r�e� � r0

ÿ2p

4ljP �e� ÿ P0j ln3 j4l�P �e� ÿ P0�jB�h� : �38�

The coefficient B�h� varies in the interval 2 � B�h� <1 with 0 � h � 1.In the limit of rather strong scattering potential �h! 1�, the magnon density of

states displays peaks at all rational points of the band, when the magnon momentumsatisfies the relation

P �e� � pm

ka; m � �1; �2; . . . ; ��kÿ 1� ; k > 1 : �39�

Equation (35) is easily solved in the region of P �e� 6� P0 � p=2a for h � 1. The densityof states r�e� is expressed for k � 1 in the following form [12]:

r�e� � r0 2p d�4l�P �e� ÿ P0�� p

4l�P �e� ÿ P0� sinhp

4l�P �e� ÿ P0�� 24 352

8><>:9>=>; :

�40�It should be noticed that the middle of the band can be approached not only by chan-ging the magnon energy but also by varying the external magnetic field. For an arbi-trary value of the magnon energy e, there exists a value of the magnetic field Bc satisfy-ing the relation P �e� � P0 � p=2a which in turn gives the following value for Bc :

mBc � eÿ 1

2S0I0 D� p2

8ÿ 1

� � ��1ÿ g2

p� �: �41�

The increase of the external magnetic field could also give rise to crossover from stronglocalization regime to weak localization.


4. Magnon Localization Length

To study the magnon localization length lloc, the density±density correlator

c0�x; x0; t� � hG��x; x0; t�Gÿ�x0; x ; 0�iimp

must be calculated, where x � na. The stationary behavior of the density correlator, i.e.

P1�xÿ x0� � limt!1 c0�xÿ x0; t� ; �42�

gives the character of the localization as

lÿ1loc � lim

jxÿ x0 j!1ln P1�xÿ x0�

xÿ x0�� : �43�

According to the Berezinskii technique [16, 10], the density correlation function is calcu-lated in a coordinate representation, where the unaveraged diagram, corresponding toG��x; x0; e� w=2�Gÿ�x0; x ; eÿ w=2�, consists of two lines going from the pointx0 � an0 to the point x � an. Each line consists of segments �x; x1�;�x1; x2� . . . �xi; xi� 1� . . . �xk; x0�. The segments represent the ªbareº GF (29) and eachscattering site xi corresponds to the impurity potential V �xi�. As a result of the aver-aging, these potential group together in correlators as Uk�n1; n2; . . . ; nk�� hV �n1� V �n2� . . .V �nk�iimp. The coordinate dependence is transferred from the lines tothe vertices due to the factorizable character of the GF (29). The vertical lines crossingthrough x0 and x can subdivide each diagram into three parts, namely right, central andleft. The sum of the contribution of all the individual right (left) parts having 2m1�2m01�lines of retarded GF and 2m2�2m02� lines of advanced GF in every cross-section is de-noted by Rm1m2

�x� � ~Rm01m02�x0��. Also the sum of the individual central parts is shown as

Zm01m

02

m1m2 �x0; x�. By shifting the point x, the following equations for Rm1m2�x� and

Zm01m

02

m1m2 �x0; x� can be obtained [13, 10]:

ÿ d

dxRm1m2

� c Ps1; s2

vm1s1v*m2s2

Rm1 � s1;m2 � s2exp f2ix��P0 ÿ P1� s1 ÿ �P0 ÿ P2� s2�g

ÿ cRm1m2� cR0�fÿ�m1 �f*ÿ�m2 exp f2ix��P1ÿ P0�m1ÿ �P2 ÿ P0�m2�g; �44�

where

P1; 2 � P �e� w=2� and vms �PkCkmC

k� sm� sÿ 1�fÿ�2k� s �1� f��2m� 2k : �45�

Also for Zm01m

02

m1m2 �x0; x�d

dxZm01m

02

m1m2 � cPs1; s2

wm1s1wm2s2

Zm01m

02

m1 � s1;m2 � s2

� exp fÿ2ix��P0 ÿ P1� s1ÿ �P0ÿ P2� s2�g ÿ cZm01m02

m1m2 � i�P1ÿ P2� Zm01m02

m1m2 ; �46�where

vms �PkCkmC

k� sm� s�fÿ�2k� s �1� f��2mÿ 2k� 1 : �47�

Introducing the new functions

P 0m1m2

� 12 �Rm1m2

�Rm1 � 1;m2 � 1� �48�


and

Q0m1m2

� �ÿ1��m1 ÿm2�=2 P1m0

1;m0

2� 0

1

l

�1x0

dx eik�x0 ÿx� Zm01m

02

m1m2 �x0; x� P 0m0

1m0

2�ÿ1��m01 ÿm02�=2

� exp f2i��P1 ÿ P0� �m1xÿm01x0� � �P0 ÿ P2� �m2xÿm02x0��g ; �49�

the density correlator c0�k; w; e� can be expressed by P 0m1m2

and Q0m1m2

as

c0�k; w; e� � 2l

v�e�P1

m1;m2 � 0

P 0m1m2�w� �Q0

m1m2�k; w� �Q0

m1m2�ÿk; w�� : �50�

The behavior of the localization length lloc can be studied near the middle of band ate! 0 by using equations (42) and (50). The more interesting results are obtained whenthe phase j0 of forward scattering amplitude becomes zero.

For small values of the reflection coefficient, i.e. h� 1, the Born approximation isrealized. The density distribution P1�x� falls as a power law at the center of the band[11, 12],

P1�x� � 1

2p2

�10

z dz eÿz2jxj=2 pz

2 sinh 12 pz

� �4

: �51�

The asymptotic form of P1�x� for jxj � 1 is

P1�x� ' 1

8��pp jxjÿ3=2 �52�

and

P1�0� � 13 :

This result shows that at the center of the band, i.e. for P �e� ) P0 � p=2a, the magnonlocalization length lloc becomes infinite. For a small but finite deviation from the bandcenter, i.e. for jP �e� ÿ P0j � 1; P1�x� must decrease exponentially at sufficiently largevalues of x [7, 13]

P1�x� � exp �ÿjxj=lloc�e�� ;where

lloc�e� ' bl ln2 l P �e� ÿ p

2a

� �� : �53�

The value of coefficient b is approximately 1/2. The singular behavior of lloc at thecenter of band becomes valid for sufficiently large values of h. For an asymptotic strongimpurity potential, i.e. h � 1, a singularity in the localization length disappears and lloc

becomes comparable with the mean free path, l. When the forward scattering shifts thephase of spin wave �j0 6� 0�, the singularity in the middle of the band turns into asmooth peak with a finite value. For a strong scattering potential �h � 1� the singularityin lloc takes place at all rational points of band as

P �e�0 �pm

ak; m � �1; �2; . . . ; ��kÿ 1� ; k > 1 :


5. Conclusions

The magnon density of states and localization length are studied for 1D disordered XYZHeisenberg ferromagnetic crystals. The randomness in the system is introduced by sub-stitutional spins at sites which cause the random distribution of the exchange couplingconstant over the systems. The disorder caused by the eigenvalues S0 of the impurityspins, the anisotropy parameters D and g and the magnetic moment m0, can be easilytaken into account.

By using the new canonical transformations (2) and (3) we exactly diagonalized andrepresented the XYZ Heisenberg Hamiltonian (1) by Bose operators. The problem wasstudied in a spin wave approximation, so many-magnon interactions were neglected. Theobtained Hamiltonian expressed by (13) to (19) is similar to the Anderson Hamiltonian.However, the possible commensurability of the magnon wavelength l�e� with the crystalperiod a, i.e. l�e� � �2k=m� a, m � 1; 2; . . . ; �kÿ 1� for k > 1 gives rise to the existingdelocalized states at the rational points of the energy band. As a result of this effect, themagnon density of states and localization length reveal a singularity at the middle of theenergy band. It should be noticed that the middle of the band can be reached not only byvarying the magnon energy but also by varying the external magnetic field.

In conclusion, it would be useful to estimate the transverse structure factors Sxx�q; w�and Syy�q; w�, and the longitudinal structure factor Szz�q; w�. These spin correlationfunctions can be expressed by Green functions in the harmonic approximation as

Sxx�q; w� �Pn

eÿiqan�

dt eiwthTr r0Sxn�t� Sx0 �0�iimp

� iS0

��1ÿ g

1� g

s P1n�ÿ1

�dt eiwtÿ iqanhG�n; 0; t�iimp ; �54�

Syy�q; w� �Pn

eÿiqan�

dt eiwthTr r0Syn�t� Sy0�0�iimp

� iS0

��1� g

1ÿ g

s P1n�ÿ1

�dt eiwtÿ iqanhG�n; 0; t�iimp ; �55�

Szz�q; w� �Pn

eÿiqan�

dt eiwthTr r0�dSzn�t� dSz0�0��iimp

� ÿPn

eÿiqan�

dt eiwthG�0; n ; t�G�n; 0; t�iimp ; �56�

where r0 is the equilibrium density matrix and

dSzn�t� � S0 ÿ Szn�t� :As is seen from eqs. (54) to (56) the transverse structure factors Sxx�q; w� and Syy�q; w�are expressed by the average value of the magnon Green function. However, the long-itudinal structure factor Szz�q; w� is expressed by the density±density correlator. There-fore, the singularities in the magnon density of states and the localization length couldbe observed experimentally by measuring the dynamic structure factors.

Recently the great achievement in synthesizing the quasi-1D molecular ferromag-nets [18, 19], namely decamethylferrocenium tetracyanothenide [DMeFe-TCNE],


(Fe�C5�CH3�5�2�� C2�CN�4�ÿ, makes it possible to study experimentally the low tem-perature properties of these materials. Because of the small value of the interchain ex-change coupling constant ��1 K) with respect to the intrachain exchange integral��30 K) the low temperature properties of these quasi-1D systems can be successfullydescribed by the properties of individual chains.

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644 E. P. Nakhmedov et al.: Magnon Localization in 1D Heisenberg Ferromagnets

magnon localization in one-dimensional disordered xyz heisenberg ferromagnets

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