Download - Frustrated impurity spins in ordered two-dimensional quantum antiferromagnets

Frustrated impurity spins in ordered two-dimensional quantum antiferromagnets

A. V. Syromyatnikov* and S. V. MaleyevPetersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300, Russia

�Received 2 August 2006; revised manuscript received 3 October 2006; published 30 November 2006�

Dynamical properties of an impurity spin coupled symmetrically to sublattices of ordered two-dimensionalHeisenberg quantum antiferromagnets �i.e., frustrated impurity spin� are discussed at T�0 �existence of asmall interaction stabilizing the long range order at T�0 is implied�. We continue our study on this subject�Phys. Rev. B 72, 174419 �2005��, where spin-1

2 defect is discussed and the host spin fluctuations are consid-ered within the spin-wave approximation �SWA�. In the present paper we �i� go beyond SWA and �ii� studyimpurities with spins S�1/2. It is demonstrated that in contrast to defects coupled to sublattices asymmetri-cally longitudinal host spin fluctuations play an important role in the frustrated impurity dynamics. We showthat the effect of the host system on the defect is completely described by the spectral function as it was withinSWA. The spectral function, that is proportional to �2 within SWA, acquires terms proportional to �2 and �T2

originating from longitudinal host spin fluctuations. It is observed that the spin-12 impurity susceptibility has the

same structure as that obtained within SWA: the Lorenz peak and the nonresonant term. The difference is thatthe width of the peak becomes larger being proportional to f2�T /J�3 rather than f4�T /J�3, where f is thedimensionless coupling parameter. We show that transverse static susceptibility acquires a new negative loga-rithmic contribution. In accordance with previous works we find that host spin fluctuations lead to an effectiveone-ion anisotropy on the impurity site. Then defects with S�1/2 appear to be split. We observe strongreduction of the value of the splitting due to longitudinal host spin fluctuations. We demonstrate that thedynamical impurity susceptibility contains 2S Lorenz peaks corresponding to transitions between the levels,and the nonresonant term. The influence of finite concentration of the defects n on the low-temperatureproperties of antiferromagnets is also investigated. Strong spin-wave damping proportional to nf4�� is ob-tained originating from the nonresonant terms of the susceptibilities.

DOI: 10.1103/PhysRevB.74.184433 PACS number�s�: 75.10.Jm, 75.10.Nr, 75.30.Hx, 75.30.Ds

I. INTRODUCTION

The problem of impurity spins in two-dimensional �2D�Heisenberg quantum antiferromagnets �AFs� has attractedmuch attention in the last two decades.1–14 The majority ofworks are devoted to defects coupled asymmetrically to sub-lattices of AFs. Among such impurities are added spincoupled to one host spin, substitutional spin and vacancy thatis the particular case of the added spin with the couplingstrength g→�. Such works are stimulated by a variety ofexperiments studying cuprates with magnetic and nonmag-netic impurities. Defects coupled symmetrically to sublatticesof AFs �see Fig. 1� are much less studied because such ob-jects are rare in occurrence. At the same time they havedifferent properties11,14 and are also of interest. For instance,attempts were made to model holes in CuO planes of somehigh-Tc compounds being in antiferromagnetic phase byspin-1

2 impurity coupled symmetrically to two neighboringhost spins.11–13

In our recent paper14 �hereafter referred to as paper I� westudied dynamical properties of impurity coupled symmetri-cally to two neighboring spins in 2D AFs at T�0. A tech-nique was proposed based on Abrikosov’s pseudofermiontechnique15 that allows one to discuss dynamics of an impu-rity with arbitrary spin value S. Meanwhile we focus in paperI on the particular case of spin-1

2 defect. Our aim was to findthe impurity dynamical susceptibility ��. We assumed thatthe impurity dynamics is governed by interaction with spinwaves. Then, two kinds of the host systems are considered inpaper I: �i� ordered 2D AFs in which the long range order at

T�0 is stabilized by a small interaction �for definitenessinterplane interaction� ��J, where J is the coupling con-stant between the host spins; and �ii� isotropic 2D AFs. Webared in mind that in isotropic 2D AFs only spin waves arewell defined with energies much larger than Jsa /�, where ais the lattice constant, s is the value of host spin, and � exp�const/T� is the correlation length. It was obtainedwithin the spin-wave approximation �SWA� that, similar tothe spin-boson model, the effect of the host system on thedefect is completely described by the spectral function whichwas assumed to be proportional to �2 in calculations of ��.We demonstrated that the spectral function is proportional to�2 in 2D AFs and in the ordered quasi-2D AFs at�Jsa /� and ��, respectively. Notice that only trans-verse host spins fluctuations contribute to the spectral func-tion within SWA. Then, only processes of absorption oremission of one spin wave by the impurity is taken intoaccount within SWA.

FIG. 1. Unit cell of a 2D AF with impurity spin coupled �a�symmetrically and �b� asymmetrically to AF sublattices. Strengthsof coupling with corresponding host spins g and g1�g2 are de-picted. The local Néel order is also shown. Only symmetricallycoupled impurities are discussed in the present paper.

PHYSICAL REVIEW B 74, 184433 �2006�

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The calculations of �� were performed in paper I withinthe fourth order of the dimensionless coupling parameterf g /J. We show that the transverse impurity susceptibility�� has a Lorenz peak with the width � f4J�T /J�3 thatdisappears at T=0, and a nonresonant term. The imaginarypart of the nonresonant term is a constant independent of T at�� and the real part has a logarithmic divergence of theform f2 ln��2+�2� /J2�. The longitudinal susceptibility�� has only the nonresonant term which differs from thatof �� by a constant. The fact that the spectral function ina 2D AF is proportional to �2 only at � �� or Jsa /�� leadsto the following restriction on the range of validity of theresults obtained: max�� , �� or Jsa /��. Notice that inthe case of an isotropic 2D AF at T�0 this relation deter-mined the range of validity of our assumption that the impu-rity dynamics is governed by interaction with spin waves.

It was observed that the static susceptibility ��0� has thefree-spin-like term 1/ �4T� and a correction proportional tof2 ln�J /T�. Quite a different behavior of ��0� was obtained inthe regime of T� �g� for asymmetrically coupled defects:5,6

the classical-like term S2 / �3T� and the logarithmic correctionproportional to ln�J /T� rather than f2 ln�J /T�.

The influence of the finite concentration of the defects non the low-temperature properties of a 2D AF was also stud-ied in paper I. It was shown that correction to the square ofthe spin-wave spectrum is proportional to nf2��. Thentwo regimes were considered, ��0 f2T2 /J and��0, at which, respectively, nonresonant and resonantparts of �� prevail. In the most interesting case of��0 we found the logarithmic correction to the spin-wave velocity and an anomalous damping of the spin wavesproportional to nf4��. Such damping was obtained also inRefs. 1 and 16 where asymmetrically coupled impuritieswere studied in 2D AFs.

In the present paper we continue our discussion of sym-metrically coupled defects in 2D AFs and �i� go beyondSWA and �ii� study defects with S�1/2. In particular, lon-gitudinal host spin fluctuations come into play outside SWA.They lead to two kinds of processes: absorption or emissionof two magnons by the impurity and scattering of one spinwave on the impurity. We find that in the case of an isotropic2D AF at T�0 our assumption that the impurity dynamics isgoverned by interaction with spin waves is wrong at any Tand �. This circumstance manifests itself in the fact that spinwaves with wavelength of the order of � are important in theprocesses of absorption or emission of two magnons andscattering of one magnon. Thus, our approach can be appliedto ordered 2D AFs only.

It is demonstrated that longitudinal host spin fluctuationslead to two important contributions to the spectral functionwhich should be taken into account. One of them is propor-tional to T2� /s and corresponds to scattering of one spinwave on the impurity. Another contribution has the form�2v�T� /s, where v�T� is a series in �T /s�ln�T / �s��, andoriginates from two processes: emission or absorption of twospin waves by the impurity and the scattering of one magnonon the impurity.

It is demonstrated that spin-12 impurity susceptibility

�� has the same structure as that obtained within SWA:

the Lorenz peak and the nonresonant term. The difference isthat the width of the peak � acquires large correction of thefirst order of f2: � f2J�T /J�3. This term dominates if thefollowing condition is fulfilled that does not depend on s:�g� /J�5. Besides, some constants in the expression for�� acquire thermal corrections. The width of the Lorenzpeak in �� is zero within our precision. We observe thatlongitudinal host spin fluctuations lead to a new logarithmiccorrection to ��0� proportional to −�f2 / �Js��ln�T / �s��.

Corrections to the spin-wave spectrum have the sameform as those obtained within SWA. The difference is thatsome constants acquire thermal corrections, � has anotherform and the quantity �0 separating two regimes becomeslarger being proportional to T2 /J rather than f2T2 /J.

It should be pointed out that the obtained remarkable in-fluence of longitudinal host spin fluctuation on frustrated im-purity spin dynamics is quite unusual. There is no such in-fluence in the case of asymmetrically coupled defects. Weshow below that longitudinal fluctuations give a negligiblysmall correction to the spectral function and there is no rea-son to expect any perceptible influence from them. Thus, weconfirm that the frequently used T-matrix approach1,16,17

based on Dyson-Maleev transformation and dealing withonly the bilinear part of the Hamiltonian is appropriate fordiscussing the asymmetrically coupled defects.

We find also �� for symmetrically coupled impuritieswith S�1/2. It is well known that host spin fluctuations leadto an effective one-ion anisotropy on the impurity site

in three-dimensional �3D� AFs, −C�T�Sz2g2 /J, where

C�T��0.18,19 Then defects with S�1/2 appear to be splitand magnetization of sublattices is a hard axis for them. Weobtain the same one-ion anisotropy for defects in 2D AFs. In

particular we observe a large reduction of the value of C�T�obtained within SWA by 1/s corrections stemming from lon-gitudinal host spin fluctuations. We show that thermal cor-

rections reduce C�T� further and can change the sign of C�T�at T�TN.

Then we find that if T is greater than the value of thedefect splitting, transverse dynamical susceptibility contains2S Lorenz peaks corresponding to transitions between impu-rity levels and a nonresonant contribution. Widths of thepeaks are proportional to f2J�T /J�3 as in the case of S=1/2.When T is lower than the distance between nearest impurity

levels �C�T�g2 /J� there are only peaks corresponding totransitions between low-lying levels. Longitudinal suscepti-

bility has only the nonresonant term. At T C�T�g2S2 /Jstatic susceptibility has the same structure as that ofS=1/2: term S�S+1� / �3T� and a logarithmic correction. If

T� C�T�g2 /J, ��0� of integer and half-integer spins behavedifferently. The imaginary part of the nonresonant term is aconstant that leads to abnormal spin-wave damping propor-tional to nf4��, as in the case of spin-1

2 impurity.It is explained in paper I that in the case of finite concen-

tration of defects the results obtained within our method havea limited range of validity. This is because the interactionwith defects modifies host spin Green’s functions and thebare spectral function acquires large corrections at energies

A. V. SYROMYATNIKOV AND S. V. MALEYEV PHYSICAL REVIEW B 74, 184433 �2006�

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around those of the Lorenz peaks. The problem should besolved self-consistently to obtain correct expressions for theimpurity susceptibility and the spin-wave spectrum aroundthe resonance energies. Corresponding consideration is outof the scope of the present paper. This large renormalizationof the spectral function can be interpreted as a resonancescattering of spin waves on impurities. A similar situationexists in the case of asymmetrically coupled defects �see dis-cussion in Sec. V�. As a result in the vicinity of the reso-nance energies one might expect strong deviation from lin-earity of the real part of the spin-wave spectrum and anincrease of the spin-wave damping.

The rest of this paper is organized as follows. The modeland diagrammatic technique are discussed in Sec. II. Pseudo-fermion Green’s functions and pseudofermion vertex are cal-culated in Sec. III. The impurity dynamical susceptibility isderived in Sec. IV. Influence of the defects on the spin-wavespectrum is considered in Sec. V. Section VI contains ourconclusions. There are two appendixes with details of calcu-lations.

II. TECHNIQUE AND GENERAL EXPRESSIONS

A. Technique

We discuss a Heisenberg AF with impurity spin S coupledto two neighboring host spins s1 and s2 describing by theHamiltonian,

H = J�i,j�

sis j + gS�s1 + s2� . �1�

It should be stressed that one can consider other symmetri-cally coupled defects �e.g., those coupled to four host spins�on the equal footing. The results would differ by some con-stants only. It is convenient to use Abrikosov’s pseudofer-mion technique15 and to represent the impurity spin as S=nmbn

†Snmbm, where n and m are the spin projections, andbn

† and bn are operators of creation and annihilation ofpseudofermions. We calculate below impurity susceptibility�� using the diagrammatic technique. First diagrams for�� and a graphical representation of the result of all dia-grams’ summation are shown in Fig. 2. Thin lines with ar-rows in the picture represent the bare pseudofermion Green’sfunctions: G

mm��0� �i�n�= mm��i�n−��−1, where � is the chemi-

cal potential of pseudofermions that should be tended to in-finity in the resultant expressions. Wavy lines in the picturedenote magnon Green’s functions. As usual, diagrams withonly one pseudofermion loop should be taken into accountbecause each loop is proportional to the small factor of e−�/T.

Within SWA discussed in paper I only one virtual magnoncan be emitted or absorbed in each vertex. Beyond SWA one

must consider diagrams in which some vortexes contain twoand three magnon lines. Then, within SWA products contain-ing many operators of host spins reduce to products of two-spin Green’s functions while there is no such simplificationbeyond SWA. Meanwhile we find below that within the firstorder of g2, the order in which all calculations of the presentpaper are done, diagrams with single and double wavy linesthat do not contain vertexes with three wavy lines give thelargest contributions. These are the diagrams shown in Figs.2 and 3. Figure 4 gives examples of diagrams that should bediscarded. A double wavy line corresponds to longitudinalsusceptibility of the host spins. Thus, the diagrammatic tech-nique used in paper I can be applied for the present task withminor modification: the single wavy line and double wavyline correspond to �−1N��Im �� with � ,�=x ,y and�−1N��Im �zz��, respectively, where N��= �e�/T−1�−1 isthe Planck’s function,

�� = − i 0

�

dtei�t��s1��t� + s2

��t�,s1��0� + s2

��0�� 2�

and �¯� denotes the thermal average; frequencies of func-tions Im �� should be taken so as they are contained inarguments of G�0� functions with positive sign; integrationover all frequencies of functions Im �� is taken in theinterval �−� ,��. Im �� is referred to throughout thispaper as the spectral function.

It is shown in Appendix A that the imaginary part of�� given by Eq. �2� has the following form in the or-dered 2D AF at ��:

Im �� = − A��

��2

sgn��d��

− Br�T�

s

�

�� T

��2

�� z �z, �3�

d�� = ��1 − �z� +v�T�

s �z �z, �4�

where A and B are positive constants given by Eqs. �A4� and�A8�, respectively, that are independent of s and which di-mensionality is inverse energy. � is a characteristic energyfor which we have Eq. �A3� and �� is a cutoff function

FIG. 2. Lower-order diagrams for the impurity dynamical sus-ceptibility �P�� and a graphical representation of the result of theoverall series summation. Lines with arrows and wavy lines repre-sent pseudofermion and magnon Green’s functions, respectively.

FIG. 3. Lower-order diagrams for the pseudofermion self-energy part.

FIG. 4. Examples of diagrams for the pseudofermion self-energy part �a� and vertex �b� containing vertexes with two andthree magnon lines. Such diagrams are negligibly small comparedto those presented in Figs. 3 and 5.

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that is equal to unity at ��, it drops rapidly to zerooutside this interval. Within the first order of 1 /s we have

r�T� = 1, v�T� =8sJ

Nk

N��k��k

, �5�

where N is the number of spins in the lattice and �k is thespin-wave energy which is equal to �8sJk at small k. Thelogarithmic infrared singularity in the expression for v�T� isscreened by the interaction stabilizing the long range order atT�0, v�T��T / ��sJ�ln�T / �s��. Notice that only the termproportional to A�2d��1− �z� was obtained in paper Iwithin SWA. It originates from transverse host spin fluctua-tions and corresponds to emission or absorption of one spinwave by the impurity. The remaining two contributions toIm �� in Eq. �3� are of the next order of 1 /s. They stemfrom longitudinal host spin fluctuations. The last term in Eq.�3� is proportional to T2� and corresponds to scattering ofone spin wave on the impurity. This term can be larger than�2. The correction proportional to v�T��2 originates fromtwo kinds of processes: emission or absorption of two spinwaves by the impurity and scattering of one magnon on theimpurity. To determine the range of values of v�T� we use thewell-known formula for the average z component of the hostspin,

�sz� = s −1

Nk

4sJ − �k

2�k−

4sJ

Nk

N��k��k

. �6�

The second and the third terms in Eq. �6� are equal approxi-mately to 0.2 and T / �2�sJ�ln�T / �s��, respectively. Thethird term must be much smaller than s within the spin-waveapproach. Comparing it with v�T� given by Eq. �5� one con-cludes that v�T� / �2s� is much smaller than unity.

As it is explained in Appendix A, higher order 1 /s cor-rections give contributions proportional to products of �2

��T2� and powers of �T / �s��ln�T / �s��. Hence, r�T� andv�T� appear to be series in powers of �T / �s��ln�T / �s��.We restrict ourselves in this paper by the first 1 /s correctionto r�T� and v�T�.

Terms in Eq. �3� proportional to v�T��2 will result in thetemperature corrections to some constants in �� obtainedwithin SWA. It will also give a logarithmic contribution to��0�. The last term in Eq. �3� will lead to the large renor-malization of the Lorenz peak widths.

We show in Appendix A that one can lead to expressions�3�–�5� considering an isotropic 2D AF at T�0. Howeverthere is no parameter in this case to screen infrared singular-ity of the function v�T�. This singularity signifies that hostexcitations with wavelengths greater than the correlationlength � play an important part in the impurity dynamics.Then spin-wave formalism is inadequate in this case and ourresults are applicable for ordered 2D AFs only.

B. Dynamical susceptibility of the impurity

We have for the dynamical susceptibility after the analyti-cal continuation from the discrete frequencies to the realaxis14,20–22

�P�� =e−�/T

2�iN −�

�

dxe−x/T Tr�P�G�x + ��P++�x + �,x�G�x�

− G*�x��P−−�x,x − ��G*�x − ��

− G�x + ��P+−�x + �,x�G*�x�

+ G�x��P+−�x,x − ��G*�x − �� , �7�

where N is the number of pseudofermions that is propor-tional to e−�/T, P is a projection of impurity spin, and G�� isthe retarded Green’s function. The trace is taken over projec-tions of the impurity spin, and signs at superscript of �Pdenote those of imaginary parts of the corresponding argu-ments �e.g., �P

+−�x ,y�=�P�x+ i ,y− i ��. An energy shift by� has been performed during the derivation of Eq. �7�. As aresult the Fermi function �e�x+��/T+1�−1 has been replaced bye−�x+��/T and the functions G and �P no longer depend on �.These are those functions we calculate in the next section bythe diagrammatic technique. It is clear that the bare pseudo-fermion Green’s functions in this case are G

mm��0� ��

= mm� /�.We derive below analytical expressions for the dynamical

susceptibility of the impurity. Perturbation theory is used forthis purpose according to the interaction. It can be done if thedimensionless constants

f2 =g2A

�and h2 =

g2Br�T�s�

�8�

are small. Two constants f and h both proportional to �g� areintroduced to distinguish contributions from the terms in Eq.�3� proportional to �2 and T2�. Notice that h=0 withinSWA. Using Eqs. �A3�, �A4�, and �A8� we have f2

= �g /J�2�� / �4s� and h2= �g /J�2�2 /s2. Then g can be evengreater than J at large enough s.

III. PSEUDOFERMION GREEN’S FUNCTIONAND THE VERTEX

A. Green’s function

We turn to the calculation of the pseudofermion Green’sfunction Gmn��. The Dyson equation for it has the follow-ing form: �Gmn��= mn+q�mq��Gqn��, where �mq��are matrix elements of the self-energy. It is easy to show thatmatrices �� and G�� are diagonal, �mn��= mn�n��,Gmn��= mnGn��= mn / ��−�n��. As we demonstrate inpaper I, there is further simplification in the case of two-levelimpurity, �� and G�� are proportional to the unitary ma-trix.

Diagrams of the order of g2 for �n�� are shown in Fig. 3.Let us represent the Green’s function in the form

Gn�� =1 − Zn��

� + cn + i�n�� + cn�, �9�

where Zn�� and �n�� are some functions, �n�� are realones, and cn are constants. Using Eq. �9� we have in the firstorder of g2


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Zn�1�� =

f2R�n�1�

��

−�

�

dx�x�N�x��x�

x + � + cn + i�n�x + � + cn�,

�10�

�n�1�� = h2R�n

�2�� T

��2

��1 + N�� , �11�

cn�1� = g2S�S + 1� − n2

�

−�

�

dxN�x�Im ��x�

x

+ g2n2

�

−�

�

dxN�x�Im �zz�x�

x, �12�

where ��=�xx��=�yy��, the principal value of the in-tegrals is implied in Eq. �12� and

R�n�1� = �S�S��nnd�� = S�S + 1� − �1 −

v�t�s�n2, �13�

R�n�2� = �SzSz�nn = n2. �14�

Summation over repeated greek indices is understood in Eq.�13� and below. The logarithmic divergence in expression�10� at real � is screened by the term i�n�x+�+cn� in thedenominator. We find that higher order diagrams both in gand 1/s �e.g., that shown in Fig. 4�a�� give negligibly smallcontributions to the self-energy part. It was obtained in paperI that within SWA the first nonzero contribution to �n�� isof the order of f4g4. As is seen from Eq. �11�, longitudinalhost spin fluctuations lead to the correction of the order ofh2g2. It is important to note that �n�� is the constant at��T,

�n�� → 0� � �0n = h2R�n�2�� T

��3

. �15�

Notice that within SWA this constant shows the same Tdependence,14 �0

�swa� f4T3 /�2.Using Eqs. �9�–�12� the number of pseudofermions N can

be calculated up to the first order of g2 with the result

N = − n

1

�

−�

�

dxN�x�Im Gn�� e−�/Tn

ecn/T. �16�

Notice that terms proportional to g2 cancel each other on theright-hand part of Eq. �16�.

It should be stressed that one cannot use in Eq. �12� ex-pression �3� for the imaginary part of ��x� because large xare essential in the integrals. We must use exact expressionsfor Im ��x� to find constants c: Eq. �A2� for Im ��x�, Eq.�A5� for 1 /s correction to Im ��x�, and Eq. �A7� forIm �zz�x�. As a result, one obtains up to an unimportant con-stant independent of n,

cn�1� = − �C −

C� + C��T�2s

�n2g2

J= − C�T�n2g2

J, �17�

where

C =J

�2��2 dk1 + cos�kR12�

J0 + Jk� 0.25, �18�

C� = C�1 −1

�2��2 dk�k

sJ0� � 0.04, �19�

C��T� =sJ

�2��4 dk1dk21 − cos��k1 + k2�R12�

�k1�k2

��sJ0�2 + s2Jk1

Jk2− �k1

�k2

�k1+ �k2

+ 4C� 4sJ

�2��2 dkN��k�

�k�

� 0.13 +4sJ

�2��2 dkN��k�

�k, �20�

where R12 is a vector connected host spin coupled to theimpurity. Jk=2J�cos kx+cos ky�, �k=s�J0

2−Jk2 is the spin-

wave energy, and integrals are over the chemical Brillouinzone. The constant C is of the zeroth order of 1 /s. It origi-nates from transverse host spin fluctuations. Constants C�

and C��T� are first 1 /s corrections stemming from transverseand longitudinal fluctuations, respectively. It is seen that theirsum is of the order of C at small s and it should be taken intoaccount. Temperature corrections are considered in C��T�only because those to C� are much smaller. We note that thethermal correction to C��T� coincides with that to �sz� givenby Eq. �6�. It should be pointed out that this correction re-

duces the value of C�T� and can change its sign at T�TN.36

Let us discuss the physical meaning of constants c givenby Eq. �17�. They describe splitting of the impurity causedby the host spin fluctuations. The distances between the lev-els are given by differences between corresponding constantsc. One infers from Eq. �16� that at T much smaller than thedistance between the nearest levels only those with maxi-mum c are populated. As is clear from Eq. �17�, these arelevels with the smallest in absolute value projections on thequantized axis. Then, the impurity splitting can be described

by the effective Hamiltonian −Sz2C�T�g2 /J that is the effec-

tive one-ion anisotropy. Such anisotropy was obtained withinSWA for frustrated defects in 3D AFs.18,19

It is also seen from Eq. �17� that the spin-12 defect remains

degenerate because c1/2=c−1/2. We have taken advantage ofthis circumstance in paper I and discarded constants c attrib-uting them to renormalization of the chemical potential �.

B. Pseudofermion vertex

Let us turn to the consideration of the pseudofermion ver-tex �P�x+� ,x�. First diagrams for this quantity are presentedin Fig. 5. It is shown in paper I that �Pmm��x+� ,x� is pro-portional to Pmm� for S=1/2. However in general it has amore complicated matrix structure.


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As is seen from Eq. �7�, we need four different branchesof �P�x+� ,x�. It is clear that �++= ��−−�* and within the firstorder of g2 one has

�Pmn++ �x + �,x�

= Pmn +f2

��l,q

Sml� PlqSqn

� d��

� −�

�

dyy�y�N�y��y�Gl�x + y + ��Gq�x + y� . �21�

It is seen from Eq. �21� that the poles of G functions are onthe one hand from the real axis. Hence, the second term inEq. �21� is small compared to the first one and we can restrictourselves by this precision. The term proportional to h2 isdiscarded in Eq. �21� being negligibly small due to the factor�T /��2.

The situation is different in the case of �+−= ��−+�*. Polesof the Green’s functions under integrals appear to be on theopposite sides of the real axis. As a result at �=0 the integraldiverges at finite x as widths �0 and constants c tend to zero.Then one must consider entire series to find �+−. After analy-sis of diagrams for �+− and evaluation of their contributionto �P�� we find that the most important diagrams withineach order of g2 are taken into account in the following equa-tion:

�Pmn+− �x + �,x�

= Pmn +f2

��l,q

Sml� PlqSqn

� d��

� −�

�

dyy�y�N�y��y�Gl�x + y + ��Gq*�x + y�

+h2T2

��2Smmz Snn

z −�

�

dyyN�y��y��Pmn+− �x + y + �,x + y�

�Gm�x + y + ��Gn*�x + y� . �22�

Notice that this equation differs from that obtained in paper Iwithin SWA. It takes into account only ladder diagramsshown in Fig. 5 whereas within SWA one must consider alsodiagrams with crossing of two neighboring rungs. Equation�22� can be easily solved if one notes that the last term inthis equation should be taken into account only when��+cm−cn � �T. Outside this interval the last term is muchsmaller than the second one. At such � the area of integra-tion near the poles of the Green’s functions is essential in thelast term in Eq. �22� and it gives the resonant contribution

�Pmn+− �x + �,x�

= Pmn +f2

��l,q

Sml� PlqSqn

� d��

� −�

�


− 2ih2Smmz Snn

z � T

��2

�x + cn�N�− x − cn��x + cn�

��Pmn

+− �� − cn,− cn�� + cm − cn + i�0m + i�0n

. �23�

Equation �23� has the same structure as the equation derivedin paper I. The difference is that �0 and the third term are ofthe order of g2 rather than g4. Solving Eq. �23� one obtains

�Pmn+− �x + �,x�

= Pmn +f2

��l,q

Sml� PlqSqn

� d��

� −�

�


+ 2ih2Smmz Snn

z Pmn�� T

��3 ��x + cn�

� + cm − cn + 2i�mn,

�24�

�mn = h2�m − n�2�

2� T

��3

. �25�

We use in Eq. �24� that the following x will be important forcalculation contributions to �P�� from �+−: �x+cn � �T.Note, the third term in Eq. �24� is much greater than thesecond one when ��+cm−cn � ��0 / f2. It should be pointedout that �mn=0 for P=Sz. Evidently, temperature depen-dences of �mn for P=Sx ,Sy and �0 given by Eq. �15� are thesame. Within SWA the constant �mn has the same T depen-dence but it is of the order of g4,14 �mn

�swa� f4T3 /�2.

IV. IMPURITY SUSCEPTIBILITY

We can derive now the impurity susceptibility using thegeneral expression �7�, Eqs. �9�–�11� for the Green’s func-tion, Eqs. �21� and �24� for the branches of the vertex, andEq. �16� for the number of pseudofermions N. It is conve-nient to discuss separately the cases of S=1/2 and S�1/2.

A. Spin-12 impurity

As a result of tedious calculations some details of whichare presented in Appendix B we have for the dynamical sus-ceptibility of the impurity up to terms of the order of g2,

�P�� =1

4T

2i�

� + 2i�+ 2R�

f2

��

−�

�

dxsgn�x��x�x + � + 2i�0

,

�26�

FIG. 5. Lower-order diagrams for pseudofermion vertex�P�x+� ,x�.


184433-6

R� =PS��S�,P�d��

2S + 1, �27�

where we introduce the notation Y =Tr�Y�, �0=h2T3 / �4�2�,�=0 for P=Sz, and �=h2T3 / �2�2� for P=Sx,y. Expression�26� differs from that obtained in paper I only by the form of�0 and � and by the thermal 1 /s correction to R�. The firstterm in Eq. �26� is the Lorenz peak with the width �. The lastterm is the nonresonant part of the susceptibility. Its imagi-nary part at �� 0 is proportional to sgn�� and the realone contains the logarithmic singularity of the formln��2+�0

2�. At T=0 and ��0 the nonresonant contributionsurvives only and the susceptibility has the logarithmic sin-gularity. It is clear that longitudinal host spin fluctuationsbecome important if �0�0

�swa� and ��swa�. These condi-tions are fulfilled when �g � /J�5, i.e., for weakly coupleddefects being considered. The contrast between results whichdo and do not take into account the longitudinal host spinfluctuations is illustrated by Fig. 6, where we plotIm �� /� using Eq. �26�.

The nonresonant term gives the main contribution to thesusceptibility �26� when

� �0 =�

s� T

��2

. �28�

Notice that �0 f2T2 within SWA.Using results of calculations presented in Appendix B we

find that static susceptibility �P�0� does not depend on � and�0,

�P�0� =1

4T�1 − 4f2UR�� + 4R�

f2

��ln��

T� , �29�

U = −4SJ2

�3/2 dxIm ��x�

x�x�

=2J2

�5/2 dk�1 + cos�kR��J0 − Jk�

�J02 − Jk

2�3/2 � 0.39. �30�

It is taken into account in Eq. �30� that one cannot use Eq.�3� for Im ��x� because large x are important in theintegral.37 Equation �A2� is used to find U as it was doneabove for calculation of constants c. It is seen from Eq. �29�that the static susceptibility contains the free-spin-like term�4T�−1 whose amplitude is slightly reduced by the interactionand the logarithmic correction proportional to f2. Static sus-ceptibility has the same form as that obtained in paper Iwithin SWA with the only difference in the form of the con-stant R�=R�

�swa�+ �v�T� / �2s��PSz�Sz , P�. Notice that R�

=R��swa�=1/4 for P=Sz and ��0�=��

�swa��0�. In contrast R�

=R��swa�+v�T� / �8s�=1/8+v�T� / �8s� for P=Sx,y and the

transverse static susceptibility has the form

��0� =1

4T�1 − f2

U2� − f2

U��s�

ln� T

s�� +

f2

2��ln��

T� .

�31�

Thus the longitudinal host spin fluctuations lead to anotherlogarithmic term in ��0�.

B. Impurities with S�1/2

1. Dynamical susceptibility

The calculations are slightly more tedious for impuritywith S�1/2. As a result of simple evaluations presented inAppendix B we lead to the following quite cumbersome ex-pression for the impurity susceptibility:

�� =e−�/T

N m,n��Pnm�2�e�cm+2i�nm−i�0n�/T − e�cn−i�0n�/T�

1

� + cm − cn + 2i�nm

+ �Pnm�2�R�n�1� + R�m

�1� ��ecn/T + ecm/T�f2

2��

−�

�

dxsgn�x��x�

x + � + cm − cn + i�0n + i�0m

+ q,l

2 Re�PnmSmq� PqlSln

�d�� + cm − cn + i�0n + i�0m

f2

��

−�

�

dx�x�xN�x��x� � � e�cm+i�om�/T

�x − � + cl − cm − i�0l − i�0m��x + cq − cm + i�0q − i�0m�

−e�cn−i�on�/T

�x + � + cq − cn + i�0q + i�0n��x + cl − cn + i�0n − i�0l�� . �32�

FIG. 6. The even-� function Im�� /��2 for spin-12 impu-

rity at T /�=0.1 and f =0.1 with longitudinal host spin fluctuationsbeing �solid line� and not being �dotted line� taken into account.


184433-7

The first term here is zero for longitudinal susceptibility �P=Sz and thus m=n�. In the case of transverse susceptibility itgives 2S Lorenz peaks corresponding to transitions betweenimpurity levels. Using Eq. �17� we obtain for the resonantfrequencies

�res�n� = �cn − cn−1� = �2n − 1�C�T�g2/J , �33�

where n=S ,S−1, . . . ,nmin and nmin=1/2 and 1 for half-integer and integer spins, respectively. Then in contrast tohalf-integer spins there is no resonance peak at zero fre-quency in the case of integer spins. Notice that the distancebetween frequencies does not depend on S. The second andthe third terms in Eq. �32� give contributions to the nonreso-nant term. The third term contains also a resonant contribu-tion. As is explained in Appendix B, this contribution shouldbe discarded because the resonant terms of the susceptibilityare calculated within the zeroth order of g2.

One makes sure after simple transformations that Eq. �32�coincides with Eq. �26� if S=1/2. It is more convenient todiscuss expression �32� further in some limiting cases inwhich �� has simpler forms. Let us consider the suscepti-bility when temperature is greater than the impurity splitting,

T C�T�g2S2 /J, and when T is much smaller than the energy

between nearest impurity levels, T� C�T�g2 /J.

T C�T�g2S2 /J. Equation �32� can be simplified greatly inthis case if one expands exponents ecn/T up to the secondterm and notes that the nonresonant terms should be taken

into account at �� C�T�g2S /J only. As a result one obtains

�P�� =1

T�2S + 1�n,m�Pnm�2

cm − cn + 2i�nm

� + cm − cn + 2i�nm

+ 2R�

f2

��

−�

�

dxsgn�x��x�x + � + i

, �34�

where R� is given by Eq. �27�. Notice that Eq. �34� coincideswith Eq. �26� if S=1/2. The imaginary part of the nonreso-nant term is greater than that of resonant terms when �� 0, where �0 is given by Eq. �28�. The real part of thenonresonant term dominates in Eq. �34� at �� f�� /T.To illustrate our results we plot in Fig. 7 the even-� functionIm �� /� using Eq. �34� for S=3/2 and S=2.

T� C�T�g2 /J. At such small temperatures only low-lyingimpurity levels contribute to susceptibility. The transversesusceptibility for integer impurity spin has the form

��int�� =

2�P01�2

� + c0 − c1 + 2i�01−

2�P01�2

� − �c0 − c1� + 2i�01

+ if2

��P01�2�R�0

�1� + R�1�1��sgn�� + c1 − c0�

− m,q,l

Re�P0mSmq� PqlSl0

�d�� + cm − c0 + i�0m

�� + cq − c0�2

� + cq − cl + i�0q + i�0l

��1 + sgn�� + cq − c0�� − �� → − �� , �35�

where ��→−�� denotes terms inside the parentheses with−� set instead of �. The first two terms in Eq. �35� describetransitions between three low-lying impurity levels:�−1↔0� and �0↔1�. The last term in Eq. �35� is the non-resonant part of the susceptibility that is zero between theresonance peaks �= ± �c0−c1�. Beyond this interval it is neg-ligibly small near the resonance peaks, i.e., at ��+c0−c1 ��T3/2 /�� and ��− �c0−c1� � �T3/2 /��, but it gives the maincontribution to the imaginary part of the susceptibility out-side these two intervals being of the order of f2. The real partof the nonresonant term is small compared to that of theresonant terms at �� .

In the case of half-integer impurity spins the transversesusceptibility has the form

��half-int�� =

�P−�1/2��1/2��2

T

2i�−�1/2��1/2�

� + 2i�−�1/2��1/2�+

�P�1/2��3/2��2

� + c1/2 − c3/2 + 2i��1/2��3/2�−

�P�1/2��3/2��2

� − �c1/2 − c3/2� + 2i��1/2��3/2�+ i

f2

4�

� n=±1/2

�m

�Pnm�2�R�n�1� + R�m

�1� �sgn�� + cm − cn� − m,q,l


�d�� + cm − cn + i�0m + i�0n

�� + cq − cn�2

� + cq − cl + i�0q + i�0l�1 + sgn�� + cq − cn�� − �� → − �� , �36�

FIG. 7. The even-� function Im�� /��2 calculated usingEq. �34� for impurity with S=3/2 �solid line� and S=2 �dotted line�at T /�=0.05 and f =0.1.


184433-8

where the first three terms describe transitions between fourlow-lying impurity levels: �−3/2↔−1/2�, �−1/2↔1/2�,and �1/2↔3/2�. The nonresonant part of the susceptibilityis negligibly small in the vicinity of the resonance peaks,i.e., at �� T2 /�, ��+c1/2−c3/2 � �T3/2 /��, and��− �c1/2−c3/2� � �T3/2 /��. Outside these three intervals thenonresonant term is of the order of f2 and it gives the maincontribution to the imaginary part of the susceptibility. Thereal part of the nonresonant term is small compared to that ofthe resonant terms at �� .

2. Static susceptibility

T C�T�g2S2 /J. Using results of Appendix B one finds forstatic susceptibility in this regime

�P�0� =S�S + 1�

3T�1 − f2U 3R�

S�S + 1�� + 4R�

f2

��ln��

T� ,

�37�

where R� and U are given by Eqs. �27� and �30�, respectively.Equation �37� coincides with Eq. �29� at S=1/2. It is seenthat static susceptibility of large-S impurities has the samestructure as that of the spin-1

2 defect.

T� C�T�g2 /J. At small temperature integer and half-integer impurity spins behave differently. Thus, transversestatic susceptibility of an integer spin is a constant,

��int�0� =

4�P01�2

c0 − c1, �38�

whereas the longitudinal one is exponentially small, ��int�0�

�0.Transverse static susceptibility of a half-integer spin has

the form

��half-int�0� =

1

T��P−�1/2��1/2��2 − f2UQ� + 4Q

f2

��ln��

T�

+2�P�1/2��3/2��2

c1/2 − c3/2, �39�

Q =1

2�n,m��Pnm�2R�n

�1� − n,m,q,l

�PnmSmq� PqlSln

� d�� , �40�

where primes over sums signify that summation is taken overprojections 1/2 and −1/2 only. It is seen that only transitionsbetween states with projections ±1/2 contribute to the firsttwo terms in Eq. �39�. The last term represents the contribu-tion to the susceptibility from transitions between states withprojections ±1/2 and ±3/2. It is of the order of 1 / f2 beingmuch larger than the logarithmic term up to exponentiallysmall temperatures T�e−1/f4

.Longitudinal static susceptibility of a half-integer spin be-

haves like that of the spin-12 defect,

��half-int�0� =

1

T��P−�1/2��1/2��2 − f2UQ�� + 4Q

f2

��ln��

T� ,

�41�

Q� =1

2�n,m��Pnm�2R�n

�1� − q,l

n,m

�PnmSmq� PqlSln

� d�� , �42�

where Q is given by Eq. �40�. In contrast to ��half-int�0� there

is no term proportional to 1/ f2 in ��half-int�0� screening the

logarithmic correction.As it is explained in paper I, there is a restriction on the

range of validity of the resultant expressions for �P��. It isthe consequence of the fact that the function Im �� has theform �3� if �� only. It is easy to see that in all calcula-tions performed above one can use the function of the form�3� if the following condition on � and widths �0 holds:max�min��0n� , ��.

V. INFLUENCE OF DEFECTS ON THE HOST SYSTEM

We discuss in this section the influence of finite concen-tration of the defects n on the spin-wave spectrum of AF. It isdemonstrated in paper I that in the vicinity of the pointsk=0 and k=k0, where k0 is the antiferromagnetic vector, thedenominator of host spin Green’s functions determined bythe spin-wave spectrum has the form

D��,k� = �2 − �k2�1 −

nf2

2��u�k�� , �43�

u�k� =1

2+

�kR12�2

k2 , �44�

where R12 is the vector connected host spin coupled to im-purity �it is assumed from the beginning that this vector isthe same for all defects�. The unperturbed spectrum is linearat k�k0 and kk0: �k=ck=�8sJk. It is seen from Eqs. �43�and �44� that the spectrum depends on the direction of themomentum k as a result of interaction of magnons with thedefects. This circumstance is a consequence of our assump-tion that the vector R12 is the same for all impurities. In fact,it can have four directions and the value �R12k�2 /k2 can havetwo different values, cos2 �k and sin2 �k, where �k is theazimuthal angle of k. It easy to realize that u�k�=1 if all fourways of coupling of the impurity with the AF are equallypossible.

Let us turn to spin-12 impurity. It is convenient to consider

separately regimes of ��0 and ��0, where �0 isgiven by Eq. �28�. In these cases the nonresonant and theresonant parts, respectively, dominate in the impurity suscep-tibility �26�. The results are summarized in Table I. They arevery similar to those derived in paper I within SWA. Theonly difference is in the form of �0, �, and R�. It is seen thatdue to interaction with defects the spin-wave velocity ac-quires negative corrections and strong spin-wave dampingarises proportional to nf4� at ��0.

As it is explained in paper I the results obtained withinour method have a limited range of validity which is alsoindicated in Table I. This is because the interaction with de-fects modifies host spin Green’s functions and the bare spec-tral function Im ��

�0�� given by Eq. �3� renormalizes asfollows:


184433-9

Im �� = Im ��0�� − snf2B Im ��d��, �45�

B =4J2

�5/2 dk�1 + cos�kR12�J0 + Jk

�2

� 0.8. �46�

The last term in Eq. �45� becomes larger than Im ��0�� at

small enough energies indicated in Table I and the problemshould be solved self-consistently. Corresponding consider-ation is out of the scope of the present paper.

This large renormalization of the spectral function atsmall energies can be interpreted as a resonance scattering ofspin waves on impurities. A similar situation exists in thecase of asymmetrically coupled defects. Let us discuss, forexample, an AF with an impurity spin weakly coupled to onehost spin. According to results obtained within the T-matrixapproach the defect leads to a resonant level inside the spin-wave band.17,23,24 The energy of this level is of the order ofthe energy cost for creation of spin excitation on the impu-rity. First perturbation corrections to observables increaserapidly near the resonant level due to the resonancescattering.17 In the case of the frustrated spin-1

2 defect theenergy of creation of the spin excitation on the impurity iszero and the resonance scattering takes place at zero energy.

In the vicinity of the resonance peak, where our theory isnot applicable, one might expect strong deviation from lin-earity of the real part of the spin-wave spectrum and an en-hancement of the spin-wave damping as in the case of asym-metrically coupled defects.17

Particular expressions for spin-wave velocity and damp-ing in the case of S�1/2 can be obtained straightforwardlyusing Eqs. �34�–�36� and �43�. Because of their cumbersome-ness we do not present here the results. We would like topoint out only that there is the abnormal spin-wave dampingproportional to nf4�� stemming from the nonresonant part

of �� when �i� ��T2 if T C�T�g2S2 /J, �ii��−�res

�n��T3/2 for n�1/2 and ��−�res�1/2��T2 if

T� C�T�g2 /J, where �res�n� are given by Eq. �33�. The range of

validity of the results obtained within our approach can befound comparing the last term in Eq. �45� with the first one

as it was done for S=1/2. The results are not valid in thevicinity of the resonance peaks due to the resonance scatter-ing of spin waves on impurities. At such energies one mightexpect strong deviation from linearity of the real part of thespin-wave spectrum and increase of the spin-wave damping.

Thermodynamic quantities of AFs with defects can befound, in principle, using impurity susceptibility. But due tothe limiting range of validity of our results and due to thefact that thermodynamic quantities are expressed via inte-grals containing susceptibility one cannot calculate them be-cause it is impossible to perform the integration over theessential energy area �in particular, over small energies�. Weexplain this situation in detail in paper I by the example ofthe specific heat. We do not obtain any noticeable correctionsto the specific heat after integration over energies at whichour results are valid.

VI. CONCLUSION

In the present paper we discuss dynamical properties ofan impurity spin coupled symmetrically to sublattices of 2DHeisenberg quantum antiferromagnets �see Fig. 1� at T�0�existence of a small interaction stabilizing the long rangeorder at T�0 is implied�. We continue our study on thissubject started in paper I where the spin-1

2 defect was dis-cussed and the host spins fluctuations are considered withinthe spin-wave approximation �SWA�. In the present paper we�i� go beyond SWA and �ii� study impurities with spinsS�1/2. We show that the effect of the host system on thedefect is completely described by the spectral function as itwas within SWA. It is demonstrated that longitudinal hostspin fluctuations lead to two important contributions to thespectral function that is proportional to �2 within SWA. Theyhave the form r�T�T2� /s and �2v�T� /s, where s is the valueof the host spin, and r�T� and v�T� are series in�T /s�ln�T / �s��. First terms in these series are given by Eq.�5�.

It is demonstrated that transverse spin-12 impurity suscep-

tibility ��

�1/2�� has the same structure as that obtainedwithin SWA: the Lorenz peak and the nonresonant term. The

TABLE I. Renormalization of spin-wave damping and velocity by interaction with spin-1

2impurities in

two regimes, ��0 and ��0, where �0 is given by Eq. �28�. Ranges of validity of the theory in eachregime are also presented �see the text�.

��0 ��0

Spin-wave damping��

nf4

2�R�u�k�

�

T

�2�

�2 + 4�2

nf2

8�u�k�

Spin-wave velocityc�1 − ln��

��2nf4

�2 R�u�k� c�1 −�

T

�2

�2 + 4�2

nf2

2�u�k�

Range of validity of the theory ��

0.1�R��nf2 ��2 + 4�2�

�3 0.004nf2�

T


184433-10

difference is that the width of the peak � becomes largerbeing proportional to f2J�T /J�3 rather than f4J�T /J�3. Be-sides some constants in the expression for �

�

�1/2�� acquirethermal corrections. The width of the Lorenz peak in longi-tudinal impurity susceptibility is zero within our precision.The resultant expression for ��1/2�� is given by Eq. �26�.We observe that longitudinal host spin fluctuations lead to alogarithmic correction to transverse static susceptibility �31�proportional to −�f2 / �Js��ln�T / �s��. It is demonstrated thatcorrections to � from longitudinal host spin fluctuationsdominate if �g� /J�5.

We derive also expressions for �� for symmetricallycoupled impurities with S�1/2. It is well known that hostspin fluctuations lead to an effective one-ion anisotropy on

the impurity site in 3D AFs, −C�T�Sz2g2 /J, where

C�T��0.18,19 Then defects with S�1/2 appear to be splitand magnetization of sublattices is a hard axis for them. Weobtain the same one-ion anisotropy for defects in 2D AFs. In

particular we observe large reduction of the value of C�T�obtained within SWA by 1/s corrections stemming from lon-gitudinal host spin fluctuations. We show that thermal cor-

rections reduce C�T� further and can change the sign of C�T�at T�TN.

Then we find that if T is greater than the value of thedefect splitting, transverse dynamical susceptibility contains2S Lorenz peaks corresponding to transitions between impu-rity levels and a nonresonant contribution. Widths of thepeaks are proportional to f2J�T /J�3 as in the case of S=1/2.When T is lower than the distance between nearest impurity

levels �C�T�g2 /J� there are only peaks corresponding totransitions between low-lying levels. Longitudinal suscepti-bility has only the nonresonant term. As it was in the case ofspin-1

2 impurity, the imaginary part of the nonresonant termis a constant that leads to abnormal spin-wave damping pro-

portional to nf4��. At T C�T�g2S2 /J static susceptibilityhas the same structure as that for S=1/2, termS�S+1� / �3T�, and the logarithmic correction. If

T� C�T�g2 /J, ��0� behaves differently for integer �Eq. �38��and half-integer �Eqs. �39� and �41�� spins.

Corrections to the spin-wave spectrum are obtained in thecase of finite concentration of spin-1

2 defects. They are sum-marized in Table I. They have the same form as those ob-tained within SWA. The difference is that some constantsacquire thermal corrections, � has another form, and thequantity �0 separating two regimes appears to be propor-tional to T2 / �sJ� rather than f2T2 /J. In particular, we findstrong spin-wave damping proportional to nf4�� for all Soriginating from the nonresonant terms of the susceptibili-ties.

ACKNOWLEDGMENTS

The authors are thankful to A. V. Lazuta for stimulatingdiscussions. This work was supported by Russian ScienceSupport Foundation �A.V.S.�, President of Russian Federa-tion �Grant No. MK-4160.2006.2� �A.V.S.�, RFBR �Grants

Nos. 06-02-16702 and 05-02-19889� and Russian Programs“Quantum Macrophysics,” “Strongly Correlated Electrons inSemiconductors, Metals, Superconductors and Magnetic Ma-terials,” and “Neutron Research of Solids.”

APPENDIX A: CALCULATION OF Im ��„�…

In this appendix properties of the imaginary part of thefunction �� are discussed. General expression for�� is given by Eq. �2�. We show that Im �� has theform �3� and find constants A and B, the characteristic energy�, and tensor d��.

One has from Eq. �2�,

�� =2

Nk

�1 + cos�kR12��s−k� sk

��, �A1�

where �¯�� denote retarded Green’s functions, N is thenumber of spins in the lattice, and R12 is the vector connect-ing two host spins coupled to the defect. Thus we must cal-culate the spin Green’s functions �s−k

� sk��.

It is demonstrated in paper I that only diagonal compo-nents of �s−k

� sk�� are nonzero. Only xx and yy components

are nonzero within the spin-wave approximation �SWA�. AtT=0 we have for them in the isotropic 2D AF,14

Im �� = − d��

s2

4� dk„�1 + cos�kR12��J0 − Jk�

+ �1 + cos��k + k0�R12��J0 + Jk�…

�1

�k� �� − �k� − �� + �k�� , �A2�

where d��= ��1− �z�, k0 is the antiferromagnetic vector,the lattice constant is taken to be equal to unity, Jk

=2J�cos kx+cos ky�, �k=s�J02−Jk

2 is the spin-wave energy,and the integral is over the magnetic Brillouin zone. If��sJ we have Jk�J0−Jk2, cos�kR12��1− �kR12�2 /2, and�k=ck=�8sJk. Notice that �k0R12�=� mod 2� if the impu-rity is coupled to spins from different sublattices and bothterms in the boldface parentheses in Eq. �A2� are propor-tional to k2. Then integration in Eq. �A2� can be easily car-ried out if one takes advantage of the approximation formagnons similar to the Debye one for phonons: the spectrumis assumed to be linear, �k=ck, up to cutoff momentum k�

defined from the equation 4�N=V�0k�dkk, where V is the

area of the lattice. As a result we lead to terms proportionalto d��1− �z� in expression �3� for Im �� in which

� = ck� = sJ8�� , �A3�

A =2�

J. �A4�

The constant A should be multiplied by 2 if the defect iscoupled to four host spins �two by two from each sublattice�.

It is easy to conclude that if a small interplane interactionof the value of ��J is taken into account the above resultfor Im �� is valid when ��. At the same time


184433-11

Im �� has another � dependence if ��. It is wellestablished25–28 that spin waves are well defined in the para-magnetic phase of 2D AFs if their wavelength is muchsmaller than the correlation length �exp�const/T�. Thus,the above result for Im �� is valid when ��Ja /�.

Let us go beyond SWA. It is well known that the spin-wave interaction gives negligibly small corrections to thespin-wave spectrum and other physical quantities in isotropicHeisenberg AFs.25,29–33 For instance, simple calculationsshow that transverse components of Im �� given by Eq.�A2� are multiplied by

�1 −1

2s�1 −1

Nk

�k

sJ0�� 1 −

0.08

s�A5�

after taking into account first 1 /s corrections. Meanwhilesmall interactions of the value of � possibly could lead to agreat renormalization of physical observable quantities at��.33,34 At the same time we derive Im �� at��. Thus we can use the expressions for xx and yy com-ponents obtained above within SWA.

We must take into account also the longitudinal spin sus-ceptibility �sk

z s−kz ��. The first correction to it is of the first

order of 1 /s. This quantity has been examined in Ref. 35.Nevertheless we derive now the corresponding expressionsin a more convenient form. Using retarded Green’s functionsg�� ,k�= �ak ,ak

†��, f�� ,k�= �ak ,a−k��, g�� ,k�= �a−k† ,a−k��

=g*�−� ,−k�, and f†�� ,k�= �a−k† ,ak

†��= f*�−� ,−k�, where ak

and ak† are operators of magnon creation and annihilation, we

have

�skz s−k

z �� = −T

N�1

k1+k2=k+k0

�f�i�1,k1�f†�i� − i�1,k2�

+ g�i�1,k1�g�i� − i�1,k2�� . �A6�

Functions f and g were calculated, e.g., in paper I. Aftersimple evaluations one obtains

Im �zz�� =�

2N2 k1,k2

1 − cos��k1 + k2�R12��k1

�k2

� ��1 + 2N��k1��sJ0�2 + s2Jk1

Jk2− �k1

�k2�

�� + �k1+ �k2

� − �� − �k1− �k2

��

+ 2�N��k2� − N��k1

��

��sJ0�2 + s2Jk1Jk2

+ �k1�k2

� �� + �k1− �k2

�� ,

�A7�

where the summations are over the chemical Brillouin zone.The first term in the curly brackets in Eq. �A7� correspondsto emission or absorption of two magnons whereas the lastterm describes scattering of one spin wave. Let us discussfirst the case of T=0. The Planck’s functions in Eq. �A7� arezero and Im �zz�� is proportional to �3. Therefore, it issmall in comparison with xx and yy components obtainedabove.

At finite temperatures we have two important contribu-tions to Im �zz��. One of them arises at ��T and origi-nates from the second term in the curly brackets in Eq. �A7�.It can be brought to the form −�T2B / �s�3�, where

B =24��

J

0

�

dxx2ex

�ex − 1�2 =8�5/2

J. �A8�

The second contribution originates from both terms in thecurly brackets in Eq. �A7� and has the form−A�2 sgn��v�T� / �s�2�, where v�T�=8sJ /NkN��k� /�k.The logarithmic infrared singularity in this expression isscreened by the interaction stabilizing the long range order atT�0: v�T��8/��T /��ln�T / �s��. There is no parameterscreening this singularity in isotropic 2D AFs. This singular-ity signifies that the spin-wave approach does not suit fordiscussing the impurity dynamics in isotropic 2D AFs.

It should be stressed that higher order 1 /s corrections cangive contributions proportional to products of �2 ��T2� andpowers of �T / �s��ln�T / �s��. As a result the obtainedabove correction proportional to �T2 should be multiplied bya function r�T�. This function as well as v�T� appear to be aseries in �T / �s��ln�T / �s��. To find general expression forr�T� and v�T� is out of the scope of the present paper. Thus,we lead to the zz component of Im �� in Eq. �3�.

Notice that in the case of impurity coupled to one hostspin, terms in Eq. �A2� containing cos�kR12� should be dis-carded and the spectral function appears to be proportional toa constant within SWA. There is no large corrections to thisconstant from higher order 1 /s terms. Hence, one could notexpect that longitudinal fluctuations play any remarkable rolein this case. Then, we confirm that the so-called T-matrixapproach1,16,17 is appropriate for asymmetrically coupled de-fects. Recall that it is based on Dyson-Maleev transformationand deals with only the bilinear part of the Hamiltonian.

APPENDIX B: CALCULATION OF THE IMPURITYSUSCEPTIBILITY

We present in this appendix some details of the impuritydynamical susceptibility calculation. We use for this generalexpression �7� and Eqs. �9�, �21�, and �24� for the Green’sfunction and the branches of the vertex. The expression for�P�� is derived up to the order of g2.

According to Eqs. �7�, �21�, and �24� the dynamical sus-ceptibility can be represented as a sum of three components.The first one, �1��, originates from Eq. �7� as a result ofreplacement of the vertex by unity. The second, �2��, ap-pears from f2 terms in Eqs. �21� and �24�. The third, �3��, isa result of replacement of the vertex by the third term fromEq. �24�. Calculations of these three quantities are similar tothose performed in paper I. Then we discuss the results only.

For �1�� we obtain


184433-12

�1�� =e−�/T

N n,m

�Pnm�2� e�cm+i�0m�/T − e�cn−i�0n�/T

� + cm − cn + i�0n + i�0m�1 − �R�n

�1� + R�m�1� �

f2

2��

−�

�

dx�x��N�x� − N�− x��x�

x + � + cm − cn + i�0n + i�0m�

+ �R�n�1� + R�m

�1� ��ecn/T + ecm/T�f2

2��

−�

�

dxsgn�x��x�

x + � + cm − cn + i�0n + i�0m� . �B1�

For �2�� one has

�2�� =e−�/T

N n,m,q,l


�d�� + cm − cn + i�0n + i�0m

f2

��

−�

�

dx�x�xN�x��x�

� � e�cm+i�om�/T

�x − � + cl − cm − i�0l − i�0m��x + cq − cm + i�0q − i�0m�−

e�cn−i�on�/T

�x + � + cq − cn + i�0q + i�0n��x + cl − cn + i�0n − i�0l�� .

�B2�

Equation �B2� has a simple form at �� C�T�g2S /J,

�2�� = −e−�/T

N n,m,q,l

Re�PnmSmq� PqlSln

�d��ecn/T + ecm/T�

�f2

��

−�

�

dxsgn�x��x�x + � + i

. �B3�

For �3�� one obtains

�3�� =e−�/T

N n,m

�Pnm�2� e�cm+2i�nm−i�0n�/T − e�cn−i�0n�/T

� + cm − cn + 2i�nm

−e�cm+i�0m�/T − e�cn−i�0n�/T

� + cm − cn + i�0n + i�0m� . �B4�

It should be stressed that we use while calculating Eq. �B4�that � is close to one of the resonance frequencies �res

�n� given

by Eq. �33�, ��−�res�n��T. Meanwhile we can use expres-

sion �B4� at all � as it is much smaller than �1�� far fromthe resonances.

It should be noted that the resonant terms in �1,2�� and�3�� are calculated in the order of g2 and g0, respectively.We would like to stress that calculation of �3�� in higherorders demands taking into account in Eq. �22� for �Pmn

+− �x+� ,x� not only the most singular diagrams in each order ofg2. Their analysis is a cumbersome task that is out of thescope of the present paper. Thus we restrict ourselves in thispaper by calculation of resonant terms in the dynamical sus-ceptibility in the order of g0 and discard the resonant terms in�1�� and �2�� that are of the order of g2. Notice that onecan keep these terms while calculating static susceptibilitybecause �3�0�=0 and the problem of the cumbersome analy-sis of the higher order diagrams does not arise.38

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184433-13

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�1990�.36 This consideration is qualitative as spin-wave formalism works

badly at such high temperatures.37 This circumstance has not been taken into account in paper I

where the f2 correction to the 1/T term was calculated incor-rectly.

38 One can make sure that �3�0�=0 using general expression �7� for�P��. Expression �B4� for �3�� is not equal to zero at �=0that is a consequence of approximations made while evaluatingEq. �B4�. Meantime �3�0� given by Eq. �B4� is much smallerthan �1�0� and can be discarded in accordance with consider-ation presented below Eq. �B4�.


184433-14

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