PHYSICAL REVIEW B, VOLUME 65, 012401

Spin-wave interaction in two- and three-dimensional antiferromagnets in a weak magnetic field

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

~Received 2 August 2001; published 29 November 2001!

A Heisenberg antiferromagnet~AFM! with easy-axis anisotropy in a weak perpendicular magnetic field isconsidered. The spin-wave interaction is studied using a 1/S expansion. It is shown that the field leads tointeractions which give infrared-divergent (1/S1) corrections to the spin-wave gap atk5k0 (k0 is an AFMvector!. So there is strong renormalization of this gap by the field and a full analysis of the 1/S expansion isdemanded for its evaluation. The dynamical chiral fluctuations mediated by the field are studied too and theirnon-trivial renormalization is considered. Qualitative consequences of these features are discussed in relationto the AFR and polarized neutron scattering experiments.

DOI: 10.1103/PhysRevB.65.012401 PACS number~s!: 75.30.Ds, 75.30.Gw, 75.40.Mg, 75.70.Ak

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Many compounds the magnetic properties of which haattracted much attention are a good physical realizationHeisenberg antiferromagnets~AFM’s!. The spin-wave inter-action in this model is studied mainly using a 1/S expansion(S is the spin value!. The total spin conservation law~TSCL!which holds in the isotropic exchange approximation forba strong interaction between long-wave spin waves.1,2 Thiscircumstance manifests itself in the so-called conspiracy,in the cancellation of terms which have infrared divergencin the expressions for the observable quantities. As a re(1/S1) terms give rise to main corrections for these quantitand (1/S2) ones are small.1–6

Meanwhile, in any real AFM there are weak interactiowhich violate the TSCL and break this conspiracy. The fievidence for that has appeared only recently in Ref. 6 whthe two-dimensional~2D! AFM is considered with magneticeasy-axis anisotropy taken into account. It is shown ththat the (1/S2) terms in the expression for the spin-wave gpossess an infrared divergence and one has to analyzfull 1/S expansion for its evaluation. This cumbersome thoretical problem has not been solved yet.

In this paper we consider 2D and 3D AFM’s with easaxis anisotropyl in a weak magnetic fieldH perpendicularto the magnetization of sublattices (gmBH!SZJ, whereZ isthe number of nearest neighbors!. As is well known,7 thisanisotropy leads to spin-wave gapsD at k50 and k5k0,wherek0 is the AFM vector. The field gives a contribution tthe gap atk50. Besides, three- and five-particle interactioappear. In the isotropic case they renormalize the canangle between sublattices and give rise to spin-wave inbility at the high-field limit.8 We will show below that thethree-particle interaction strongly renormalizes the spwave gap atk5k0. We evaluate the (1/S1) correction whichreveals the infrared divergence. We evaluate also in thisder the spin susceptibilities and demonstrate that the fiprovokes the nondiagonal chiral spin fluctuations which cbe investigated by inelastic polarized neutron scattering9 Itshould be mentioned that atl50 there are no large corrections to the spectrum of the order of 1/S1. In all cases (l50 andlÞ0) a complete analysis of the full 1/S expansionis out of the scope of this paper.

The Hamiltonian of a Heisenberg AFM with anisotropexchange in a uniform magnetic field has the form

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H5 (^ l l 8&

@JSlSl 81lSlzSl 8

z#1gmBH(

lSl

y , ~1!

wherel.0, z is the easy axis, and summation over all paof nearest-neighbor spins is assumed. The sublatticescanted backwards in the field direction by the angleu (u50 at H50) as illustrated in Fig. 1. We will use below thfollowing representation for the spin operators:10

Sl5Slxx1~Sl

zz1Slyy!eik0Rl cosu1~Sl

zy2Slyz!sinu,

Slx5AS

2S al1al†2

al†al

2

2S D ,

Sly52 iAS

2S al2al†2

al†al

2

2S D ,

Slz5S2al

†al , ~2!

whereal are Bose operators andl labels all lattice sites. For3D and 2D AFM’s k05(p,p,p) and k05(p,p), respec-tively. Let us substitute Eq.~2! into Eq. ~1!. The classicalenergy is given by

E052N@~S2ZJ/2!cos 2u2gmBHSsinu1~S2Zl/2!cos2u#,~3!

whereN is the total number of spins in the lattice. Minimzation of Eq.~3! with respect to the tilting angleu yields

FIG. 1. Canting of the sublattices in the field directed opposto they axis. Local coordinates for both sublattices are also sho

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BRIEF REPORTS PHYSICAL REVIEW B 65 012401

sinu52gmBH/(2SZJ1SZl). Below we use this classical expression for sinu, unless otherwise specified. However, tdeviation from it is very important for us and will be consiered later.

The term in the Hamiltonian linear in Bose operators dappears. As a result we haveH5E01( i 52

6 Hi , where

H25(k

FEkak†ak1

1

2Bk~ak

†a2k† 1aka2k!G , ~4!

H35 iAS

2sin 2u

1

N1/2 (112135k0

FJ112a1†~a232a3

†!a22

1l112

2a1

†~a232a3†!a221

l0

4a1

†a22a23G , ~5!

H451

2N (11•••1450

$a1†~2J3a222J312a2

†!a23a24cos2u

1a1†a2

†a23a24@~J1132J1!sin2u2l113#%, ~6!

H552 isin 2u

2A2S

1

N3/2 (11•••155k0

S J113

11

2l113Da1

†a2†a23a24a25 , ~7!

H65cos2u

8S

1

N2 (11•••1650

J11314a1†a2

†a23a24a25a26 ,

~8!

where j [k j , Jk52J( i 51d coski (d is the dimension of the

AFM!, lk5lJk /J, Bk5SJk cos2u, Ek5SJ01SJk sin2u1Sl0, and each sum is over the chemical Brillouin zone. Walso omit interactions of the order ofl sin2u in Eqs.~4!, ~6!,and ~8!.

For investigation of the spin-wave interaction wwill use an approach similar to Belyaev’s one developfor a diluted Bose gas.11,12 This approach is more convenient for this purpose than Bogolyubov’s transfomation.6,10 We determine three Green’s functionsGll 8(t)52^Tal(t);al 8

† (0)&, Fll 8† (t)52^Tal

†(t);al 8† (0)&, and

Fll 8(t)52^Tal(t);al 8(0)&. Because of the non-Hermicitof the Hamiltonian, we haveF†ÞF. There is a system oBelyaev’s equations for these Green’s functions, twowhich are presented in Fig. 2.6,10 The solution of this systemis given by

FIG. 2. Belyaev’s equation for the Green’s functionsG andF†.The left and the right dashed arrows represent the bare Grefunction G0k(v) and G0k

† (v)5G0k(2v), respectively, where

G0k5(v2Ek)21, S(v)5S(2v), andP(P†)5Bk1P(P†). The

valuesEk andBk are defined in Eq.~4!.

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G~v,k!5~SJ01SJk sin2u1Sl01v1S!D21,

F~v,k!52~SJk cos2u1P!D21,

F†~v,k!52~SJk cos2u1P†!D21, ~9!

where

D5v22ek21~SJ01SJk1Sl0!~P1P†!

1v~S2S !2K~v,k!, ~10!

K~v,k!5~SJ01SJk sin2u1Sl0!~S1S1P1P†!

1SS2PP†. ~11!

Here S(v)5S(2v) and ek is the spin-wave spectrum inthe linear spin-wave theory:

ek25~SJ01SJk!~SJ02SJk cos 2u!12S2J0l0 . ~12!

In the vicinity of k50 andk5k0 for ek we have the well-

known result7

ek25H ~ck!21DH

2 1D2, if k!k0 ,

~ck!21D2, if k;k0,~13!

where k5k2k0 , c5SJ0A2/Z is the spin-wave velocity,DH

2 5(2SJ0)2 sin2u, andD252S2J0l0.We proceed with a discussion of the first 1/S terms in the

expressions for the self-energy parts. In the exchangeproximation they correspond only to the single-loop HartreFock diagram shown in Fig. 3~a! and, as was obtainedbefore,1–6 give rise to the main corrections to all observabquantities. In particular, the renormalized spin-wave sptrum has the formekR

2 5ek2Zc , whereek

25(SJ0)22(SJk)2,

Zc5112(b12b2)/S, and

b151

~2p!d

1

2E ddkSJk

2

J0ek, ~14!

b251

~2p!d

1

2E ddkSJ02ek

ek. ~15!

The integrals in Eqs.~14! and ~15! are over the chemicaBrillouin zone. For example, for 3D AFM’s we haveb1'0.127 andb2'0.078.

As is shown in Ref. 6, anisotropy leads to violation of tconspiracy and gives rise to large corrections for the spwave gapD. As follows from Eqs.~5!–~8!, at H50 all in-

n’s

FIG. 3. ~a! The Hartree-Fock diagram of the first order of 1/S;~b! the diagram giving the first divergent terms in 1/S2 order atH50 andlÞ0; ~c! the diagram of the order of 1/S1 which appears atHÞ0 and gives rise to infrared divergent contributions for the seenergiesS, P, P†.

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BRIEF REPORTS PHYSICAL REVIEW B 65 012401

teractions contain an even number of Bose operators andfirst large contribution is of the order of (1/S2). It corre-sponds to the diagram shown in Fig. 3~b!. It was obtained inRef. 6 that for 2D AFM’s the expression for the square of tgap atq&D/(SJ0) has the formD2(11X). HereX is pro-portional to (T/SD)2 in the region D!T!TN and toln(SJ/D) at T!D.

When HÞ0, the odd interactions~5! and ~7! come intoplay and, as we demonstrate below, the three-particleleads to stronger divergences for the self-energy partslarge correction to the gap atk5k0 arises even in (1/S1)terms. It stems from the diagram presented in Fig. 3~c!. Letus consider it now in detail.

If the external frequencyv!SJ0 and the external impulsek is in the vicinity of 0 ork0, then, as is clear from Eqs.~5!and~9!–~13!, at smallH the most singular terms are propotional to u2R, where

R5T (v1 ,k1

~SJ0!3

D~v1 ,k1!D~v2v1 ,k2k12k0!. ~16!

HereD(v,k)52v22ek2 andv, v1 are the Matsubara fre

quencies. After summation overv1 and analytical continuation to realv1 id we get from Eq.~16!

R51

~2p!d2E ddk1

ek1ek2

S ek12ek2

~v1 id!22~ek12ek2

!2~Nk1

2Nk2!

2ek1

1ek2

~v1 id!22~ek11ek2

!2~11Nk1

1Nk2!D , ~17!

wherek25k2k12k0 andNk5(eek /T21)21 is the numberof magnons with impulsek.

If T50, we have to putNk50 and obtain

R521

~2p!d

1

2E ddk1

ek1ek2

S ek11ek2

~v1 id!22~ek11ek2

!2D .

~18!

If k;k0, we get from Eq.~18! using Eq.~13! that a self-energy part contains a divergence of the type@u2 ln(q2

1D2/c2)# and @u2(q21D2/c2)21/2# @whereq5(v/c,k)# for3D and 2D AFM’s, respectively. Fork!k0, we have fromEq. ~18! for 3D and 2D AFM’s @u2 ln(q21DH

2 /c2)# and@u2(q21DH

2 /c2)21/2#, respectively, where nowq5(v/c,k).For TÞ0, when ck(ck), DH!T, one can replaceNk

'T/ek and we obtain from Eq.~17!

R521

~2p!d

1

2TE ddk1

ek1

2 ek2

2 S ~ek12ek2

!2

~v1 id!22~ek12ek2

!2

1~ek1

1ek2!2

~v1 id!22~ek11ek2

!2D . ~19!

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In the case of 3D AFM’s atk;k0 we have a divergence othe type@Tu2(q21D2/c2)21/2# and, atk!k0, Eq. ~19! gives@Tu2(q21DH

2 /c2)21/2#.At l50 for 2D the AFM integral in Eq.~19! diverges in

the vicinity of k0 even at qÞ0. It is a consequence oabsence of a long-range order in 2D AFM’s atTÞ0.Taking into account a gap atk5k0 we get for the self-energy parts terms which are proportionalT(u/max$cq,DH%)2 ln(max$cq,DH%/D) if k!k0 and toTu2(q21D2/c2)21 if k;k0.

We begin with an evaluation of Green’s function denomnatorD in the first order of 1/S in the case ofl50. For thispurpose one has to take into account renormalizationthe tilting angle. In accordance with Ref. 13 we have tfollowing relation between renormalizedu and classicalu angles: sinu5sinu(11f/S), where f 5b22b12cos2u(2p)2d*d d kSJk /(2ek). Taking into account thisquantum correction a new term inH2 arises after substitutionof Eq. ~2! into Eq. ~1!:

H28522 f J0 sin2 u(k

ak†ak . ~20!

It leads to the replacement ofS(S) in formulas~9!–~11! by

S(S)22 f J0 sin2 u.In the first order of 1/S one has to consider only the firs

term in the expression~11! for K. We discuss first the case ok;k0. Direct calculations show thatK(0,k0)50 andK(v,k)524v2(R/S)sin2 u cos4 u. Here and below wetake into account the most singular contributioonly. Eval-uation of the odd part ofS gives v(S2S)528v2(R/S)sin2 u cos4 u For the combinationP1P† weget P1P†58J0R sin2 u cos6 u Substituting all these expressions into Eq.~10! we obtain, forD,

D5@124~R/S!sin2 u cos4 u#@v22~ck!2#. ~21!

Simple but tedious calculations show that in the casek!k0 all terms proportional toR in the expression for thegap cancel each other and the denominator has the form

D5v22@~ck!21DH2 #Zc . ~22!

So one can see from Eqs.~21! and ~22! that there are nodivergent (1/S1) corrections to the spin-wave spectrum atl50. We show now that this situation changes if the anisropy l is taken into account.

At k!k0 one can use expression~22! for D. Evaluation ofGreen’s function denominator atk;k0 gives

D5@124~R/S!sin2 u#@v22~ck!22D2ZD#, ~23!

whereZD5128(R/S)sin2 u. From Eq.~23! we see that thespin-wave gap is renormalized by the factorZD which be-comes large asq,D→0. This renormalization can be studieby AFR and polarized neutrons.

We represent now corresponding spin susceptibilitiestermined as follows:xab(Rl l 8)52^TSl

a(t)Sl 8b (0)&. At H

50, the x and y directions are physically equivalent an

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BRIEF REPORTS PHYSICAL REVIEW B 65 012401

there are no nondiagonal components. However, the mnetic field gives chiral spin fluctuations, nondiagonal compnents appear, and we havexzx52xxz . In this case one hato take into account, along with the change of the denonator D in Eqs. ~22! and ~23!, the corresponding renormaization of the numerators of theG andF functions as well asadditional contributions connected to the last terms indefinition of operatorsSx andSy, Eqs.~2!. As a result we getfor k!k0

xxx~v!52S2J0Zx sin2u

v22~ck!22DH2

, ~24!

xyy~v!5S2l0ZD

v22~ck!22D2ZD

, ~25!

xzx~v,k!5 iSv sinu~12b2 /S!

v22~ck!22DH2

, ~26!

where Zx511(b122b2)/S ~see Ref. 6 and referencetherein!. Here we see that there is a strong renormalizationthe gap in thexyy channel which can be investigated in AFexperiments. Corresponding expressions atk;k0 have theform

xxx~v!52S2J0Zx

v22~ck!22D2ZD

, ~27!

xyy~v!52S2J0Zx

v22~ck!22DH2

, ~28!

xzx~v,k!5 iSv sinu~122R!

v22~ck!22D2ZD

, ~29!

where

R5T (v1 ,k1

SJk1~SJ01SJk1

!2

D~v1 ,k1!D~v2v1 ,k2k12k0!. ~30!

We have also taken into account in Eqs.~24!–~29! that ourresults are valid forR!1 only. One can see now that stronrenormalization appears in thexxx channel. Analysis of the

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expression~30! for R similar to that forR reveals thatR atT50 is proportional to ln(q21DH

2 /c2) and to (q2

1DH2 /c2)21 for 3D and 2D AFM’s, respectively. ForTÞ0

the corresponding expressions areT@(cq)21DH2 #21 and

TSJ0@(cq)21DH2 #22. So R would be larger than unity a

small fields.The nontrivial renormalization of the chiral fluctuation

which can be successfully studied by polarized neutrscattering should be noted.9 It is a result of mixing longitu-dinal and transverse spin fluctuations which appear inmagnetic field and the divergence inR reflects the infrareddivergence of the longitudinal spin susceptibility in AFM’s.14

Hence, investigation of the chiral fluctuations using polized neutrons would be very important for understandingrenormalization of these fluctuations.

According to Ref. 9 the chiral part of the cross sectionschdepends on the imaginary part of2 ixzx . So one can seefrom Eq. ~29! that, atk;k0 , sch is proportional toH@d(v2ek) 1 d(v1ek) #@1 22 ReR# / 222Hv Im R/(v22ek

2),whereek5@(ck)21D2ZD#1/2. Investigation of the nonresonant contribution of Im(2 ixzx) provides direct informationof the infrared divergence in longitudinal spin fluctuationApparently it is the easiest experimental method for itsvestigation.

Based on the results above obtained and on those of6 we can make the following qualitative predictions whican be verified experimentally:~1! the spin-wave gap atk5k0 in 2D AFM’s should strongly depend on the valuethe field in the region (gmBH)2*DSJ0 and on the temperature atD&T!TN , and~2! in 3D AFM’s the gap depends onthe field at (gmBH)2*(SJ0)2/ln(SJ0 /D) and it depends bothon the field and onT at T(gmBH)2*(SJ0)2D.

In summary, we demonstrate that the field leads tostrong infrared renormalization of the spin-wave gap inand 3D antiferromagnets atk5k0 (k0 is the antiferromag-netic vector! and it provokes chiral spin fluctuations.

We are grateful to D.N. Aristov and M.E. Zhitomirsky fointerest in the work and useful discussions. This work wsupported by the Russian State Program for Statistical Pics ~Grant No. VIII-2!, the Russian Foundation for BasResearch~Grant Nos. 00-02-16873, 01-02-06241, and 00-196610!, and the Russian State Program Quantum Macphysics.

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10D. Petitgrandet al., Phys. Rev. B59, 1079~1999!.11S.T. Belyaev, Sov. Phys. JETP7, 289 ~1958!; 7, 299 ~1958!.12A.A. Abrikosov, L.P. Gor’kov, and I.E. Dzyaloshinsky,Quantum

Field Theory Methods in Statistical Physics~Pergamon, NewYork, 1965!.

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