PHYSICAL REVIEW B VOLUME 52, NUMBER 2 1 JULY 1995-II

Bosonic mean-field theory for frustrated Heisenberg antiferromagnets in two dimensions

R. ChitraPhysics Department, Indian Institute of Science, Bangalore 560012, India

Sumathi Rao~Institute ofPhysics, Sachivalaya Marg, Bhubaneswar 751005, India

Diptiman Sen~Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India

S. Suresh Rao~Institute ofPhysics, Sachivaiaya Marg, Bhubaneswar 751005, India

(Received 7 February 1995)

We use a recently developed bosonic mean-field theory to study the ordered ground states of frustratedHeisenberg antiferromagnets in two dimensions. We emphasize the role of condensates in satisfying themean-field variational equations and their relation to spin-correlation functions at low temperatures.This mean-field theory is closely related to a spin-wave theory that enables us to obtain the spin-wavespectrum easily (without any rotation of axes) for an entire class of three-dimensionally ordered states.For the frustrated Heisenberg antiferromagnet on a triangular lattice, we use this theory to compute thespin-wave spectrum for all values of J2/J& and demonstrate the phenomenon of order from disorder.

I. INTRODUCTION

The ground state and low-energy excitations of Heisen-berg antiferromagnets (AFM) in two dimensions havebeen extensively studied, ' ' particularly after thediscovery of high-T, superconductors. The various tech-niques used for this purpose include nonlinear fieldtheories, spin-wave analysis, bosonic and fermionicmean-field theory ' (MFT) and numerical methods. ' 'For unfrustrated AFM such as Heisenberg models on bi-partite lattices with just nearest-neighbor couplings, it isnow well-established that the zero-temperature groundstate is ordered for all values of the spin S. At any finitetemperature T, the models are disordered with a correla-tion length which grows exponentially with 1/T for smallT. However, for frustrated AFM (e.g., Heisenberg mod-els on bipartite lattices with next nearest-neighbor in-teractions as well), the situation remains interesting andunresolved. Besides the conventionally ordered states ex-pected by extrapolation from the classical limit, manyother interesting ground states, such as spin nematics,Aux phases, and chiral spin liquids have been proposed.The issue of which of these states is actually a groundstate for different ranges of parameters (i.e., the phase di-agram) of frustrated spin models remains open.

In this paper, we do not address the question of thecomplete phase diagram of frustrated models. Instead,we concentrate on studying the properties of the hel-icoidal (ordered) phases that are obtained by using a bo-sonic mean-field approach. Our main aim is to furtherdevelop a recently introduced bosonic MFT (Ref. 17) andshow how various properties of the mean-field state canbe easily obtained. We also compare our MFT, whichhas three bosons at every site (3BR), to the more conven-

tionally used Schwinger boson MFT which uses two bo-sons at each site (2BR), and explain both the merits andthe limitations of our approach. Our 3BR has the furtheradvantage that it is closely linked with a new spin-wavetheory, which can be apphed even when the spin orderingis not along any line or plane, but is fully three dimen-sional. We explore this connection in this paper and alsouse the spin-wave theory to find the spectrum anddemonstrate order from disorder, in the most generalcase, for the frustrated Heisenberg AFM on a triangularlattice.

The rest of the paper is organized as follows. In Sec.II, we study simple examples of the kinds of classicalground states possible for frustrated models. This ismainly a review of the models considered by Villain. '

Using our new spin-wave formalism (inspired by the3BR), we work out the general spin-wave spectrum anddemonstrate order from disorder. ' In all the cases, thestates with the minimum quantum energy always turnout to be colinear spirals. As a particular example, wepresent the complete calculation of the spin-wave spec-trum and the phenomenon of order from disorder for afrustrated model on a triangular lattice. A previousanalysis' of this model did not include the two-parameter family of degenerate canted states. In Sec. III,we introduce the 3BR and develop the mean-field ap-proach. The 3BR uses a representation of the spins bythree bosons transforming under the adjoint representa-tion of the SO(3) group. A simple mean-field picture thenleads to variational equations. We argue that in general,the equations require condensates of the three bosons atthe Goldstone modes (i.e., zeros of the spin-wave energy)for a solution. These condensates are related to long-range order (LRO) in the system. In Sec. IV, we explicit-

0163-1829/95/52{2)/1061(9)/$06. 00 1061 1995 The American Physical Society

CHITRA, RAO, SEN, AND RAO

ly solve the variational equations for S =1/2 for the un-frustrated models on both square and triangular latticesand the frustrated model on a triangular lattice. We findexcellent numerical agreement with earlier results for theunfrustrated cases, provided we drop the extra factor of3 which comes from the spurious tripling of spin-wavemodes in the 3BR. This multiplicity is an unfortunatefeature of a11 bosonic MFT. The commonly used 2BRalso produces a spurious doubling of modes. We end thepaper with a brief discussion in Sec. V, where we brieAypoint out directions for future research.

II. CLASSICAL GROUND STATESAND SPIN-WAVE THEORY

Lattice

Square

Coupling strengths

J, &J, /2Jq= Ji/2Jq) Ji/2

Category ofground state

{ii)(iv)(iii)

Triangular J2 & Ji/8J, /8+ J2+J,Jq) Ji

(i)(iv)

(i)

TABLE I. Categories of classical ground states on the squareand triangular lattices for various ratios of the coupling con-stants J& and Jz.

Let us consider a general Bravais lattice in d dimen-sions and a Heisenberg spin Hamiltonian of the form (S„)=Su,cosQ.n=+Su, , (2.4)

=1H =—QJsS„S„+s .2

n, 5(2.1)

Here the sum over n runs over the positions of all thesites and 6 denotes the position of a neighboring site. Weassume that the couplings J& +0 for all 5 and that J&goes to zero suFiciently rapidly for large 5. In part A ofthis section, we tabulate the different kinds of classicalground states of this model on square and triangular lat-tices, for various ratios of the nearest-neighbor and next-nearest-neighbor couplings. In part B, we introduce anew Holstein-Primakoff representation of spins to obtainthe spectrum in all the cases tabulated in part A. Finally,in part C, we explicitly compute the dispersion for thefrustrated model on a triangular lattice. We show howthe zero-point fluctuations always pick out the colinearstates as the lowest energy states even though, classically,planar canted and three-dimensional canted states are al-lowed as ground states.

=S(+u,cosP+uzsinP) . (2.&)

These configurations describe planar canted states.Category (iv): When the minimum of Ep(Q) occurs for

three different vectors Q&, Qz, and Q3, all of which satisfy2Q,. =~„ then there exists a two-parameter family ofground-state configurations given by

(S„)=S(u,cosQ, nsinOcosg+uzcosQ2 nsinOsing

+u3cosQ3 n cosO)

which describes a colinear spiral.Category (iii): When the minimum of Ep(Q) in Eq.

(2.3) occurs for turbo different vectors Q, and Qz, such that2Q, =~, and 2Q2=~2 but Q, +Q2%&3, then there exists aone-parameter family of configurations which haveminimum energy, given by

(S„)=(u, cosQ, n cosP+uzcosQ2 n sing)

A. Classical ground states=S(+u, sinO cosP+u2sinO sintI)+u3cosO), (2.6)

The classical ground states of the Hamiltonian in Eq.(2.1) have been studied in detail in Ref. 18. Here we lookat examples relevant to frustrated models on square andtriangular lattices.

Category (i): In general, the classical ground stateshave the spin configuration

(S„)=S(u,cosQ n+uzsinQ. n), (2.2)

where u, and uz are two orthonormal vectors defining aplane, and the pitch vector Q is obtained by minimizingthe classical energy of the Hamiltonian given by

Ep(Q) =—QJscosQ 5,NS

(2.3)

where X is the total number of sites. This configurationis called the planar spiral configuration.

Category (ii): Let us use r as a generic symbol todenote a reciprocal-lattice vector so that w. n=0mod 2~for all sites n. Then for the special case 2Q=~, theconfiguration in Eq. (2.2) reduces to

where (u„u2, u3) form an orthonormal triad and0~8~m. , 0~/~+/2. These configurations describe anordered state in three dimensions and are called three-dimensional canted states. Notice that the planar cantedstates in (iii) are a special case of these canted states for8=sr/2 or for /=0 or m/2. The colinear spiral in (ii) is,in turn, a special case of the planar canted state for /=0or ~/2.

All these kinds of ground states occur for the J& —Jzmodels on square and triangular lattices. (J, is thenearest-neighbor coupling and J2 is the next-nearest-neighbor coupling. ) In Table I, we show the differentkinds of classical ground states that occur for different ra-tios of the coupling constants.

B. Spin-wave spectrum

We now introduce a variant of the Holstein-Primakoffrepresentation of the spins to obtain the spin-wave spec-trum. This formalism can be used to expand about anyclassical configuration as shown below. Consider a spinwith classical components given by

BOSONIC MEAN-FIELD THEORY FOR FRUSTRATED. . . 1063

(S, ) =S cosa,

(S +iS ) =Se —'~sina .(2.7)

(and ferromagnetic) order parameters, respectively.From Eq. (2.10), we get

Bosonic operators b and b are introduced (similar toHolstein-PrimakofI) to represent deviations away fromthe classical configuration by

~Xs ~

=S sin (Ps/2)

~ Ys ~

=S cos (Ps/2) .

(2.14)

S, =cosa(S b—b) ——,'sina[(2S —b b)' b +H. c.],

(2.8)

S„+iS =e —'~[sina(S btb—)

Thus, their magnitudes equal S (and 0) if Ps=+ and 0(and S) if ps =0, justifying their nomenclature.

Finally, let us define

+—,'cosa[(2S bb)—'~ b +H. c. ]

+—,' [(2S btb—)'i b —H. c. ] ] . and

A, = —SgJs costs5

(2.15)

s=(s) 1—b b

5 b(a —&* b'(a—), (2.9)

where ( a ) denotes the vector with components given by

and

(a, ) =v'S/2( —cosa cosp+i sinp),

(az ) =v'S/2( —cosa sinP —i cosP),(2.10)

(a3 ) =V'S/2 sina .

The reason for introducing the vector a will become clearin Sec. III, where the three a; s will be identified withthree bosonic operators and (a; ) will denote theground-state expectation values of these bosons. Here, ais just a convenient notation. Notice that ( S ) /S,+2/S Re(a) and &2/S Im(a) form an orthonormaltriad so that

It can be easily checked that these spin operators satisfythe usual commutation relations if [b, bt]=1. Noticethat the usual Holstein-Primako6'bosons are recovered ifwe set a =0 or m..

To obtain the spin-wave spectrum, we expand Eq. (2.8)to next-to-leading order in 1/S. We then find that

h„=+Js(S„+s) .

Here h„may be interpreted as the local magnetic field atthe site n. In the classical ground state, (S„)at the site naligns itself antiparallel to the local magnetic field h„.Thus the orthogonality of (S) and (a) in Eq. (2.10) im-

ply that

h„(a„&=h„&a„&'=0 . (2.16)

Substituting for S„and S +s from Eq. (2.9) in the Hamil-tonian in Eq. (2.1) and using Eqs. (2.14)—(2.16), we findthat to 0 (S) in an expansion in 1/S,

H = — +kgb„b„n

+ QJs (Xs—b „+sb„+Ysb „+sb„+H. c. ) .1

2(2.17)

The first term in Eq. (2.17) is the classical energy E,&of

0 (S ) and the remaining terms are of 0 (S).To find the spectrum, we transform to momentum

space and diagonalize Eq. (2.17) by a Bogoliubov trans-formation. This yields

&S&.&a&= &S&.(a)'=0. (2.11)

In part A, we classified states of the spin model where

(S„) (S„+s) is independent of n and only depends on 5through the angle Ps between the classical spins at n and

n+5, i.e.,

0= — +1V Dk cok BgB~+—,'NSA,

where the dispersion relation is given by

—(

2i

i2) /

(2.18)

(S„) (S„+s)=S costs .

In this part, we make the further assumption that

and

(2.12) with

and

pq=A, +QJs Yse'"'5

(2.19)

Y, =(a.&.&a„+,&*

are also independent of n and depend only on 5. [Weshall see later that this assumption is always valid incategories (i)—(iii), but in (iv), it holds only if there existsan extra condition relating the vectors Q„Q2, and Q3.]Xs (and Ys ) are called the short-range antiferromagnetic

v, =yJsx, e"' .5

The Bogoliubov transformed bosons are

Bi,=coshObi, +sinh8b

and (2.20)

1064 CHITRA, RAO, SEN, AND RAO

TABLE II. Table of the angles Pq and the short-range orderparameters X& and F~ as a function of 5 for category (iv)—thethree-dimensional canted state.

cosQi. 5 cosQz 5

1—1

1

cosgg

1

cos20sin 8 cos2$—cos 0—sin'8 cos2$—cos~O

Xg /S

0sin 0sin P —cos 8 cos P+ i cos8 sin2$cos P —cos 8 sin P

i co—s8 sin2$

Fg /S

1—cos 0sin 8cos P

sin 8sin P

costs=cosg 5 .

For category (iii),

costs=cosg, .5cos P+cosgz. 5 sin P .

(2.21)

(2.22)

For category (iv), however, Xs and Ys in Eq. (2.13) arenot independent of n, unless cosg, n cosgz n cosQ3 n= 1

for all n. Since Q;.n =0, this implies that

B&=sinhOb &+coshOb k

and the integration measure is Dk =vd k/(2m ), where v

is the area per unit cell.The expi~~i'. . forms for the dispersion in the four

categories defined in part A can now be found. Forcategories (i) —(iii), Xs and Ys as defined by Eq. (2.i3) areindependent of n and hence, it sufFices to find costs. Forcategories (i) and (ii)

extend those results to the most general case, where theclassical ground state has an arbitrary three-dimensionalspm ordenng.

The sites of a triangular lattice are located atn=a [(n, +nz/2)x+&3/2nzy ], where (ni, nz) are apair of integers, a is the lattice spacing and v =+3a /2 isthe area per site. The hexagonal Brillouin zone is shownin Fig. 1. Define e=Jz/J, . For e ( I/8, the classicalground state is an example of category (i), with the pitchvector Q given by Q=(4m/3a, O), denoted by point 2 inFig. l. Other values of Q related to this one by vr/3 rota-tions in the (k„,k~) plane (i.e., the other five corners ofthe hexagon) are also allowed. For e & 1 too, the classicalground state falls into category (i) with Q=(O, Q~) withcos(+3aQ /2)= —(I+ I/e)/2. This is represented bypoint C in Fig. 1, with other possible values of Q beingrelated to it by n/3 rotations. In both these cases,costs =cosQ 5, Xs and Ys are correspondinglyidentified.

However, for I/8 ~ e ~ 1, the classical ground state be-longs to category (iv) because Q lies at the center of oneof the six edges, such as the point B in Fig. 1, and henceis half of a reciprocal-lattice vector. Let us choose thethree vectors as Q, =(m. , m/v'3)/a, Qz=( m, n/&—3)/a. ,and Q3=(0, —2m. /&3)/a. The other three possible vec-tors are equivalent to these up to reciprocal-lattice vec-tors. Besides 2Q;, we can check that Q, +Qz+ Q3 is alsoa reciprocal-lattice vector. Hence, Ps, Xs, and Ys can befound from Table II. The spin-wave spectrum is given byEq. (2.19) with

pi = I+a+sin O(cos PDi+sin PDz) —cos OD3,1

Qi+Qz+Q3=& . (2.23)

[Fortunately, this is true for the two examples of category(iv) in Table I.] With this condition satisfied, we have

costs=cosg, 5sin Ocos P+cosgz 5sin Os.in P

=[sin P—cos Ocos /+i cosOsin2$]D,

1

+[cos P —cos Osin P —i cosOsin2$]Dz

+Sin OD3 (2.25)

+cosg3'5 cos O . (2.24) where

Since Q„Qz, and Q3 are not independent, Eq. (2.24) maybe further simplified. Using Eqs. (2.22) and (2.24), Xs,Y&, and A, and consequently co& can be computed.

In Table II, we present the values for the angle Ps[given by Eqs. (2.6) and (2.12)], Xs and Ys for the fourdifFerent possible values of cosQ, 5 and cosQz-5. Thus,we see that this method can be used to And the spin-wavespectrum about any classical ground state, even when ithas no linear or planar order, but is fully three dimen-sional.

C. Spin-wave spectrum of the J&

—J2 modelon a triangular lattice

We now apply this new spin-wave theory to the J, —J2model on a triangular lattice and show that thephenomenon of "order from disorder" occurs. Althoughthis phenomenon has been demonstrated for the triangu-lar AFM for some ratios of J2/Ji, ' we have been able to

FIG. 1. Brillouin zone for the triangular lattice. The pointsA, 8, and C denote the pitch vectors of the classical groundstates for e=J2/Jl lying in the ranges e( 1'/8, 1/8 ~ e ~ 1, ande & 1, respectively.

BOSONIC MEAN-FIELD THEORY FOR FRUSTRATED. . . 1065

akD =cos '

v'3 3 &3ak~ +e cos —ak„+ ak

2 " 2

(2.26)

and

af.a=s

a~.a~a.a=o(3.2)

ak„ 3D =cos —— + ak +icos —ak—2 2 2 X

D3 =cos(ak )+e cos(&3 ak ) .

Thus, the quantum energy

V3ak

on all physical states. Since a a is a number operator,this clearly indicates that the 3BR only works for integervalues of S. In terms of these bosons, the Hamiltonian inEq. (2.1) can be written as

H =—QJs[:Y„sY„s.—X„sX„s]n, 5

Eq p dk" (2~)2

++[A,„(a„a„—S)—p at a„a„.a„],

where we have used the identity

(3.3)

is a function of 8, P, and e. [We will henceforth use thesymbol Dk for Ud k/(2') .] We have numerically stud-ied this for a large number of values of e from 1/8 to 1

and (0,$) for 0~8~+/2, O~g~vr/2 Othe. r values of(8,$) can be related to this range by rotations andrefiections. [A symmetric tetrahedral spin configurationarises in the special case O=cos '(1/&3), P=m/4. ] Forall e, we find that E is minimum for the three cases(0=0, any value of P) and (8=m/2, /=0 or vr/2) These.are precisely the colinear spirals of category (ii). Thus wehave established that the order from disorderphenomenon occurs for triangular AFM, where the gen-eral classical ground state has three-dimensional order-ing.

III. THREE BOSON RKPRKSKNTATION OF SPINS

The spin-wave spectrum presented in Sec. II is an ex-pansion about an ordered classical ground state and isonly applicable when the system has LRO. Moreover, itexplicitly breaks rotational invariance and is only valid inthe large-S limit. To overcome these limitations, differenttechniques using bosonic and fermionic operators havebeen tried. The most well-studied of these is theSchwinger boson MFT. ' '" The Schwinger boson MFTinvolves a representation of the spins in terms of two bo-sons (2BR) with a constraint on the number of bosons ateach site. This representation has the disadvantage thatthe first step in the analysis involves a rotation of all thespins to a "ferromagnetic" alignment. [There are paperswhich do not rotate to a ferromagnetic state. Howeverthey do not get the correct spin-wave spectrum forS—+ oo (Ref. 12).] However, a different representation ofthe spins was recently introduced' in terms of three bo-sons (3BR) with two constraints at each site. This doesnot involve any such rotation and is technically simplerto use, particularly when the classical ground state be-longs to the noncolinear categories (iii) or (iv).

The 3BR represents the spins as

S= —ia~Xa, (3.1)

where a are three bosonic operators satisfying[a, ,at]=5,". To satisfy the relation S =S(S+1), weneed to impose two further Hermitian constraints on thebosons given by

Sn m ~nm nm +nm+nm

with

(3.4)

rn m =a„am

(3.5)

A. Hartree-Fock mean-field theory

We make a mean-field decomposition of the Hamiltoni-an by writing

AtA =(At)A+At(A ) —( A )(A ), (3 6)

where A =X„„+sor Y„+s. [Such a decomposition canbe justified in the large-1V limit, by generalizing the spinHamiltonian from SO(3) to SO(N) (Ref. 17).] Next, wemake the mean-field ansatz that (X„+s), ( Y„+s), &„,and p„are all constants independent of n. With this an-satz, the Hamiltonian can be diagonalized in momentumspace by a Bogoliubov transformation [analogous to Eq.(2.18)] to obtain

""= —XS+pX2+ —yJs(~Xs~2 —~ Y, [2)

3 3+fDk co A ~ A+ —m ——pk k k 2 k 2 k (3.7)

where

and

V~=A, +XJs Ys5

vt, =px0+ gJsXse '

5

(2

i

i2)1/2

(3.8)

Unlike the spin-wave spectrum, we now have three

X„=a„.aThe colons indicate normal ordering, and k„and pn areLagrange multiplier fields introduced to enforce the con-straints.

1066 CHITRA, RAO, SEN, AND RAO 52

+—fDk o)q3 (3.9)

by extremizing it with respect to k, p, X&, and Y5. Weget

decoupled bosons for each k, because the two constraintsat each site have been relaxed by the mean-field ansatz totwo global constraints.

The variational equations of motion are obtained fromthe mean-field ground-state energy

"' = —X S+— +pX', + —yJ, ((X,(' —fI, J')

1 1

2Ni 2COk

2~'+ ga;5 (k —k, )l

(3.12)

on the LHS. Thus Eqs. (3.11) are replaced by

—fDk +3+a;pk =S+—,3 Pj 3

2 cok . ' 2

3 Vk Ik. .5Dk e' +3+a;vq e ' =Xs,

@Ok I

(3.13)

in the bosonic language and it is implemented in theequations by replacing

32pXO+ —fDk p=0,

2 cok

3 Pa 3ak =S+—,2 Q)i 2

3 Dk e'" =X2 coi

—fDk e'"' =F2 coi

(3.10)

(3.1 1)

—fDk e'" +3+a;pq e ' = Fs .2 cok l

(S, S„)=I„' —X'„,where

(3.14)

To solve these equations to leading order in S, the in-tegrals which are of O(1) are dropped, and Xs, Ys, I,, and

p are substituted from Eqs. (2.14) and (2.15). We then ob-tain the values of e;. Note that when the condensate pa-rameters a; in (3.13) are introduced, we must also add anequal number of equations co& =0.

I

These condensates are related to LRO in the system. "To see this, note that we can write

Equations (3.10) imply that pX& =0. As a result, vk and

coi, and, consequently, Eqs. (3.11) do not depend on pX0.Hence, we only need to solve Eqs. (3.11).

r„=(a,'a„)

X„=(a,a„) .

(3.15)

B. Condensates and long-range order in two dimensions

Let us first understand what we expect from the varia-tional equations. To leading order in S, we expect to re-cover the spin-wave results, because the di6'erence be-tween X& and Y& as defined here and in Sec. II, is onlythat here they are expectation values of bilinears in bo-sonic operators, while in Sec. II, they were products ofexpectation values of single bosonic operators. These twodefinitions must coincide as the importance of quantumfluctuations decreases, i.e., as S~ ao. Hence, in this lim-it, spin-wave results tell us that X&, Y&, p&, vk, and co& areall of O(S). This means that Eqs. (3.11) cannot besatisfied because the left-hand sides (LHS) are of O(1),whereas the right-hand sides (RHS) are of O(S). (In onedimension, the LHS were at least singular when cok van-ished. Hence, the equations could be satisfied by allowingfor the formation of a gap. ' This is consistent with theexpectation that AFM are disordered in one dimension. )

In two dimensions, the LHS are not singular (except forspecial values of Js). Moreover, the expectation is thatAFM in two or more dimensions are generally ordered atzero temperature.

The mismatch between the LHS and RHS of Eqs.(3.11) is resolved by realizing that the bosons Ak con-dense at the momenta k, ,i = 1,2, . . . , where co& is exact-

l

ly zero. This is the signature for an ordered ground state

In terms of Fourier components,

Pa+&kY.+X.=-' Bk "" " e'" "43+a;(pq +vt, )e

(3.16)

To 0 (S), this gives

(Ss S„)=S cosg n (3.17)

as expected for an ordered ground state. Finally, weshould point out that the energy coj, can vanish to O(S)for other values of k also. The values of k where conden-sates are formed are called Goldstone modes. Here, cok isexactly zero for reasons of symmetry breaking. At theother values of k, the condensate density is zero and coi, iszero only to O(S) and quantum effects produce a gap athigher orders in 1/S, called the quantum exchange gap.We refer the reader to Ref. 7 for further details.

In a disordered state, co& is not zero for any k and thereare no condensates either. So the parameters a; aredropped together with the equations co& =0. For a given

l

set of couplings J&, we try to find solutions to the mean-field variational equations both with and without conden-sates. If both solutions exist, we see which of the twosolutions has the lower energy EM„.

BOSONIC MEAN-FIELD THEORY FOR FRUSTRATED. . . 1067

IV. NUMERICAL RESULTS FOR S = 1/2 where Dk =vd k/(2n) and

In this section, we discuss the numerical solutions ofthe variational equations (3.13) after dropping the factorsof 3 in those equations. In part A, we solve the variation-al equations for the Heisenberg model on a square latticeand compare the energy and sublattice magnetizationwith the values obtained by other methods. In part 8,we do the same for the Heisenberg model on a triangularlattice. Finally, in part C, we obtain the solutions of thevariational equations for the J& —J2 model on a triangu-lar lattice and plot the energies as a function of e=Jp/J, .

y1,=

—,' [cos(ak„)+cos-,'-(ak„+ &3ak )

+cos-,'-(ak, —&3ak )],Pg —k+ 6Yyg

v~ =6Xy~,

(2 2)1/2

Q)p —Q)Q —0 .

(4.6)

A. Heisenberg model on a square lattice

For the ordered state, co1, vanishes at 0 = (0,0) andQ=(7r, m)/a. The variational equations for S =1/2 aregiven by

1 PaDk +o.ppo+a p =1,

Q Q

1 PaDk cosa(k +k )+aauo —a&@&=F, (4.1)COg

= —0.5567J, (4.7)

and the sublattice magnetization

Bl = 3A,Q(XOCtg= 0.226 . (4.8)

On solving these equations with Ave variables, we ob-tained

E= —A, +3J(X —P )+—fDk ~1,

1

2

—' fDk "2 Q)k

where Dk =a d k /(2' ) and

p1, =A. +4Y cosa (k +k ),vt, =4X cosa (k„+k~ ),

(2 2)1/2

andcop —coQ —0

cosa(k +k )+o,ovo—agvg=X,

(4.2)

The value for energy agrees well with those given in Ref.20. The good agreement for the nearest-neighbor Heisen-berg models on the square and triangular lattices en-couraged us to proceed with the numerical analysis forthe frustrated model. This is presented in part C.

We have also solved the above equations for arbitraryS in order to determine the critical value S, below whichthe condensates are zero and there is no LRO. We Andthat S, =0.1966 for a square lattice and 0.3395 for a tri-angular lattice.

C J& J2 model on a triangular lattice

= —0.6704J, (4.3)

while the sublattice magnetization I is given by

m = 2A,Q aoa& =0.303 . (4.4)

These values compare very well with the results fromother studies which yield E = —0.6693J andm =0.308.

B. Heisenberg model on a triangular lattice

The variational equations for S =1/2 are given by[with Q=(4'/3, 0)/a]

We solve these equations numerically. The energy isfound to be

E = —k+ 2J (X —Y ) +— Dk co1

2 k

The spin-wave theory of Sec. II tells us that there arethree regimes to be examined:

(a) @=J2/J, (1/8 where the zeros of the spin-waveenergy lie at the momenta 0=(0,0) and Q=(4n/3,0)/a,

(h) 1/8(e(1 where the zeros are at O, Q, =(m, ~/&3)/a, Q2=( m, vr/U'3)/—a and Q3=(0, —2n/&3)/a,and

(c) e& 1 where the zeros are at 0, and (O, Q ) withcos(&3aQ~ /2) = —(1+1/e) /2.

The variational equations for regime (a) (e(1/8) aregiven by

1 PkDk +o;pup+ 2o;QpQ = 1,

2 cok

1 PIDk y )g+ cxopp 0,'QPQ —F)

2 Q)]j

1 Pj—fDk +a+12,o+2aqP&=1,2 cog

1 PkDk y]I, +appp —uQPQ

=+, (4.5)

1 Dk y ik+ ovo ~QvQ =+&2 cog

Pay2I +aopo+2cxQPQ= +2,

COk

(4.9)

1y, +novo —c QvQ=X,

COg

1 Dk y2g+ Aovp+ 2CXQVQ X2C9g

CHITRA, RAO, SEN, AND RAO

y,k= —,'[cos(ak, )+cos—,'(ak —~3ak )

+cos—,'(ak„+ V'3ak )],y2&= ,'[c—os(~3ak~)+cos—,'(3ak, —~3ak )

+cos—,'(3ak, +~3ak )],Pk=~+6Y1 y1k+«Y2y2k

vk =- 6X,y,k+ 6eX2y2k,

(4.10)

Ov5

OG

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

/

//

//

/

///

//

//

//

//

//

//

//

//

6)Q =coq —00.2 0,4 0.6 0.8

epsilon1.2 1.4

Note that the ground state is a planar spiral and we havecondensates at 0 and +Q, so that the SO(3) spin symme-try is completely broken. These equations with sevenvariables were solved, and the results for the energy

E= —A, +37, (X, —F, )+37,E(X~ —F2)+ —fDk co~1

(4.11)

FIG. 3. Magnetization as a function of e=J2 (J1=1). Hereagain, the solid line denotes the results from regime (a) and thedashed line the results from regime (b).

1 PkJDk +cxppp+cxgpq —12 cok

and sublattice magnetization

m =3X+a,aq (4.12)1 Pk

Dk y12+ O.QPQ+ eqPQ = Y12,2 Q)k

are plotted in Figs. 2 and 3, respectively, as a function ofe. We took values of e up to 0.12 where the magnetiza-tion went to zero.

For regime (b) (1/8(@&1), one could have conden-sates at Q„Q2, or Q3. However, the phenomenon of or-der from disorder chooses one of the three values for con-densation. (The three choices are related to each otherby rotations. ) We choose the condensate to be at Q=Q3.So the ground state is a collinear spiral. Since this leavesa residual U(1) symmetry, there are only two condensatesat 0 and Q3. The variational equations are then given by

1 Pky 14+QPQ QPQ = Y142 QPk

1Dk y12+ CXQVQ+ CXQVQ X122 cok

1Dk y14+OQVQ &QVQ=X14 ~2 ctPk

1 PkDk y22+ CXQPQ+ o;QPQ —Y222 cok

1 PkDk y24+ CKQPQ CKQPQ Y242 Q)k

(4.13)

1y 22+ novo+ ~qvq —X222 QPk

—jDkv

2k

y24+ aQVQ—

aQVQ X24COk

-0.65

-0.7

-0.75

-0.8

with

Pk ~+2 Y12y12+ Y14y14+26Y22y22+4E' Y24y24 ~

Vk 2X12y12+ 4.X14y 14+2~X22 y 22+ 4~X24y 24 &

y,2=cos(ak„),(4.14)

-0.90 0.2

I I

0.4 0.6I

0.8epsilon

1.2

ak„y14 =cos cos

2

y22=cos(~3ak»),

&3ak,

FIG. 2. Energy as a function of e= J2 (J1=1). The solid linedenotes the results from the equations in regime (a) and thedashed line the results from regime (b).

3y24 =cos —ak cos

&3ak,

52 BOSONIC MEAN-FIELD THEORY FOR FRUSTRATED. . .

cop =cog —0

These equations with 11 variables were solved numerical-ly and the results for the energy and sublattice magneti-zation are plotted in Figs. 2 and 3 along with the data forregime (a) (e(1/8).

Similar equations can be written down for regime (c)(e) 1). However, we were unable to obtain solutions forthese equations numerically.

What do we learn from the results for regimes (a) and(b)? The energy graphs clearly show that the spiral phasewith Q=(47r/3, 0)/a is stable up to @=0.113 after whichthe spiral phase with Q=(0, —2m/&3)/a becomes lowerin energy. Since we do not have results for e ) 1,it is dificult to say how far the phase with Q=(0,—2m/&3)/a is stable. However, both the energy and themagnetization show bumps at @=1.0, which could indi-cate a phase transition to a new phase beyond a=1.0.We also note that we only found ordered ground statesand no disordered ground states (i.e. , states with zerocondensates) within our MFT.

V. DISCUSSION

We have used a three-boson representation for spins inorder to develop a Hartree-Fock MFT for frustratedHeisenberg AFM in two dimensions. At zero tempera-ture, the mean-field variational equations are satisfied byforming condensates of the bosons at the Goldstonepoints which implies LRO. Our numerical results for theunfrustrated models on square and triangular lattices arein agreement with the results obtained from othermethods. For the frustrated model on the triangularlattice, we have explicitly obtained the energies of the

ground state and the phase diagram for e ( 1.An interesting feature of this MFT is its connection to

a new spin-wave theory which we have introduced in thispaper. This new theory is applicable when the spin or-dering is fully three dimensional. Another interestingfeature of our formalism is that it can also be used to ad-dress the question of disordered ground states. For in-stance, in certain regions of the space of couplings (e.g. ,J2 =J, /2 on the square lattice), the question of whetherthe ground state is ordered or disordered even at T =0 isstill open. Within our formalism, this would depend onwhether the energy cok vanishes faster than quadraticallyas k approaches a Goldstone point. If that happens, theintegrals would diverge and the variational equationswould be satisfied, not by condensates, but by gaps in thespectrum. Such gaps could imply the absence of LROand, perhaps, spin-liquid states. This formalism can alsobe used to investigate the possibility of ground stateswhich break translational symmetry (e.g., dimerizedstates) by allowing the variational parameters to havedi6'erent periodicities.

However the 3BR has a disadvantage as well. The ana-lytic expressions for the gaps di6'er by an overall factor of3 from similar expressions derived from othermethods. ' This disagreement stems from the basicproblem that the 3BR overcounts the number of spin-wave modes by a factor of 3. Thus, all our numerical cal-culations require that we drop the extra factor of 3 byhand.

ACKNO%'I. KDGMENTS

We thank D. M. Gaitonde and H. R. Krishnamurthyfor useful discussions.

'Electronic address: chitra ~cts.iisc.ernet. in~Electronic address: sumathi ~ iopb. ernet. inElectronic address: diptiman@ cts.iisc.ernet. in

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