quantum frustrations in quasi-2d antiferromagnets

Z. Phys. B Condensed Matter 8I, 433~440 (1990) Condensed

Zeitschrift M a t t e r for Physik B �9 Springer-Verlag 1990

Q u a n t u m frustrations in quas i -2D ant i ferromagnets

D.N. Aristov and S.V. Maleyev

Leningrad Nuclear Physics Institute, Gatchina 188350, USSR

Received May 30, 1990

An influence of localized holes on magnetic properties of the CuO2 planes in quasi-2D antiferromagnets (La2CuO,, YBazCu3Q+x etc.) is studied. It is demon- strated that the hole localized on the bond connecting Cu 2+ ions gives birth to frustration which in a wide range of parameters behaves as the two-level system with the energy E< J, where J is the exchange interaction of the neighbouring copper spins. As a result for T < E the spin-wave energy may noticeably decrease even at a low concentration of the holes c if the parameter cJ/I E[ is not small. When this parameter is greater than unity, the reentrant transition may occur in the system. Experi- mental data presently available are in a qualitative agreement with the theoretical predictions.

1. Introduction

As it is well known, systems La2_xSrxCuO4 and YBa2CuaO6+x, being at large enough x high-temperature superconductors, with the decrease of x become quasi-two-dimensional Heisenberg antiferromagnets. Up to now a lot of experimental results regarding properties of such antiferromagnets are accumulated (see [14 ] and references therein). Apparently, one of the brightest phe- nomena is fast destruction of antiferromagnetism with the increase of x. For instance, in the system La2_xSrxCuO4 antiferromagnetic order disappears already at x ~-0.02. In [-5, 6] this phenomenon is discussed in a classical way. Here we consider a quantum mechani- cal problem, using essentially the same model.

The studied systems contain CuO2 planes with Cu 2 + ions having spin S = 1/2. The superexchange in-plane interaction of the neighbouring spins is very large (J,-~100 meV [-1, 4]), while the long-range magnetic order at T# 0 exists due to the weak interplane exchange J• According to [-1, 4], J • for La2CuO4 and YBazCu306+ x. Below the temperatures T~ TN, where TN is the Neel temperature, are of our interest. In such a case the presence of an interplane

interaction is essential only for the excitations with the energy

E < E3D = 2(J J• 1/2 <= 10 K (1)

This criterion is derived below. We deal with the energies and the temperatures greater than this value and therefore restrict ourselves with the consideration of two-di, mensional antiferromagnetic planes, forgetting that the antiferromagnetic order is maintained only by means of the interplane exchange.

Assuming only the nearest neighbour interaction let us write the exchange Hamiltonian as

Ho = Z S, Sm (2) <ira)

Here spins Sz, m, arranged in a plane square lattice with parameter a, form two sublattices labelled by indices l and m respectively.

According to Emery's model [7] the holes appear on some oxygen ions 0 2- in the CuO2 planes for x ~ 0, i.e. O 2- transforms into O 1-. For small x the holes are localized in the middle of the bonds connecting adjacent Cu z + ions [-1, 5]. This localization seems to be produced by a spatial disorder originating in the system under doping. As a result the interaction of the copper spins S 1 and S~ nearest to the hole is replaced [-5, 6] by

V-.~JNNS 1 S 2 - W S 1 $2 -~- J s (S 1 S 2 ) s (3)

where the second them describes the variation of the exchange of the neighbouring Cu 2 + spins caused by the hole, and the last term describes the exchange interaction of the copper spins with the spin of the hole s. Some arguments from [5] imply J~ >> J, and, besides, it is shown there that in the classical limit the last term in (2) gener- ates ferromagnetic interaction between $1 and Sz. How- ever at present we do not know the real values of W and J~, but there are all grounds to assume that at least one of these energies is comparable with J. To under- stand qualitatively the possible role of the interaction

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(3), let us assume it to be large in comparison with the interaction of spins S t and $2 with the other spins of the system. Then it determines the spectrum of the complex of three spins $1, $2 and s. The eigenvalues of the Hamiltonian (3) are classified by the total spin I and by the spin S of the pair of the spins St and Sz, and are given by

VQ (3/2,1) = (1/4)(J- W ) - Jj2

Vm (1/2,1) = (1/4)(J- W) + 4 (4)

VDO (1/2, 0) ---- (3/4)(W-- J)

Thus, we have three groups of levels: quadruplet Q with S = 1 and I = 3/2, doublet D1 with S = 1 and I = 1/2 and doublet DO with S = 0 and I = 1/2.

We assume the energies separating levels (4) to be so large that only the lowest state is of interest. So in this paper, as in [5], it is supposed that the interaction (3) leads to the triplet coupling of spins St and $2. Hence, frustration appears in the antiferromagnetic matrix. Later on such a bond is called the frustrated bond (FB).

The spin direction in the sublattices determines the uniaxial anisotropy in the spin system. Taking into account as perturbation the interaction of spins St and $2 with other spins of the matrix we obtain the additional anisotropic interaction

H a = E 1 S 2 (5)

where S = St + $2. This interaction splits the quadruplet levels into sublevels with I~= _+ 3/2 and I== _+ 1/2. At the same time, it does not split doublets. Therefore if the quadruplet is the ground state of the three-spin system, then the two-level system (TLS) appears. Further we show that S~=I for I~= +3/2 and S 2=1/3 for I~ = + 1/2. Thus the distance between the levels of this TLS is equal to 2E1/3. If 4=0, the spin of the hole does not interact with St and $2, so these two spins form degenerate triplet and singlet. So the interaction (5) splits the triplet into sublevels with S~ = + 1 and Sz = 0 separated by the energy Et, and, if J - W< 0, then only these two sublevels participate in thermal motion. How- ever, if d - W> 0, then this pair of spins escapes thermal motion, as if vacancies appear in the sites 1 and 2. Note that the problem of one and two static vacancies in two- dimensional antiferromagnets has recently been considered in [9].

Thus we see the appearance of the TLS within a certain range of parameters of the interaction (3) in the system of three spins St, $2 and s. We deal below with the problem of spin-waves interacting with such TLS and show that the energy Et is small. In particular, we demonstrate that if E~ ~ J, then even a small concentration c of the localized holes destroys the antiferromagnetic state. Concluding this section we note again that when the doublet forms the ground state of the 3-spin system, the interaction with the matrix does not remove degeneracy. In. this case quite a different problem of spin-wave interaction with the degenerate center arises. We come to this degenerate problem, in particular, when W=0 and J~ ~ d. The behavior of a degenerate center in the field of 3D-phonons was treated in [10] where it was

shown that in 2D case the problem was more complicated. We suppose to perform later a detailed study of this problem.

2. Frustrated bond in the linear theory of spin-waves

In this section we consider the interaction of the localized hole with spin-waves by means of perturbation theory�9 The last term in (3) in the lowest order of perturbation gives a zero contribution, and in the second order it renorms W. Therefore, the Hamiltonian of the system may be written in the form

H = H o - WS1 S2 (6)

where the Hamiltonian Ho is given by (2). Using linear approximation for spin-waves, we represent the spin operators as

Sf = - s + a { al Sm =s-b+~ bm

S~ + = (St-) + = ] /~az + S + = ($2,) + = ~ s b m (7)

where S+ = Sx +_ iSr, az and bm are Bose-operators while indices 1 and m enumerate spins in sublattice. Using this representation, we rewrite (6) as follows

H = s Z [30 (a; a k + b; bk) + ~k (a; b +_ k "q- ak b_ k) ] k

Ws N • (a~, eikllq-}-b_k, e ik'llz) kl,k2

�9 (ak2 e -ik21q +b_+k2 e -ikER2) (8)

where ~k = SS a Ja exp(ik6), c5 is the vector connecting the spin with its neighbours, J~ is the corresponding exchange integral, N is the number of spins in a sublattice, and R1 and R2 are coordinates of the spins $1 and $2.

We introduce four retarded Green functions, which are conveniently combined in a matrix

Gkkl(t) . . . . [([ak(t),a+l]), ( [ak( t ) ,b-k l] )

/([b-+k(t), a~-~] ), (l-b-+k(t), b-kt] )] (9)

With the help of the Hamiltonian (8) for this function in co representation we obtain

0 '~0 " G k ' k = ~ k ' k - - N - g k ' Z(Uk'k, Gk'k) (10)

Uk = (exp(ikR1), exp(ikR2))

Its solution has the form

Gkk' = GO ~kk ' - -2~ ( GO Uk) (Uk' GO') (11)

where

) 6~ --CO+ 0

Ao (co) = 2 Ws(1 + ~ r s ~(~o -- ~k COS (k a)) (co 2 -- 22) - 1) I 1

k (12)

Here ~k=(~2-42) 1/2 is the spin-wave energy and a is the vector connecting the neighbouring spins. Using the expression for ~k in the case of the weak interaction between the layers, we find

~ = ~2 { 1 -- cos 2 [ (k:~ + ky) a/Z] cos 2 [ (kx - Icy) a/Zl }

q-(J• --cos(k• d)) V 2 k 2 + ~2 (jz/j)(1 - cos (k• d)) (13)

where 4o=4SJ, d is the distance between the layers and V= 4oa" 2-1/2 is the spin-wave velocity along the plane. To derive (13), we took into account that the interaction between planes was essential only for small k a. The condition (1) follows from (13) where s = 1/2. The poles of the function Ao(co) entering (11) determine excitation spectrum near the FB. It is convenient to transform this function to the following form

D(co)= 1+(co2 2/N) k

(14)

where 2 =(2sW/~o) (1-2sW/~o)-1. Using (13) for ek, one finds

D(co) = i + 2 c o ( z c ~ + i)) - I K((I -~z) -I/2)

1 + (2/2~z) z- 1/2 ln(16/z)

where z = (co2/~o2)-1 and K is the complete elliptic integral; the approximate equality in the right-hand side holds in the z ~ 0 limit. From (15) for the small negative W we have a local level with the energy

coo = ~o [1 + (22/2~r 2) ln2 (2~/2)3 (16)

This expression is obtained with the logarithmic accura- cy. Under the change of the sign of Wthe level becomes quasilocal one. Corresponding energy is still determined by (16) and the level damping is 7o=~o(22/~)ln(8;c/2). With the increasing of Wthe quasilocal level energy de- creases, and for 2 >> 1 we have

coo = ~o(~/2 In 2)1/2; 70 = rcco2 sgn(co)/(2 coo In 2) (17)

Thus for the large 2 the energy c~ o is proportional to 2-a/2. As a result the spin deviation at the ends of the FB become large. For instance, for the site 1 deviation is

(a~ a l ) = - - r c -1 ~ dxN(x) I m G l l ( x ) (18) -oo

Using (11)-(14) for Gl l we find:

G ll (co) = 1~1~ (co) -- 2 ~o (2 D) -1 F 2 (co)

C~ (co) = y,(co + r - k

Fl(co)=N -1 ~eik' '(co+~0--~keiklh~)(co2--e2)-l. (19) k

The substitution of (19) into (18) at T = 0 when coo~4o gives

(a;- a l ) = N -1 2 ( 4 0 - 8k)(28k) -1-1-IF1(0) ~o/212 (2coo/4o) k

(20)

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Here the first term is the zero-point contribution into the spin deviation. For the square lattice and the nearest neighbour interaction it equals to 0.20 [9]. The second term, according to (17) diverges as (2/ln 2) 1/2 in the 2 ~ Go limit. But, in any case, it must be hold the inequality (a~ a l ) < 1 (we take s = 1/2). Therefore, at 2 ~ 1, i.e. for W ~ o = 2 J , the linear spin-wave theory becomes inapplicable.

As a result we obtain the following picture for the quasilocal excitation near the FB. For W~ ~0 one has the quasilocal harmonic excitation with the energy coo -~ ~o. Then with the increasing of Wat W< 4o the problem becomes strongly nonlinear, and for ~ ~o, instead of the quasilocal oscillation, we have the TLS with the energy E ~ ~o, as it has been explained in Sect. 1.

3. Spin-waves in the presence of the two-level systems

Since strong nonlinearity reveals itself only in the vicinity of the FB, at the large distance from it the coordinate dependence of the Green function has to be the same for both the quasilocal level and the TLS. (We demonstrate it below in a more accurate way). Hence, for small k (11) remains to be valid with the factor Ao(co) replaced by the function A (co) describing the TLS. The situation, however, differs from the usual one (see, for instance, [111). Indeed, in the case of the conventional TLS two nondegenerate levels exist. Now we have two possibilities which lead to different results. If the TLS was born by the splitting of the quadruplet levels, then both levels, with lz = -t- 3/2 and Iz = _+ 1/2, have the same degree of degeneracy. Since spin-waves generate transitions with AIz = _+ 1 only, the factor A (co) has the conventional from [11]. But if the TLS appears as a result of the splitting of the triplet then the degrees of degeneracy of the levels with Sz = + 1 and Sz = 0 are different. In both cases (see the next section) the factor A (co) may be written in the form

A(co) = _g2 ~2(E_i7 )

�9 [co• - ( E - i ? ) 2] -1 tanh (E/2 T) r(E/T) (21)

where g • 1, and factor r(E/T) takes into account rela- tive degeneracy of the levels. For the quadruplet r (x)= 1, and for the triplet

rdx ) = [1 + e x p ( - x)]/[1 + 2 e x p ( - x ) ] (22)

The factor E - i T in the nominator of (21) provides the spin deviation to be finite even in the E --+ 0 limit.

Let us discuss now the problem of the damping 7. It is known [111 that in 3D-case for interaction with phonons ?~co3 [2N(E )+ 1]. Spin-waves are excitations with the linear dispersion law, similar to phonons. There- fore in 3D-case 7~co 3, while in 2D-ease 7,-~co 2 sgn(co), where sgn(co) is the sign function. To determine temperature dependence of 7, one must consider again the rela- tive degree of degeneracy of the TLS levels. Actually, factor 2 N + 1 originates from the addition of probabili-

436

ties of absorption and emission of a quantum (phonon, spin-wave). Consequently, if the TLS is of the quadruplet nature, then again 7 ~ 2 N + 1, but in the triplet case the situation is more complicated. Namely, for E > 0 the probability of emission is proportional to 2 [N(E) + 1] and the absorption probability proportional to N(E) so 7 ~ 3 N ( E ) + 2 . If E < 0 , then ~ 3 N ( E ) + I = - [ 3 N ( I E I ) + 2]. Thus, combining both cases, we obtain:

7(co) = f 2 ((2)2/40) sgn(co) coth(E/2 r ) r - 1(_ E/T), (23)

where f2 < 1 determines the interaction of the TLS with the spin-waves. Note that the dependence of this expression on (2) is the same as in (17) obtained in the linear theory.

Consider now small concentration c of the FB. Using the standard method [12] in the linear approximation, instead of (11) we obtain

G k = G ~ (1 - c) - (cA/2)(G ~ Uo + )(Uo G ~ (24)

Here the factor (1 -c ) in the first term reflects the fact that the spins at the ends of the FB do not participate in the spin-wave motion after the TLS formation. The solution of (24) has the form:

G l l ((2)) = G22 ( - (2)) = Z - 1. [((2) + 40)( 1 _ c ) - cA/2]

G a 2 ((2))= G21 ((2)) = Z - 1. [ _ ~k (1 -- C)+ cA~2]

z = o~ ~ - ~ + C(4o- 4k)A (25)

The combinations of G,r of physical interest are those describing the response of the system to an external action. In agreement with the general rule [121 this response is determined by the retarded Green function of the operator

S, = (2 N)-1/2 Z Sj exp (i q. R j) (26) J

where j enumerates all the spins of the system and q is the wave-vector of the action (scattering vector for neutrons). This Green function may be represented in the form:

(]q (t) = - iO ( t ) ( [S~ (t), S~] } = 1 / 2 ~ 8 (q + v , . - k)[G~.~l 1 Cm

+ G~Pz2 + cos (z,, a)(G~l 2 + G~,~21)] (27)

where 1 and 2 label two magnetic sublattices, zm is the reciprocal magnetic lattice vector and k belongs to the first Brillouin zone of magnetic lattice. The value cos(zm a) equals unity for r , coinciding with the reciprocal vectors of the chemical lattice and equal ( - 1) other- wise. Thus the Green functions of real interest are as follows

G~ ~ = 1/2[G]Cx + --,22 n- -,12 + G ~ ] ~ ~ r - G ~ ] G] ~ = 1/2[G~ +-- ,22- ,o12 (28)

Here G y describes the reaction to external magnetic field, and G a describes inelastic neutron scattering near magnetic Bragg reflections. We study the spin-wave fluctua-

tions which are transverse excitations. Since external field is zero, we have

Gxx = Gyy = 1/2G_ + = 1/2G+ _ = G•

Using the definitions (7) and (9), from (25) we derive

G { , k = (40 -- 4k)(1 -- C) Z - 1

Gi.k = (40 + 4k)[1 --C-- cA~(2 4o)3 Z-1

Z = (2)2 _ e~[1 - cA~(2 40)] (29)

The function G{o(0) is related to transverse antiferromagnetic susceptibility by the equality Z• - - S ( g p ) 2 N O G{o(0), where No is the magnetic atom den- sity. At c = 0 this susceptibility has its usual value )~o =(gp)2 N0/(8 4o). However for c # 0 the susceptibility )~• depends on the temperature and is given by

Z• = X ~ (1 - c). [1 -(C/Co) r (E/T) tanh(E/2 T)] -1 (30)

where Co=2E/(g24o). Later on we demonstrate that [Col~l and a strong dependence of )~• on T for T < E arises already in the region of small FB concentration. If leo] < c ~ i, the susceptibility becomes negative for T < T~, determined by the equality

(C/Co) r (E/T~) tanh(E/2 T~)= 1 (31)

Thus at T= T~ one has a reentrant transition from the antiferromagnetic state. This result is obtained in the lowest order with respect to the FB concentration. It is obvious that at T ~ T~ this approximation is inapplicable, and further theoretical investigations are required. The zeros of the denominator Z in (29) determine the spin-wave spectrum in antiferromagnets with the FB. It is dear that even at small c in the frequency range 1(2)1 ~-E the value cA/2 4o becomes large and our approximation is invalid. We do not further pay attention to this energy region and deal with low frequencies only, when ek~E. In this case, taking into account (23) for the renormalized spin-wave velocity and the damping one obtains

V 2 = V 2 [1 -(C/Co) r (E/T) tanh (E/2 T)]

rk = ce~ 1/1 r(E/T)[2]Co El V r ( - E/T) ] -1 (32)

The T-dependence of 1/12 is identical to that in the denominator of (30) for the susceptibility. Therefore it is not necessary to discuss its properties. Damping Fk for k ~ 0 is proportional to k 3, i.e. it has the same k-dependence as the scattering by static inhomogeneities in the 2D-case. In real antiferromagnets there always exists magnetic anisotropy and the gaps in the spin-wave spectrum determined by it. Both in L a z C u O 4 and in YBa2Cu306+x is has been experimentally observed El, 4] that spin-waves polarized in the CuO4 plane and in the perpendicular direction had different gaps, which reveals the biaxial anisotropy. We analyze here only the uniaxial one. In this case the additional term - 6 J Z ST S~, to Hamiltonian (2) appears as well as similar term in the FB energy. This last term gives the small correction to the energy levels (6) and may be neglected below.

As a result the value 40 in (8) and all subsequent expres- sions must be replaced by ~0+A2/240, where A = 4s(J 6J)t/2~ 4o. In particular, the spin-wave energy is given now by the expression 2 2 2 1/2 g k : [ ~ O - - 4 k - } - A ] . SO from (29) we see that A 2 is renormalized by the same way as V2:

A ~ = A 2 [ 1 - (C/Co) r (E/T) tanh (E/2 T) ] (33)

In [1] it was observed the strong softening of the spectrum of scattered neutrons with the scattering vector q - 0 at T<30 K in a sample of La2CuO4. The Neel temperature of this sample was comparatively small, (TN = 195 K) and, hence, the FB concentration was not very low. In [-151 this phenomenon was assumed to be related to the decrease of the gap, while in [11 to the reentrant behavior. In our theory the decrease,of the gap is one of the characteristic features of the reentrant behavior. However, to clarify this question completely one needs more detailed experiments.

In conclusion of this section let us present the expression for the sublattice magnetization in the presence of the FB. Using the expression similar to (18), after some calculations we find

KS=)I = 0 . 3 ( 1 - c ) + ~ { 1 - [ 1

c r(E/T)tanh(E/2T)]l!2~ Ic01 1 J

2 T V 2 rc~o V 2 ln(T/E3

(34)

where Ec=max[A, E3o] is the cutoff energy and E3D is given by (1). It should be noted here that the c-depen- dent contribution to sublattice magnetization does not contain the large factor 1~Co, therefore the influence of the FB on (S ~5 is not so prominent as in the case of Z• and ~k-

4. Frustrated bond in the field of spin-waves

In the previous section we considered phenomenologi- cally the TLS and analyzed their influence on the spin- wave spectrum. Here we investigate in more details the spin-wave interaction with the TLS and obtain, in particular, the expression for its energy. We assume that two spins at the ends of the FB are bound into the quadruplet or the triplet by the interaction (3) and consider the problem of the interaction of this complex with spin-waves. The Hamiltonian of the system may be written as H = H o + Hin t , where Ho is the usual Hamiltonian (2) con- taining, however, two vacancies at the FB ends and Hin t

is the exchange interaction of the FB's spins $1 and $2 with their neighbours,

H~., = (S1 + S2) b + + (S, - S2)h_

he =(J/2)[(Sll +Sl2 +S13)---(Sml --~-Sm2 q-Sin3) 1 (35)

where St, and S ~ are the nearest neighbours of the spins St and $2 belonging to the sublattices I and m respective-

437

ly. This interaction splits the levels of the quadruplet (or the triplet if J~= 0) thus creating the TLS and then causes transitions between the levels of this TLS.

Operators h_+ are still related to the spin-deviations by (7), but the spin-wave Hamiltonian H0 is not of the form (8) now, due to the presence of the vacancies. As it was already noted, the problem concerning vacancies had been considered in [9]. It was shown there that the zero-point deviations for the spins next to the vacancy were somewhat smaller than those in the perfect lattice. This result can be easily understood. Indeed, the vacant sites are inaccessible for spin-waves. Therefore if the vacancies form a large domain, the spin-wave wave-function is zero at its boundary. Then for the domain consisting of a few vacancies the wave-function magnitude must decrease near its boundary. Moreover, no bound states can appear near the vacancies and the spin-wave spectrum is the same as in the case of the perfect lattice. Therefore the wave function of spin-wave can be written in the form:

@k, lm : exp (ikRt,m) + (pk (Rt,m) (36)

where the first term is the incident plane wave and the second is the scattered one. For the sites next to the vacancy we have Jq~k,t,~J < 1. Spin-waves are similar to phonons. This means that in the 2D-case if k ~ 0, the cross section for the scattering by a vacancy is proportional to k 3 and therefore we have ~Oo,tm-0. In other words, for small k and all Rt,~ the second term in (36) should be small. We do not solve the cumbersome problem of the calculation of ~k,e,, for a pair vacancy here and use later only its properties mentioned above. As it is known, the Hamiltonian (8) is diagonalized by a linear transformation. Its generalization in the presence of vacancies has the form

a t = N - 1/2 Sk(U k i]/k,l Ck _~ Uk @~k,l d-+k) b m = N - 1/2 Sk (U k ~/k, m dk -t-/)k ~/* k,m C _+ k) (3 7)

where Ck and dk are new boson operators and Uk and Vk are the usual coefficients of the canonical transformation: Uk = [(40 + ~k)/(2ek)] 1/2 and Vk = -- [(40 - ~0 / (2 e k ) ] t a .

We also introduce the wave functions of the quadruplet ~P~) and *he two doublets g*~) where S = 0, 1 is the total spin of two copper ions. These functions may be written as

~ ( Q ) - - ~ • 1 ~ • " • 3/2 - - 1/2 ,

~(Q)_+ 1/2 ---- 3-1/2(q~+1_ q~ r- 1/2 +2t/2 #o q~+ 1/2) I//(1) _ _ ] _ 3 - 1 / 2 ( ~ o q ~ _ + 1 / 2 _ 2 1 / 2 ~ + l~bT.1/2) ; _+1/2-- - -

~o)= 0o ~. (38)

where ~M and Oo are the triplet and singlet wave functions of the copper spins and ~b u is the spin-wave function of the hole.

Using (35) and (38) we obtain in the second order of perturbation theory an expression for the splitting

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of energy E1 in (5). It has the form

E1 = N -1 Y-~{2/3 [h+,kl2 [ek -1 --(4ek + 6 J~) -1]

- 1/21h_+,kl2(ek+ W--J+JJ2) -1 } - N - 2 Z {1/3 " 2 ]h+,k,+~.l

k l , k 2

�9 [ ( S k ' .~_ 8 k a ) - 1 __ (g 'k , "~- 8k2 "~ 3 J J2)- 1]

--IhZ-,k,+kzl2(ek~-t-ek2-F W--J + Js/2) -1 } (39)

Here + z h• and h• +k2 are matrix elements of h• +'~ corresponding to the emission of one and two spin-waves, calculated using (7) and (37). The value of E1 depends on the two differences of the energies (4): VD 1 - Va = 3 J J2 and V o o - V Q = W - J + J J 2 . The two-magnon term in (39) is numerically small and the upper boundary for E~ we obtain when both these differences go to infinity. In this case E l > 0 . Since h ~ o = 0 while h%o~0 , in the W - J + J ~ / 2 ~ O limit, the energy E 1 tends to - ~ as In [ ( W - J + JJ2)/J]. This means that transitions mixing the triplet (S--1) and the singlet (S--0) become essential, and the classification of the levels given in the Introduc- tion is now meaningless. Moreover, if in this case we replace in (35) J by V~ J, perturbation theory for evalua- tion of E 1 remains inapplicable. Therefore the interaction between the spin-waves becomes strong. In Fig. 1 the function E(W) for the case J~=0 is plotted. It was calculated in one-spin-wave approximation using (39) and plane-wave part of the functions (36). We see that E ( W ) ~ oo when W ~ o / 2 = J . In Fig. 1 it is also shown the W-dependence of the quasi-local energy coo(W) discussed in Sect�9 2. It was shown there that the strong interactions of spin-waves near the FB took place, when W~=~o/2=J. Therefore in the region where W ~ J we have the crossover from the quasilocal level to the TLS. The description of this crossover is rather complicated problem not intended to be solved in this paper. One of possible versions of dependence of the energy on W

2,00 L

1 " i, 00 ~ "i" ""-,,\\

! ' - . o , o o /F . . . . . . . . . . ~ - = - - - ~ . . . . . . . _~- - -22~: : :~-~- - . . . . . . . . . . . . . . . . . . ~j

i y ~ - I k .--'- J I / I t I I / " I

,/ I / 1 r / 1 I /

. . . . . I I

, ~ . _ I i I : i i i i J I [

0.00 i.00 2.00 S.00 ~.00 5.00 6.00 7.00 "~1 , i J

Fig. 1. Dependence on W for Js = 0 of the TLS energy E, calculated in the one-spin-wave approximation and the quasilocal level energy O)o. One of the possible dependence of the level energy on W in the exact theory is shown by the dotted line; its segment to the left of the point marked by a cross corresponds to the quasilocal level, and that to the right - to the TLS

is shown in Fig. 1 by a dotted line. A similar picture takes place for Js ~ 0 and IV-- 0. In this case the crossover takes place for Js,,~J also. Thus, in general, the TLS energy has to be considered as a parameter of the theory. Besides, one should expect that E ~ 4o.

As it was mentioned above, the expression for A depends on whether spin-waves split triplet or quadruplet�9 It seems to be natural that this difference survives all over the parameter range where the TLS exists�9 We demonstrate below that A is proport ional to the Green's function of the TLS which is determined by

oo

K_ + (co)= - i S d t exp( ico0([S_ (t), S+ (0)]) 0

(40)

where S =$1 "~- S2. In the case of the Hamiltonian (5) matrix elements

of operators S_ (t) and S+ (0) are easily found both for the triplet and the quadruplet (for the latter case one has to use the wave functions (38))�9 After some simple calculations one gets

K_ + (co) = 4 E tanh (E/2 T) rt (E/T)(co2 _ E 2)- 1;

E = E1 (triplet)

K _ + (co) = (4 E/3) tanh(E/ZT)(co 2 - E 2) - 1;

E = 2 E 1/3 (quadruplet) (41)

where rt(E/T) is determined by (22). We are interested in longwave spin-waves. In this case, as we already explained, the contribution of the scattered wave into (36) is small and therefore the interaction energy responsible for transitions between the TLS levels may be written a s

3 J r s ]1/2 eikR2) S + H i n t : ~ - [ ~ ] Z E ( a k e i k R l ~ - b - + k

k

"Jv ( a k - e - i k R 1 .~_ b _ k e - i k R 2 ) S - ] (42)

As a result taking into account that K_ + = K + _ , we obtain

A (co) = - 9/32 ~ K_ + (co) (43)

Comparing this expression with (21) for nonrenormalized coupling constant g2 we get: gt2=9/8=1.12 and g~-g2 /3=0 .38 . It should be stressed that due to the interaction with spin-waves these values of g2 may be noticeably renormalized, just like in the case of the interaction between the TLS and phonons considered in [13]. Obviously, this renormalization is especially strong, if W, J~~~o.

5. Discussion

Let us discuss here to what extent the above theory agrees with existing experimental data. But before doing this, one should clarify two points: first, that our theory is applicable only if c ~ l , if the difference R = I -c/[col.r(T=O) is not too small. The last condition,

439

Table 1. Susceptibility parameters for YBa2Cu306+x determined, using the data from [15] (ceramic sample), by fitting the expression Z=c~x• where X• is given by (30). The quantity Icol =21El(g2~o) - I was calculated using nonrenormalized ga values given in the text, 40=4000 K [4]; c was determined by using = c/]col ratio, obtained from the experimental data, and calculated I%1; R = l - - c / l c o l r ( Z = O ) . The susceptibility is given in units 10 -4 emu/mol

x ~Z ~ X1 E , K ~ Icol c R

Quadruplet

0.05 3.6 - 1.8 78 0.63 0.10 0.065 0.37 0.22 3.6 2.0 78 0.89 0.10 0.091 0.11

Triplet E > 0

0.05 1.95 0 115 0.74 0.051 0,038 0.26 0.22 1.95 0 120 0.91 0.053 0,048 0.09

Triplet E < 0

0.05 1.9 0 - 85 0.75 0.038 0.029 0.25 0.22 1.9 0 - 90 0.92 0.040 0.037 0.08

however, fails for the majority of experiments. Second, most experimental data available are either not accurate enough, or consist of results of measurements of the quantities hardly comparable with our theory. Therefore, the following comparison has a qualitative character.

We start with the analysis of susceptibility data. Both for LazCuO4 and for YBazCu306+~ an increase of susceptibility in antiferromagnetic phase with the decreasing of temperature was systematically observed (see, for instance, [14, 15]). The results of the most accurate measurements of susceptibility for Y B a z C u 3 0 6 +x for various x are presented in [-15]. Unfortunately, experiments were held with ceramic samples, what makes it difficult to discuss absolute values of the susceptibility. We tried to describe the results of this paper for two smallest values of x equal to 0.05 and 0.22 by means of the expression Z = c~ Z.c +)~ , where Z• is determined by (30). Corre- sponding results are shown in the Table 1.

From this table first of all one can see that for all the three versions of the theory (quadruplet and triplet with two signs of E) the energy E is almost independent on x. Then the calculated values of concentration c are of order of a few percents. According to modern concepts [5, 7, 8] additional oxygen atoms between the CuO2 planes serve as a source of holes. Therefore the localized hole concentration c is bound by the condition c < x/2. None of the calculated values of c for x = 0.05 presented in the Table 1 satisfies this criterion. The triplet with E <0 is the closest to it. However, we exploited nonrenormalized 2 gQ,t. Renormalization may increase these constants. Thus the discrepancy may be considered as not a very serious one.

In 1-4] it was stated that for x < 0.2 the hole concentration in CuOz planes was zero. This statement was based on the observed weak x-dependence of the sublattice magnetization. As it was shown in the end of Sect. 3 the magnetization depends on c much weaker than other physical quantities such as Z• V2 and A 2. Therefore the weak x-dependence of the magnetization is not, in

our opinion, an evidence against presence of the localized holes in CuO2 planes. Then in [4] it was estimated, using neutron scattering data for x = 0.37, that the hole concentration c ~ 0.02 destroyed AF order. This estimation was based on the assumption that interaction between magnetic defects might be neglected. We believe that it could hardly be true for such large x because we have seen that there was the strong renormalization of some physical quantities (see R-factor in Table 1) for much lower x, and as a result the interaction of the magnetic defects was to be important.

Comparatively small variation of c/Ico] with an increase of x by more than four times should not surprise, since additional oxygen ions produce not only holes in the CuO2 planes, but also ions Cu 2+ with spin 1/2 in the basis plane. The latter gives an addition contribution to the susceptibility which may also depend on T. How- ever, at least for x = 0.05, it is small. In the case of the triplet, background susceptibility ZI =0, while for the quadruplet Xl <0. It is quite unclear how this negative value of )~1 could be explained. Summarizing, one can state that our theory does not contradict the experimental data on the susceptibility of YBazCu3 O6+x, and that the triplet version of the theory with E > 0 seems to be most preferable. At the same time, the triplet version requires J~=0. But in [5] some arguments were provided in the favour of the large value of J~. Besides, in L a z C u O 4 it was experimentally discovered [16] that the hole spins interacted strongly with the copper spins, i.e. J~r

Below T= 100 K a strong enhancement of the magnetic diffuse scattering in Y B a 2 C u 3 0 6 + x was detected [3, 4]. Under the decrease of Tboth the spin-wave velocity and the gaps in the spin-wave spectrum are decreasing strongly and the scattering must increase. Unfortunately, experimental results were obtained only for x=0.3, where our theory can hardly be applicable. Note also that in [4] for x=0.37, i.e. near the boundary of the antiferromagnetic state the 2.5 time decreasing of the spin-wave velocity was observed. This result is also in a qualitative agreement with this theory.

In [1] spin-waves were studied in a sample L a z C u O 4 with Neel temperature Tu = 195 K using neutron scattering. It was shown at T= 80 K that the gaps in the spin- wave spectrum corresponding to oscillations both in the CuO2 plane and in a perpendicular direction equaled to 1 meV and 2.5 meV respectively. Besides that, a strong increase of quasielastic scattering intensity (transferred energy lower than 1 meV) for T<30 K was observed. This phenomenon was associated in [17] with possible reduction of the spin-wave gap down to energies smaller than 1 meV. However in [1] this phenomenon was explained qualitatively as a consequence of the frustration. From our theory it follows that it is the frustrations that lead to the gap reduction with the decreasing of T. Unfortunately, the results of [1] are not sufficient even for rough estimates of E and c.

In conclusion we would like to stress that, in our opinion, presently available experimental facts are in a qualitative agreement with theoretical predictions given here. Further experimental investigations, are undoubt-

440

edly necessary for a quant i ta t ive verif ication of our theory.

We thank Y. Endoh and J. Rossat-Mignod for preprints of their papers.

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quantum frustrations in quasi-2d antiferromagnets

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