ELSEVIER

22 May 1995

Physics Letters A 201(1995) 247-251

PHYSICS LETTERS A

Magnetic states and spin-wave excitations in strongly frustrated 2D Heisenberg antiferromagnets

R.S. Gekht, A.M. Epikhin, V.I. Ponomarev

Kirensky Institute of Physics, Russian Academy of Sciences, Siberian Branch, Krasnoyarsk 660036, Russian Federation

Received 21 February 1995; accepted for publication 3 March 1995

Communicated by V.M Agranovich

Abstract

It is shown that frustrations caused by additional spins result in a greater variety of phases with commensurate and incommensurate periods as well as in the appearance of nonperiodic phases. It is established that a single-Q modulated phase is unstable against zero-point motions. As a result, a disordered state with zero magnetization at a site can exist in a wide range of values of the frustration parameters.

1. It is now well known that the effects of frustra- tions due to either competition of exchange interac- tions or the lattice geometry play a very important role in various magnetic systems (see, for example, Ref. [l], and references there). Experimental and theoretical investigations have shown that in many respects their properties are fundamentally different from the corresponding nonfrustrated systems. This difference is reflected primarily in the rich diversity of phases and phase transitions which is the result of the strong degeneracy and high sensitivity of the systems to different types of perturbing interactions.

Quantum effects in frustrated systems can alter the ground state decisively, essentially because the effective spin length can vanish completely [2]. For this reason, in low-dimensional systems where these effects are most pronounced, disordered states with zero site magnetization are possible in addition to the conventional ordered states (NCel structures, helix phases, etc. [3-51). Experimental evidence for the

existence of such states is presented in numerous studies [6-lo]. Chandra and Doucot were the first to

study this question theoretically [ll]. It has been established that in square frustrated lattices quantum fluctuations can destroy long-range order at zero temperatures even for systems with classical spins. (Similar results have been obtained for the two-di- mensional frustrated KagomC lattices [ 121.1

A great deal of attention has also been devoted in various studies to the character of the magnetic structure in compounds in which additional 3d-ele- ments are either embedded in the crystal lattice or replace some cations of the main elements [13-151. In the present paper the structures and properties of the states of such systems are investigated. It is shown that mixed phases of such materials induce an additional frustration channel. The result is that sepa- rate structures are strongly degenerate and unstable with respect to zero-point motions in a wide range of values of the parameters of two-dimensional mag-

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248 R.S. Gekht et al. /Physics Letters A 201 (1995) 247-251

netic systems. The problem considered here is de- scribed by the following Hamiltonian,

z= C CJnS, ’ si+a,, (1) i n

where J, (> 0) is the antiferromagnetic interaction

between the main magnetic ions along the edge of a square and J, and J, are interactions of either sign

along the diagonal of the square. The first constant

corresponds to interaction of the main spins and the second one corresponds to the interaction between the main and additional ions located at the center of

the square. For J, = 0 the additional spins are free, and the

main spins form a simple square lattice in which

frustrations are induced by the antiferromagnetic in-

teraction J, . For jz < 0.5 where j, = J/J,, the ground state for large S is the conventional NCel

ordered state with two antiparallel sublattices. Con- versely, for j, > 0.5 each of these sublattices decom- poses into two antiparallel sublattices forming a NCel state with four magnetic sublattices. For jz = 0.5 the

ground state is strongly degenerate: quantum fluctua-

tions destroy long-range order, and as a result the N6el structures can be separated from one another by an intermediate disordered phase [11,16,17].

For J, # 0 additional frustrations arise in the

system irrespective of the sign of the exchange con-

stant. For this reason it should be expected that such a system will have a richer phase diagram.

2. Let us consider the possible states in the large-s limit. The energy of the spatially nonuniform struc-

tures in the ground state is

E = NS2J( Q)/2, (2)

Fig. 1. Phase diagram of the ground state in the large-d limit: ICI:

incommensurate (0, Q) phase, IC,: incommensurate (Q, Q) phase.

where

J(k) = J,(cos k, + cos ky) + 25, cos k, cos k,

+ 4 J3 cos(+kx) cos( +ky )

(the lattice spacing is equal to unity). Analysis shows

that in the general case a frustrated Heisenberg sys- tem has five different phases whose wave vectors Q satisfy the condition V,J(Q> = 0. The energy and

the region of existence of these phases are given as

follows ( j, = J,/J,): (i) ferromagnetic phase (Q = 0):

E, = 2J,( 1 +j, + 2 j3)S2N,

j2 G 0, j,G -1;

j, 2 0, j3> -2j,-1, (3a)

(ii) antiferromagnetic phase (Q, = 0, Q, = 27r):

E, = 2J,( 1 + j, - 2 j3)S2N,

j2 G 0, j, > 0;

j2>0, j3>2j2+l, (Sb)

(iii) antiferromagnetic phase (Q, = QY = n):

E, = -2J,(l- j2)S2N,

jzGO, -lGj,<l;

0 Gj, < 0.5, 2j2 - 1 d j3 Q (1 - 4ji)1’2, (SC)

(iv) single-Q incommensurate phase IC, (Q, = 0,

cos(iQ,) = - j3/(l + 2 j2)):

E4= -2Jl( j2+ &)s’N.

O(j,<O.5, (1-4j~)1’2<j,<2j2+1;

j, > 0.5, -2j,-1<j,<2j2+1, (3d)

(v) double-Q incommensurate phase IC, (Q, =

QY = Q, cos Q = -(l + j,)/2j2):

E, = -J, $ [(I +jd2 - 4j2j3]S2NT 2

0 Gj, < 0.5, -2j,-1djf<2j2--1. (3e)

The phase diagram of different states is displayed in

Fig. 1. Note that the antiferromagnetic (n, rr) phase is not unique in its region of existence. Since the field acting on an additional spin at the center of the

R.S. Gekht et al. /Physics Letters A 201 (1995) 247-251 249

square is zero, the system is locally degenerate and has a set of nonperiodic states. We shall see below that the incommensurate phase IC, also is not unique in its region of existence.

3. We now investigate the energy spectrum of the states found. Let the magnetic structures of the

ground state lie in the xz plane. In a helix coordinate system the pair interaction Si - Sj is represented as

follows,

Si * s, = gcos Blj + 1)s; ’ s; + gcos Qij - 1)

x[+S;-2(.S;.k)(SJ*k)]

+sin oij k-($X5$), (4)

where oij = Q * (Ri - Rj> is the angle of rotation of

the helix axis between the sites i and j, and k is here a unit vector along the y axis. The starting Hamiltonian in the leading order in l/S has the

form

where Nd is the number of particles with spin S in

the magnetic subsystem; the indices cr and p in the Bose operators u,‘, ui and in Jms characterize the number of magnetic subsystems.

The simplest nonuniform collinear structure to study the energy spectrum is the (0, 27r) phase. In this phase the magnetic sublattices of the main spins

((Y, /3 = 1) and the secondary spins (a, /3 = 2) have an antiparallel orientation. The excitation spectrum obtained after diagonalizing the quadratic Hamil- tonian (5) is linear for the lower branch e,(k)

(= hw,(k)/J,S) near the points k = (0, 0) and k = (27r, 2n):

sl(k)=1 l- 2 ( y)‘“t_

As the interface between the initial phase and the

(6)

Fig. 2. Spectrum of spin-wave excitations: (a) &&k) for the

(0,27r) phase with j, = 1.5, k, = 0. Curve (1) j, = 0.15; curve

(2) j, = 0.24; (b), (c) &,(k) and &I for the (?r, T) phase with

jj = 0.5 and jz = 0.2 (solid lines); curves (1) k, = 7r/4; curves

(2) k, = a/16; curves (3) k, = 0; dashed line: jX = 0.5, jz = 0.4.

and k, = 0; (d) lower branch of the excitation spectrum of the

(0, Q) phase for jX = 1, jz = 0.7. k, = n/2 (1). r/6 (21, and 0

(3).

modulated phase IC, is approached additional soft- ening of the spectrum occurs: the spin-wave velocity

c _ ( jX - 2 j, - l)r”

decreases and vanishes on the critical line j3 = 2 j2 - 1 on which, as a result, the spectrum near small k

becomes quadratic (Fig. 2a). The spin contraction AS, = (a,’ (Y, > for the main and additional ions is

the same. As j, -+ 2 j, + 1 the quantum fluctuations

grow and diverge logarithmically in the asymptotic limit.

The spin-wave spectrum for the (n, ~1 structure is displayed in Figs. 2b, 2c. The curves c,(k) are presented as a function of k, for three values of k,: n/4, n/16, and 0. The lower branch cl(k) charac-

terizes the energy spectrum of the additional spins. For j, = 0 this branch vanishes, as it should. For j2 # 0, however, the entire branch can vanish if k, or

k, is zero (curve 3, Fig. 2b). It is obvious that this fact reflects the local degeneracy of the classical ground state with respect to continuous rotations of the additional spins. The upper branch s2(k), how- ever, characterizes the energy spectrum of the main spins and vanishes only at symmetric points of the Brillouin zone k = 0, k = (r, T>. Nevertheless, due to frustrations additional softening of the spectrum at

250 R.S. Gekht et al./Physics Letters A 201 (1995) 247-251

the points k = (n, 0) (dashed line in Fig. 2c) and k = (0, 7~) is possible as j, increases. But if j, + 0 holds the spectrum c*(k) does not vanish due to a first-order phase transition to the neighboring phase.

In accordance with this behavior of the dispersion curve, the spin deviations of the main and additional ions at the lattice sites are significantly different. Calculations show that, as expected, the magnetic subsystem of the additional spins is completely dis- ordered as a result of local degeneracy. At the same time, the magnetic subsystem consisting of the main spins is ordered everywhere except at the point (j,, j2) = (0, 0.5) in the phase diagram in Fig. 1 where like the N&e1 (T, 7~) state [ll] quantum fluc- tuations have a logarithmic divergence.

Let us consider now the spin-wave excitations of the single-Q incommensurate phase IC,. The energy spectrum for this phase with wave vector Q = (0, Q) is presented as follows,

&,(k)=[f(~ri9~)1’2]1’2.

Here

(7)

&=A2,-C;+4[(B;)‘-(8;)‘] +D2,

vk = (A; - C; - D2)’

+8(AZ,-C:+D2)[(B;)2-(B:)2]

+16D[Ak(B;)2-2B;B;C,+A,(B;)2],

(8)

where

A, = 2[2 j, + cos k,

+ S*( 1-t 2 j, cos k,) cos ky] ,

Bc = - j3( S + 1) cos( $kX) cos( $ky ),

C,=2(a2-1)(1+2j, cos k,) cos k,,

D =2j,S, S= j3/(1 + 2j,).

Analysis of the dispersion relation cl(k) = c_(k)

shows that for k, = 0 the expression for & in (8) is equal to 7):” and thus the energy of the spin waves is zero for all values of k,. Fig. 2d shows the lower branch of El(k) as a function of k, for different values of k,. One can see that as k, decreases the gapless spectrum cl(k) is further softened near k, =

_--_d-------

b

s-1

Fig. 3. (a) Local degeneracy of the (0, Q> phase along the y axis

for Q = 4~/3. The arrows mark the direction of simultaneous

rotation around the local axes. (b) Phase diagram in the S-’ -j,

plane for j, = 0.625. The central peak ( j3 = 0) corresponds to the

(0, n) state and the antiferromagnetic phase corresponds to the

(0, 2~) state.

0. As k, decreases further the region of mode soften- ing becomes larger and in the limit k, = 0 it covers the entire range of k,.

This circumstance is due to the fact that the classical ground state (0, Q) is strongly degenerate because of frustrations. As a result both globally and locally degenerate classical states are possible. Fig. 3a shows as an example a state with j, = 1 and j, = 0.5: the initial 120-degree structure (Q = 47r/3) is locally degenerate along the y axis, i.e., along the axis for which the dispersion branch E,(k) vanishes for all values of k,. The spins in the layer between the dashed lines can be simultaneously turned con- tinuously around axes parallel to these lines without a change in the energy. For this reason classical states which are nonperiodic along the y axis are possible together with periodic states.

Such degeneracy results in divergence of quantum fluctuations in the entire range of existence of the incommensurate phase and local degeneracy disap- pears only when Q = rr, i.e., when the main spins form a simple square lattice with an antiparallel structure in which, naturally, there is no local degen- eracy along the y axis because this structure is

R.S. Gekht ef al. /Physics Letters A 201 (1995) 247-251 251

collinear. As it should be, the (0, T) state has long- range order. The phase diagram in the S-‘-j, plane is shown in Fig. 3b: the NCel (0, T) state is sepa-

rated from the ferromagnetic and antiferromagnetic (0, 2~) states by a phase with zero average spin at a site. Calculations carried out by the Schwinger boson method [l&19] show that in the disordered phase the

spectrum of spin waves has a gap and short-range order rather than long-range order appears.

We have also examined spin-wave excitations of

the double-Q incommensurate phase IC, (Q, = QY = Q). Analysis of the behavior of the spectrum

g,(k) shows that for k, = 0 (or for k, = 0) the lower

branch exhibits weak dispersion near the interface of

the <Q, Q> and ( r, T) phases. This occurs because

the transition between these two phases is continuous and the system, just as in the (T, 7) phase, is close to local degeneracy as before. Far from the interface, however, it becomes rigid, the energy of the wave

excitations increases and for this reason the quantum fluctuations cannot destroy long-range order in the

(Q, Q> phase.

Note that in the classical ground state the (Q, Q>

phase is globally degenerate: separate magnetic sub- lattices can be turned continuously relative to other sublattices. However, quantum fluctuatons remove the continuous degeneracy. At the comer point of the phase diagram in Fig. 1 (j, = - 1, j, = 0) states with any Q are possible. Such states are locally

degenerate as a result of which the quantum fluctua- tions diverge at this point.

In conclusion, we emphasize that in the case of a strongly frustrated system considered here a non- magnetic (spin-liquid type) state may be realized in a

wide range of the parameters of the system due to a local degeneracy of the classical state, in contrast to Ref. [ 111, where instead there is only a global degen-

eracy.

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