Solid State Communications, Vol. 87, No. 12, pp. 1141-1143, 1993. Printed in Great Britain.

0038-1098/93 $6.00 + .00 Pergamon Press Ltd

EXCHANGE RENORMALIZATION IN ONE DIMENSIONAL CLASSICAL ANTIFERROMAGNETS IN THE CONTINUUM LIMIT

A.S.T. Pires, M.E. Gouv6a and S.L. Menezes

Departamento de Fisica, ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, CEP 30161, MG, Brazil

(Received I 1 February 1993; in revised form 6 May 1993 by P. Burlet)

In this paper we study the effects of the anharmonic terms, usually neglected when a continuum limit is taken, to the exchange part of the continuum Hamiltonian for the classical easy-plane one dimensional antiferromagnet. This correction leads to a dependence with temperature of the soliton energy.

NONLINEAR excitations in one dimensional mag- netic systems as, for instance, the antiferromagnetic compound TMMC, have been studied by mapping the Hamiltonian

H=JESnSn+I-DESZSnZ+I+BE(SnY) 2 (l) n n n

into the continuum classical Hamiltonian [1]

jS2Jdz{ (o0' 2+ _l¢Oq)' 2 1 (00" 2 n=-y + - Y \ o t ]

cos 2 0 2 } + 7 \ Ot } + 26 sin 2 0 + 2b cos 2 0 sin 2 q~ (2)

where c = 2JS, 6 = D/J, b = B/J. In fact, in the study of soliton like excitations it is

usual to include in Hamiltonian (1) an external magnetic field applied perpendicular to the chain. However, our main concern here is the exchange term and the inclusion of such a field would not affect our results - for this reason, we will not include it in our equations. We note that, in the mapping of equation (1) into equation (2), the terms in 6 and b in equation (2) keep the full nonlinearity of the initial problem but the same does not hold true for the exchange term where anharmonic terms have been neglected. This can be easily seen as follows. For motion in the X Y plane the exchange part is given by JS 2 ~ cos (~b, - 4~,+ l). Expanding the cosine term and keeping the quadratic part, we obtain j S 2

2 E(~b" - ~b"+')2 + O(~b, - q ~ n + l ) 4. (3) ?t

* Present address: Los Alamos National Laboratory, Theoretical Division T-l l , MS-B262, Los Alamos, New Mexico, 87545, U.S.A.

In the continuum approximation, equation (3) becomes

J : f 2, .... 2 J kOz] (4)

the static part of the ~b component in equation (2), given by the 0 = 0 limit.

One way to include the effect of the anharmonic terms, i.e. the O(~b,-4~,+1) 4 and higher terms in equation (3), is to write, instead of that equation, the following expression

JS27 _ (5)

2 I f

and then calculate 7 in a self-consistent way. This procedure has been used in two dimensional magnets where the main interest was to study phase transi- tions [2-4].

Of course for the exact spin chain Hamiltonian the anharmonic terms have been treated properly in all the different theories existing in the literature [5-10], but since these theories are perturbative they cannot include non-linear excitations like solitons. On the other hand, in the continuum approximation [11-13] not all anharmonic terms are taken into account, as discussed above. As pointed out by Wright et al. [8], the magnon-mass renormalization for the exact spin chain differs substantially from the continuum model mass renormalization. It is not the aim of this paper to develop a spin wave theory for the antiferromagnetic chain, neither to treat quantum corrections to the classical continuum model because these topics have already been investigated [5-13]. What we intend to do here is to show that when we write the Hamiltonian in the continuum limit, we must use a correction -y for the exchange part of

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1 142

the 4~ component in order to include the anhar- monic terms that were neglected when the continuum limit was taken. Then it follows from our theory that, for instance, the Hamiltonian used by Mikeska [13] should be multiplied by the correction factor -y. In this sense, our work complements Mikeska's calculations.

To apply this technique to Hamiltonian (1) (with B = 0, for convenience) we will use the polar representation for the spin at site n

S, = S 1 - ~ S ) c o s ~ b . , S 1 - ~ S ) s i n 4 ~ , , S ] ,

(6)

where for the antiferromagnet q~, must be redefined according to

~b, + [1 - ( -1)" ]2 . (7)

For the sake of clarity, we have chosen to use a classical representation but the final result [i.e., equation (17)] also holds in the quantum approach. In this representation, Hamiltonian (1)becomes

- ~.) - ,~szsz+l] (8) (~.+1 COS

where )~= 1 - 6 . In order to obtain the long- wavelength limit of Hamiltonian (8) we proceed as follows: if we calculate the time derivative of ~b, through equation (8) we find (setting ,~ = 0 for simplicity)

~, ~ - 4 J S z. (9)

This is exactly the same result that we find con- sidering the quadratic Hamiltonian obtained by expanding Hamiltonian (8) into powers of (SZ/S) 2 and (q~n+l--(~n) 2 wi th no need to introduce any renormalizing factor. Otherwise, the time derivative of S z is related to the angle difference (q~,+l - q~n) and its analytic expression depends on which Hamiltonian is used; whether it is the one given by equation (8) or its quadratic expression. This means that in obtaining the correct long-wavelength limit of equation (8), we must be aware of the coefficient to be put on the ((~n+l --(~n) 2 term. The behavior of S~ with both Hamiltonians of interest leads to the conclusion that the renormalization factor can be determined by the calculus of SZ with Hamiltonian

ONE DIMENSIONAL CLASSICAL ANTIFERROMAGNETS Vol. 87, No. 12

(8). We find, in the reciprocal space, "Z °Z

(Sq S_q) = 4JS 2 T(1 - cos q)

x - ~ - ) ~ / 1 ( - ~ - ! ) 2 cos (q~,,+ 1 -

(10)

where T is the temperature. Following this reasoning, the quadratic form of Hamiltonian (8) is found to be

H0 = J T T ( ~ . + l - ~.12 + (S~) 2 + ~S~S;+ l ,

(111 where the renormalization factor 7 is given by

(12)

Using equation (9) we will obtain that the term in (S~) 2 in equation (11) leads to the time derivative term in Hamiltonian (2).

Hamiltonian (11) can be diagonalized using standard methods [14], introducing the following canonical transformation,

, r,+ cosql,,. ~bq - (2S)l/2 [3'(1 - cos q)J (a+ + a_q), (13)

S q : i L 1 + A c o s q J ( d ; - - a _ q ) , (14)

+ and a_q are the boson creation and where aq annihilation operators, respectively. We find

no = ZWq(a+aq +1), (15) q

where

OJq = 4JS [7(1 - cos q) (1 + A cos q)]l/z. (16)

Now for calculating ~/we approximate (...) by (.. ")0. In this approximation ~bn and S z are uncoupled variables, leading to

S~+l 2 - , , :

x exp [-½ ((~bn+ , - q~,))02]. (17)

Using equations (13) and (14) we obtain

1 1 1

x [(1 - cosk)(1 + )~cosk)] t/2 (18)

Vol. 87, No. 12 ONE DIMENSIONAL CLASSICAL ANTIFERROMAGNETS

io S~-- , /7(I - cos k) coth ( _ ~ ) (19)

V T ; . (20)

In the classical limit we have

7 = (1 - tl)e -t/7, (21)

where t = T/4JS 2 and

1 f ~ d k 1 I = ~ J-,, 1 + )~ cos k = x/1 -- )~2" (22)

For small t, which is the region where our theory should hold, we obtain 7 = I - t(I + 1).

The magnon energy given by equation (16) and using equation (22) agrees with the exact transfer matrix result [15] for the second moment for the dynamical z correlation function for Hamiltonian (l) (with B = 0).

If Es ° is the classical soliton energy for Hamil- tonian (2), then the renormalized energy is given by

Es = 7E °. (23)

For TMMC, J = 13K, S = 5/2, 6=0.009, and equation (23) becomes

Es = (1 - (0.026T)Es °. (24)

We notice here that, in spite of the small values for the anisotropy in TMMC, it is a well known fact that for low temperatures the spins are largely confined to the XY plane [8]. In the region where, up to now, the experiments have been done [16] ( T < 5 K ) the correction given above is negligible and can be ignored. However, for higher temperatures, the renormalization of the soliton energy could be detected experimentally.

We do not claim here that all anharmonic

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corrections are contained in our calculation for 7. Out aim in this paper was only to show that for a proper treatment of the soliton model we should include the effect of the anharmonic terms, neglected when we perform the continuum limit, to the exchange term on the continuum Hamiltonian.

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