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PHASE TRANSITION IN AN ANISOTROPIC HEISENBERG ANTIFERROMAGNET IN TWO
DIMENSIONS
E. Rastelli, L. Reatto
To cite this version:
E. Rastelli, L. Reatto. PHASE TRANSITION IN AN ANISOTROPIC HEISENBERG ANTIFERRO- MAGNET IN TWO DIMENSIONS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-884-C1-886.
�10.1051/jphyscol:19711313�. �jpa-00214345�
JOURNAL DE PHYSIQUE Colloque C I, supplkment a u no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 884
PHASE TRANSITION IN AN ANISOTROPIC HEISENBERG ANTIFERROMAGNET IN TWO DIMENSIONS (*)
E. RASTELLI and L. REATTO (**) Istituto di Fisica dell'Universit8, Parma, Italy
RbumC. - On etudie la phase ordonnee d'un antiferromagnetique de Heisenberg en couches avec une faible aniso- tropie en se fondant sur la theorie des ondes de spins, modifiee pour tenir compte des effets hydrodynamiques pour les faibles longueurs d'onde et avec une anisotropie calculee de manikre Self-consistante. On applique cette theorie a K;.NiF8.
On peut faire colncider la tempkrature de transition Tc calculee avec celle relevee experimentalement grsce a un choix raisonnable du vecteur d'onde limite pour les effets hydrodynamiques, alors que la theorie habituelle des ondes de spin donne une valeur de Tc bien plus grande (60 % environ).
Abstract. - The ordered phase of a two-dimensional Heisenberg antiferromagnet with small anisotropy is studied on the basis of the spin wave theory corrected for hydrodynamic effects at small wave vectors and with anisotropy self- consistently calculated. The theory is applied to the layer antiferromagnet KzNiFs. The predicted transition temperature Te can be brought to coincide with the experimental one by a very reasonable choise of the cut-off wave vector for hydro- dynamic effects, whereas Tc results much higher (about 60 %) on the basis of the conventional spin wave theory.
Recent measurements of inelastic neutron scat- tering [I] and of Ratnan scattering [2] on K2NiF, show that over almost the whole Brillouin zone the system is characterised by spin wave excitations with frequency a n d width essentially independent of temperature in the whole antiferromagnetic phase and even for temperatures slightly above TN, the transi- tion temperature. This compound, which has a magne- tic layer structure, seems to be described very well by the spin isotropic Heisenberg model with first neighbor antiferromagnetic interaction a n d aniso- troPY PI.
where the prime in the summation means that the sum is extended to first neighbor pairs, (9 runs
U
over the sites of the sublattice with the ( ( u p ))
(ct down YJ) spins, and the anisotropy field HA is proportional t o the sublattice spontaneous magneti- zation [4]. The interaction term explicitly written in (1) contains therefore only the coupling between spins in the same plane, whereas the interaction between planes is included in H', which contains the remai- ning coupling betwcen more distant neighbors. II' results very small [ I ] and its main effect is that of causing the magnetic ordering of the planes, as soon as the order in the plane is established.
The experiments we have discussed above suggest that the standard spin wave theory [5] has a much wider range of validity in this system than in three- dimensional systems. This theory should indeed be appropriate in the whole antiferromagnetic region for those properties which d o not depend strongly on the very small k excitations. We have computed the
sublattice spontaneous magnetization as a function of the temperature o n the basis of the spin wave theory ; neglecting H', the Hamiltonian (1) reduces t o the sum of Heisenberg Hamiltonians for a quadratic lattice and following Lines [6] we have :
-
dK,,'fD(S) coth ( o(K, 31,~" T ) (3)
- n / u COW, S )
where a is the lattice parameter in the plane, D ( 9 measures the anisotropy field : D(g) = ggB HA(S)/4 JS, which is linear in S [4] so that D(S) = RS, and o(k, S)
is the spin wave spectrum
o(k, S) = 2 J S { [ 1 + ~ ( 3 1 1 ~ - + y 2 (k)}, (4)
~ ( k ) = cos ( k , a) + cos (k, a ) . ( 5 ) In (Fig. I ) we show the sublattice magnetization S(T), solution of (eq. 2), as a function of the temperature, for J = 112 OK1, S = 1, and R = 2.3. This value
FIG. 1. - Reduced sublattice magnetization S(T)!S(O) (solide
(*) Work supported in part by Consiglio Nazionale delle curve) as a function of reduced temperature T!TN for spin 1
~icerche. quadratic layer antiferromagnet with anisotropy on the basis
(**) Also : Istituto di Fisica dell' Universiti, Milano, Italy. wave theory (J = 112 OK).TN is the transition temperature [8]
present address : Research Institute for Theoretical Physics for K2NiF4 (TN = 97.1 OK).Experimental data are taken from university of Helsinki, Helsinki, Finland. (ref. [8]) (dots) and from (ref. [71) (crosses).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711313
PHASE TRANSITION I N A N ANISOTROPIC HEISENBERG ANTIFERROMAGNET IN TWO DIMENSIONS C 1 - 885
for R has been obtained using the measured aniso- tropy field [4] gp, HA = 0.85 OK at 4.2 OK and the value S(0) = 0.82 which is the selfconsistent solution of (eq. 2) at T = OOK. From (Fig. 1) we notice the very good agreement between the theoretical S(T) and the experimental data [7, 81 for K2NiF, for temperatures T 5 0.7 TN. This is in agreement with the results de Wijn et al. [8] have obtained on the basis of a similar calculation in which R and S(0) are consi- dered as fitting parameters. At temperatures higher than 0.75 TN the agreement is very rapidly lost and the theory predicts the phase transition at Tc = 1.65 TN. We notice also that the transition is first order. This is because X(T, S) becomes infinite for S -, 0, so that (eq. 2) does not have solution for S arbitrarily small.
We may understand this disagreement on the transi- tion temperature if we take into account the parti- cular role played by the long wavelength fluctuations in the isotropic Heisenberg system. In fact in a two- dimensional system these fluctuations become so large that at any finite temperature the spontaneous magnetization is depressed to zero [9]. This can be seen also on the basis of the spin wave expression (3).
In the isotropic limit, D = 0, the spectrum w(k), linear in k at small wave vectors, gives rise to the singular behavior l/k2 in the integrand of (eq. 3), thus implying a divergent X(T) and the absence of order. This indicates that it is very important to include the correct behavior of the system at small k. A three- dimensional isotropic system at long wavelengths is described by hydrodynamic spin wave [lo] instead of the usual spin wave. We expect that hydrodynamics apply also in the anisotropic two-dimensional system for wave vectors in the range k,, < k < kc, where kc is a cut-off for hydrodynamics and k,, is the wave vector at which the small anisotropy term D can be neglected with respect to the y2(k) term in the expres- sion (4) for the spin wave frequency. We know that the conventional spin wave theory understimates the small k fluctuations. In fact, the fluctuations are characterized by a to/5(T)k2 behavior [lo] in the hydrodynamic regime, where 5(T) and 5, are the correlation lengths at temperature T and 0 OK respec- tively, instead of the simple k-2 behavior predicted by the conventional spin wave theory. Another aspect of this effect consists of the fact that the hydrodyna- mic spin wave spectrum has essentially the behavior :
- --
o(k, T)> v i 5 0 / 5 ( ~ k . Since the correlation lengths ((T) diverges as the transition temperature is approa- ched, we expect that hydrodynamic effects strongly depress the transition temperature. To see if this could explain the discrepancy on the transition temperature, in the absence of a detailed hydrodynamic theory of a Heisenberg antiferromagnet with small anisotropy, we have modified in a heuristic way the spin wave spectrum (4) to the following form :
where, to reproduce the results of hydrodynamics and of the dynamic scaling laws [ll], the factor g(k, T)
must have the following behavior : g(k, T ) = 1 for k % kc ; if t-'(T) > kc, g(k, T ) = 5,/5(i'? for k < kc ; if t-'(T) < kc, g(k, T ) = t0/5(T) in the hydro- dynamic region k < <-'(T) and g(k, T ) = kt, in the critical region kc + k 9 r - ' ( 0 . In the absence of experimental information on kc and t o , for simplicity we assume that kc = 50' and the simple formula :
g(k, T) = 1 for k > kc
g(k, T) = (k + 5-' (T))/(k + 5-A ) for k < k , (6) has the desired behavior. An upper bound to the cut- off kc is given by the wave vector -(n/10) a-'. In fact this is the smallest wave vector at which the spin wave frequency has been measured by neutron diffrac- tion [ I ] and there it has been found no frequency renormalization with temperature. In the absence of experimental information on the correlation length, we have used the expression
This choisc was suggested by the following considera- tions : two correlation lengths are present in (6), one of them, t o , is the correlation length at T = 0, the other one, ((T,), is the cut-off correlation length of the hydrodynamic theory. Some results [12] on three- dimensional antiferromagnets in the critical region yield ((T) = 0.3 a(l - T/T,)-'.~'. However we are interested in an expression for ((3") which works in the whole temperature region from T = 0 to T = Tc.
Clearly the above expression does not work at T = 0 because we would obtain 5, - 0.3 a which is not realistic since we know that 5, should not be smaller than the lattice space a. The discrepancy disappears if we choose the proposed formula for &T). The nume- rical values of v employed in the calculation are v = 1 and v = 213, the value of the Ising model in two and three dimensions respectively.
Using (5) and (6) in (2) and (3), we have solved the resulting equation for S a n d for the transition tempera-
FIG. 2, - Reduced sublattice magnetization S(T)/S(O) as a function of the reduced temperature TITc (Tc transition tempe- rature for the model) on the basis of spin wave theory with hydrodynamic corrections as described in the text for kc = (n/20)a-1, v = 213 (solide curve) and v = 1 (dashed
curve). Experimental data as in (fig. 1).
C 1 - 886 E. RASTELLI A N D L. REATTO lure T, which appears in t(T). The transition tempera-
ture turns out to be strongly depressed by the hydro- dynamic effects as included in (5, 6). The transition temperatures is found to be independent of the expo- nent v and results : Tc = TN for kc = (7~120) a-', where TN is the experimental transition temperature,
TN = 97 OK. In (Fig. 2) the theoretical sublattice magnetization S is shown as a function of the reduced temperature TIT, for kc = (~120) a - ', v = 1 and v = 213. This calculation shows that spin wave theory with hydrodynamic correction is able to explain the observed transition temperature in K,NiF4.
The agreement between experimental data and hydrodynamic spin wave theory is shown in (Fig. 2).
We conclude that the experimental data and this cal- culation suggest that it is possible to give a complete description of the ordered phase of K,NiF4 and of other layers antiferromagnets on the basis of the con- ventional spin wave theory in presence of a small anisotropy. The correct hydrodynamic theory should also change the order of the transition from the first order predicted on the basis of (eq. 3). I n fact, the spin wave expression for the spin reduction is not correct in presence of strong fluctuations [5].
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