Effect of the magnetic field on quantum fluctuations in quasi-one-dimensional hexagonal antiferromagnets

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  • Solid State Communications, Vol. 84, No. 5, pp. 573-576, 1992. 0038-1098/92 $5.00 + .00 Printed in Great Britain. Pergamon Press Ltd


    I.A. Zaliznyak*

    Centre d'Etudes Nucleaires de Grenoble, DRF/SPSMS/MDN, 85X, 38041 Grenoble Cedex, France

    (Received 7 July 1992 by P. Bur&t)

    Effect of zero-point spin motion on the ground-state properties of quasi-one-dimensional antiferromagnets CsNiCI 3 and CsMnBr3 is considered. Comparison of the low-temperature magnetization curves measured in these compounds with the results of spin-wave calculations reveals two principal features:

    (I) The measured magnetic torques are strongly reduced with respect to the values calculated in classical approximation due to zero- point spin oscillations. This is found to be in a good agreement with the neutron scattering results. The value of the spin reduction at H = 0 appears to be well reproduced by the harmonic approximation of the routine spin-wave theory.

    (2) Strongly nonlinear growth of the magnetization with magnetic field is observed. It can be regarded as a direct evidence for the gradual supression of quantum fluctuations by the increasing magnetic field.

    THE GROUND-STATE magnetic properties of the quasi-one-dimensional Heisenberg antiferromagnets are for a long time the topic of great theoretical and experimental interest. To describe the low-tempera- ture behavior of such systems one usually starts with the microscopic Hamiltonian

    )~v = J ~ t Sf iSf j W Jt y ~ " J i ~ j ij i d

    + D L . ~ i: - T H id i

    where the first sum involves the scalar products of the spins which are the nearest neighbors in the z-axis direction (or inside the 1 D chains), the second one, of the nearest spins from the neighboring chains, third and fourth account for the anisotropy and Zeeman energy respectively. Quasi-one-dimensionality implies the smallness of the ratios J ' / J and D/J.

    One of the principal features of these systems is the crucial influence of zero-point spin oscillations on their low-temperature behavior. This in particular makes the usual semiclassical spin-wave approach inapplicable to them in a certain range of parameters of the Hamiltonian (1) near the purely one-dimen-

    * On leave from the Kapitza Institute for Physical Problems Russian Academy of Sciences, 117334 ul. Kosygina 2, Moscow, Russia.

    sional isotropic limit J ' , D = 0. As was suggested by Haidane [1], quantum fluctuations can even modify the ground state to possess no N6el order, being a singlet with the finite correlation length for integer spins in a finite range of parameters. However, the presence of the anisotropy and the interchain exchange J ' beyond that range can induce the development of the long-range order at finite TN, leaving, nevertheless, the applicability of the spin- wave treatment questionable. Namely, this "inter- mediate" situation will attract our attention in the present study.

    The most transparent evidence for the failure of the spin-wave theory is obtained through the calculation of the difference between the average spin at each site in the ground state and spin value S involved in the Hamiltonian (1) (or the spin reduction, ~S). In harmonic approximation it is given (settingh = 1) by

    ~S = S - (6:~') = (ai a+}

    = 2 (27r)" e(k) 1 . (2) Brill one

    Here z' is the local quantization axis for the spin 6: i, n is the dimension of the space, ek = v'~A~ + B~) is the magnon energy and Ak, Bk are the standard coefficients of the Fourier-transformed Hamiltonian


  • 574 Q U A N T U M F L U C T U A T I O N S IN H E X A G O N A L A N T I F E R R O M A G N E T S Vol. 84, No. 5

    which are involved in the u-v transformations. For n = 1 ( J ' = 0 ) and D-- -0 we have Ak = 4 J S , e(k) = 4 JSs ink and equation (2) diverges. Account for the nonzero J ' and (or) D in the Hamiltonian (1) cuts the divergence extending the integration in (2) to 3D and (or) introducing the gaps in e (k). For H = 0 the exact expressions are known (D _> 0):

    [ J(k + k) + J(k - k)] A k = S D + J(k) - 2J(k0) -+ 2

    e ( k ) = 2 S ~ ( [ D + J ( k ) - J(k0)]

    x [ J ( k + k ) + J ( k - k ) - J ( k ) ] ) 2 , (3)

    where, as usual, we designate J ( k ) = ~-'~iJicos(kri) over the nearest neighbors and k 0 is the wavevector describing system's ground state in terms of the helicoidal spin arrangement ( J (k0 )= min{J(k)}). It is easily seen now that the easy-plane anisotropy alone does not remove the divergence of equation (2) for n = 1 at k = 0, introducing only the gap at k = k0 = 7r in the magnon spectrum e (k). This is in agreement with the phase diagram of Haldane-gap spin chains: increasing D > 0 does not restore the N6el ordering, imposing the transition from the Haldane state to a singlet single-ion-like state. On the other hand, the easy-axis (or Ising-like) anisotropy creates the gaps both at k = 0 and 7r, thus cutting the divergence of equation (2) at the value -,~ ln(ID[/J ). This implies that the estimate given in [2] is valid only for D < 0 while for D > 0 the leading contribution to 6S is always ,-, ln(J'/J).

    In the present paper we shall consider the hexagonal antiferromagnets CsNiC13 (S = 1, D < 0) and CsMnBr 3 (S = 5/2, D > 0) which are believed to be described by the Hamiltonian (1). This is supported by magnetization, magnetic resonance and neutron scattering measurements, see [2-7] and references therein. Both of these compound possess non-collinear triangular ground state spin arrange- ments which develops below finite T s. It corresponds to a plane helicoid with k 0 = (27r/3, 27r/3, 7r) which minimize the exchange part of the Hamiltonian (1) on a hexagonal lattice. Anisotropy fix the orientation of the spin plane with respect to the crystal axes (z [I C6). However, D < 0 also cause a distortion of the exchange structure, --~ D/J'. In CsNiC13 it has been found to be small [3]: the angle between the nearest spins in the neighboring chains is ~ 119 instead of 120 for D = 0. Strong quasi-one-dimen- sionality (jr~j, D/J,~ 1 0 - 2 - 10 -3) leads to a sub- stantial spin reduction due to zero-point oscillations.

    As was established experimentally by the means of elastic neutron scattering (ENS) [3,4] the average spin value of each magnetic ion at H = 0 extra- polated to T = 0 is reduced by 33% in CsMnBr 3 and by 50% in CsNiCI3.

    Naturally, large spin reduction should also reveal itself in the net magnetization of the system. As was established in our previous experiments on CsMnBr 3 [8], the magnitudes of the magnetic torques measured at T < < TN appear to be about 1.5 times smaller than the classically calculated values, in a very good agreement with the ENS result. Nevertheless, their functional dependencies at low fields (H < < H e = 8JS) agree well with those calculated in classical approximation starting with the Hamilto- nian (I). The magnetization curves measured on CsNiC13 at T = 1.8K, which is believed to be sufficiently small to "freeze" the temperature fluctua- tions [3], demonstrate essentially similar behavior (see Fig. 1). They have been measured with the same samples as in [7] in the most straightforward way, using computer-controlled SQUID magnetometer "Quantum Design" at D R F M C CEN-Grenoble. Classical calculations in this case can be easily performed for two directions of the magnetic field and yield simple formulae

    H M(H) = g#sNA H"-~e

    {'El I x 2 -~-7+4---ff)~J , H I I z . 5 1 _ 2 D H 2 ] (4)

    0 . 6 / /

    / /

    I/II I

    " i i I

    0 0 0 . 4

    S : / * i I I / 0 0

    i i O o [ /

    i / I

    "* P Q Q @ . 2 j. ," Q , ', oO

    /// I 0 0

    ~ ' ~ . e J ~J~ mu I L [ p l t e d by sp Ln r e - 8 .e ~ ' d u c L L o n

    8 2e 40 6e 88

    H, k Oe

    Fig. 1. Magnetization curves in CsNiC13. Open circles are the experimental points for H parallel to z, closed for H perpendicular.


    Table 1. Constants of the Hamiltonian and ground-state spin reductions 6S = S - (~z') at zero field for the substances discussed

    Compound J, GHz J ' , GHz D, GHz (6S)ENS (6S)cal c

    CsNiCI 3 345 8.2 1 -0.6 0.5 0.57 CsMnBr3 214 0.5 + 1.95 0.85 0.86

    Account for the spin reduction through a simple multiplication of these expressions by the ratio (S)/S~-0.5 determined with ENS at H = 0 , as proposed in [8], again leads to a fairly good fit at low fields. This again supports the conclusion that the main effect of zero-point motion on the magnetiza- tion of quasi 1D antiferromagnet is the renormaliza- tion of its magnitude by the factor (S)/S.

    In view of the arguments presented above it seems interesting to check whether this effect can be reproduced within a framework of the harmonic approximation of the spin-wave theory as discussed at the beginning of the paper. The obvious theoretical construction is the following: first we determine the directions of the quantization axis classically mini- mizing equation (1) and after that - the average


    spin values on these directions ( ~ z ) , giving the sublattices magnetizations, which we then sum up to get the net magnetic moment.


    For H = 0 one can perform calculations of (5~) for given compounds using expressions (2)-(3) and constants of the Hamiltonian determined from the neutron scattering and magnetic resonance measure- ments (Table 1). Montano et al. [9] have done a model estimate of this kind for CsNiCI 3 restricting the integration in equation (2) to n = 1 and introducing large D < 0 in a simple 1D spin-wave theory to cut its divergence. The value of D ~ 8 GHz they used to fit the ENS result 6S ~- 0.5 is of the order of J ' in CsNiC13 which makes it clear that the finite value of (Se~) in this compound is essentially due to the 3D effects.

    To get the actual values of spin reduction in the framework of the spin-wave theory for the com- pounds involved we have performed numerical calculations of 6S using expressions (2) and (3). The results together with the experimental values are summarized in a Table 1 (they are exact* for CsMnBr3 with D > 0, and approximate for CsNiCI3 neglecting D, ID < < J, J ' ) . Surprisingly, the calcula- tions fit well with the experiment despite the large relative spin reduction which in principle makes the applicability of the harmonic approximation ques-

    * In the harmonic approximation, together with equation (2).

    + tionable: the higher order terms in ak, a k are known to be large [10]. What one can suggest on the basis of the present analysis, is that they eventually cancel each other, leaving the leading contribution due to


    a k a k .

    It should be mentioned that usually it is possible to treat the effect of the anharmonicities in terms of renormalization of the constants involved in the expressions for the experimentally measurable quan- tities. For instance, the functional dependencies of the magnon energies calculated in the harmonic approximation are known to remain valid upon the account for the anharmonicities but involve J, J ' and

    which are different from the values J, J ' , in the Hamiltonian (1). Major renormalizations in this case are believed to be [2]

    [ rr- 2 rc2-10 +O(l/S3)] J = J 1+--~-~--~ 47r2S2

    o=o(, In fact, /) and ) ' contain also corrections incorpor- ating 6S which could not be explicitly taken into account. Using the constants determined from the measurements of magnon energies (Table 1) to fit the magnetization data one imply them to be renorma- lized in the same way for both quantities. Our experiments support this presumption at low fields for the systems in hand: as it is already mentioned, one obtains a good fit to the experiment calculating the magnetization in the way described above.

    Another principal feature that has been observed in both compounds is the visible deviation of the functional dependencies of ' the measured magnetiza- tions from the classical formulae at higher fields. This effect is most clearly and unambiguously seen if we consider the magnetization in the field, perpendicular to the initial spin plane (i.e., H ]l z for CsMnBr3 and H _1_ z for CsNiCI3), which theoretically should be linear and in both cases is given by the same expression, see equation (4): instead of being a straight line it has a noticeable upward curvature. In CsNiCI3 this effect is much more pronounced than in CsMnBr3 despite the smaller fields attained in experiment. Similar observation for CsNiCI3 have


    1 . 0



    ~ 0 . 8


    ~ 0 . 4



    O 0 . 2

    0 . 0

    2 . 5



    13 2 . 0



    ~ 1 . 0 0

    { _

    ~ 0 . S

    0 . 0


    o.O,. o o -'~ o -e ' ap,4.4 ~ o-o.-O

    e - m " o ~ - e o 4 4

    . . . . , . . . . i , , r , , , , , , i . . . . . . . . .

    . . . . . . . . . 2~ 4e 60 ae H , k Oe


    , , , , . . . . . i , , , , , , , , , t . . . . . , , , ,

    . . . . . . . . . 20 40 60 fi~

    H, t o e

    Fig. 2. Field dependencies of the average spin value at each site as obtained by division of the measured magnetization by the classically calculated one: (a) for CsNiCI3 with H z; (b) for CsMnBr 3, with H [[ z. Dashed curves are the lowest order polynomial extrapolations, plotted to be guide for the eye.

    been made by Palme et al.* and in our earlier experiments with three-axis magnetometer, however, Johnson et al. [11] did not report them.

    In view of the whole picture developed above one can explain this curvature as being essentially caused by the effect of the magnetic field on zero-point vibrations which strongly renormalize the average magnetization of the system. Increasing magnetic field suppress zero-point fluctuations and thus the average sublattice spin grows. This picture can be visualized if one divides the experimentally measured magnetization by the classically calculated one, as shown in Fig. 2. From the formal side the picture is the following: magnetic field changes dispersion (3)

    * Preprint to be presented at a conference in Darmstadt.

    introducing the gaps in the magnon spectrum and thus decreasing the 6S value given by equation (2). However, it is not clear whether this effect can be described in the harmonic approximation as it appeared to be the case for H = 0: the effect of the magnetic field on the anharmonicities can also play the crucial role. To answer this question one has to calculate integral in equation (2) using the magnon spectrum in the field e (k, H) which, at the time being, is not available for the complex structures we are dealing with.

    In conclusion, a simple picture for the ground state of the quasi 1D antiferromagnets CsNiCI 3 and CsMnBr3 under magnetic field involving large average spin reduction has been developed. It explains all the features of the magnetization measurements which are presented. However, it needs further experimental verification, for which the best would be the dir...


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