CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET

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CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET

D. Betts, R. Ditzian, C. Elliott, M. Lee

To cite this version:

D. Betts, R. Ditzian, C. Elliott, M. Lee. CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-356-C1-358.

�10.1051/jphyscol:19711122�. �jpa-00213939�

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JOURNAL DE PHYSIQUE Colloque C I, supple'ment au no 2-3, Tome 32, Fe'vrier-Mars 1971, page C 1 - 356

CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET

D. D. BETTS (*), R. V. DITZIAN (**), C . J. ELLIOTT and M. H. LEE (***) Theoretical Physics Institute, University of Alberta, Edmonton, Canada

Rbumk. - Le modkle X Y de spin 112 pour ferromagnktique se dkfinit par 1'Hamiltonian de I'interaction

Les a's sont les matrices de Pauli definies aux sites i et j d'un reseau regulier. Ce modele est approprik aux isolants magn4 tiques dans iesquelles g, S 6,. Les developpements en skries exacts aux tempkratures BlevCes des fluctuations dans 1e paramktre d'ordre a longue distance donnent une estimation des temperatures critiques pour les rkseaux f.c.c., b.c.c.

et LC., et un index critique de la susceptibilite y - ^413.^Laskrie de fluctuation d'ordre quatre donne Yestimation A s 513 a partir de laquelle, en utilisant la loi d'kchelle, tous les autres indices sont determines. Une autre hypothkse permet de prkdire les valeurs des indices du modele Heisenberg en utilisant les indices d'lsing et du modkle XY.

Abstract. -The spin one half XY model of ferromagnetism is defined by the interaction Hamiltonian

X, = - J (a? ^0:+ ^a?my)

< i j >

where the o's are Pauli matrices defined at the sites I and j of a regular lattice. This model is appropriate for magnetic insulators for which ^g,> gll. Exact high temperature series expansions for the fluctuation in the long range order yield estimates of the critical temperatures for the f.c.c., b.c.c. and S.C. lattices and a susceptibility index y GS 413. The fourth order fluctuation series yields the estimate A = 513 from which by the scaling hypothesis all other indices are obtained.

A further linearity hypothesis leads to predictions for the Heisenberg model indices from the Ising and XY indices.

The spin one half XY model is a model of an insu- lating ferro- or antiferromagnet defined by the interaction spin Hamiltonian

X o = - J C (a; ô;+ â:â;)

< i j >

where the o's are Pauli matrices defined at the sites i and j of a lattice and the sum is ordinarily restricted to nearest neighbour pairs of sites on a regular lattice.

We may of course add Zeeman terms to obtain the full XY Hamiltonian for a ferromagnet

where N is the number of magnetic ions, pB is the Bohr magneton and Hz, Hz are applied magnetic fields.

The spin one half XY model arises in magnetic systems in essentially the same way as the spin one half Ising model, and the two models should a priori be expected to occur equally frquently in nature.

Consider a crystalline substance containing both non-magnetic atoms and magnetic ions of high half odd integral spin such as Gd3+ (s'= 712) or ~y~~

( s = 1512). In isolation the most important interaction between a pair' of such ions is the Heisenberg interaction

X H = - J t s 1 . s 2 . (3) The crystalline environment introduces an extra crystalline field term in the Hamiltonian which we may write

(*) During 1970-71 on sabbatical leave at Department of Physics, King's College, Strand, London W. C. 2, England.

(**) Now at Department of Physics, University of Toronto, Toronto, Canada.

(***) Now at Department of Physics, Massachusetts Xnsti- tute of Technology, Cambridge, Mass., ^U. S. A.

where all possible time reversal invariant single ion terms may in principle occur. The exact form of (4) is not so important. The important point is that the crystalline field can be much stronger than the effective field of the Heisenberg interaction so that for T % J ' f k only the lowest Kramers doublet of levels for each magnetic ion is appreciably populated.

Then in the critical region the system if essentially a spin one half system. It is equally likely that the populated doublets will have either ST ⁼+ ^sor s: = + +.

In the former case the matrix elements of the Ising part of (3) are much larger than those of the XY part, so the system is well described by the spin one half Ising model. If, on the other hand, the s: = rt_ 3 doublets have the lowest energy the matrix elements of the XY part of the Hamiltonian (3) are much larger than those of the Ising part and the spin one half XY model results with the effective interaction Hamiltonian (1).

The XY model arises in another connection as a lattice model of liquid He4 in which the phase transition of the model is identified with the helium 1 transition, as was first shown by Matsubara and Matsuda [I]. From the point of view of the experi- mentalist, then, perhaps the most interesting result is that there should exist magnetic substances which are analogues of superfluid helium. In particular the order parameter of an XY ferromagnet, M x (or MY), is the analogue of the order parameter of superfluid helium, Y , where ( Y l 2 = p,, the superfluid density.

The superfluid-magnet analogy has recently been discussed in some detail by Fisher 131. So far a good example of an XY magnetic substance has not been found, but Gd2(S0,),.8 H,O is somewhat XY-like in character [2].

The thermodynamic properties of the XY model have been studied recently by the present authors

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711122

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CRITICAL BEHAVIOUR OF THE XY MODEL OF A FERROMAGNET C 1 - 357

using the exact high temperature series expansion approach [4, 51. Like the Ising and Heisenberg models the XY model orders below some temperature, Tc, characteristic of the lattice (and the model). As T approaches Tc from above, the critical region is cha- racterized by large fluctuations in various thermodynamic properties. Typically the fluctuations in the order paramater is strongest. Thus Tc is most preci- sely determined from the divergence of the high temperature series for the fluctuation in the order parameter. For the XY model the square of the order parameter fluctuation is

Y E [ < M : > - < M , > ~

inZ N I ⁽⁵⁾

which, for zero external field, becomes

For the Ising and Heisenberg models the square of the fluctuation in the long range order is simply pro- portional to the susceptibility. However, in the XY model this is not the case because the order paramater operator, M,, does not commute with the Hamilto- nian. It turns out to be simpler to compute the series for the fluctuation, Yo, than that for the initial susceptibility, x0. Further, it has been shown [6] that x0

and Yo have the same critical index, y. Technical details concerning calculation of the high temperature series expansion for Yo for the XY model, the actual coeffi- cients for the simple cubic, b.c.c. and f.c.c. lattices up to and including the coefficient of Kg, and analyses of the series, can be found in [4].

Standard ratio and Pad6 approximant analyses agree in predicting critical temperatures,

k, T,/J = 4.520 -t 0.010, 2.902 + ^0.010

and 2.02 + 0.02 for the f.c.c., b.c.c. and simple cubic lattices respectively. The critical index for the second order fluctuation and the susceptibility,

y = 1.35 + ^0.02.

If y is really a simple fraction then y = 413 seems most likely.

For the specific heat on the f.c.c. lattice we have derived eleven terms in the exact series expansion, which represents a very large amount of information.

Nevertheless, in spite of strenuous efforts we have been unable to obtain a meaningful direct estimate of the specific heat index, a.

For the Ising and Heisenberg models series expansions for a sequence of initial even order field deriva- tives of free energy have been calculated in order to establish the existence of a gap index, A , [7] and hence support the scaling hypothesis. For the XY model it is simpler t o compute the series for the higher even order fluctuations. We have calculated six terms in the series expansion of the fourth order fluctuation 151,

Analysis of Y, yields an estimate of its critical index y , = 4.64 + 0.10. A possible simple fractional value commensurate with y is y, = 1413. Then together with the estimate of y = 413 the scaling hypothesis yields all other static critical indices. These are dis- played in Table I.

It has been argued by Jasnow and Wortis [8} that the value of a critical index for the anisotropic Heisen- berg model depends only on the symmetry of the order parameter but not otherwise on the anisotropy. Thus the Ising values of the indices should be realized for all J I I > JL, the X Y model indices for all J,, < J , and the isotropic Heisenberg model indices for J,, = JL only. The corresponding ferromagnetic order parameters M,, M , and M have one, two and three degrees of freedom or dimensions respectively. This conjecture is supported by their estimates for y for the spin oo anisotropic Heisenberg model.

If we make the stronger conjecture that the critical indices depend linearly on the dimensionality of the order parameter we may predict all indices for the isotropic Heisenberg model from those for the king

Critical Indices for the Thee Dimensional Spin Half Ising, XY and Isotropic Heisenberg Models (")

Index Defining Relation

- -

H = O , T > Tc

Y XT ( T - TC)-'

P4F/dH4 - ^{( T}^-^T,)-^{y - 2 A}

a C , - ^{( T}^-^TC)-'^f^B

T = Tc 6 I H I - l M 1 6

H = O , T < T ,

P ^M- ^{( T ,}^-^T)@

Y' xT - ^{( T ,}^-^T)-'I

a 3 ~ / a ~ 3 - ^{( T ~}^-~ ) - y ' - ~ '

a' C , - ^(T,^-^T)-"'⁺^B'

(") Notation as in ref. 131.

(b) See ref. [9].

(C) See ref. [lo].

Ising Heisenberg

(indirect)

Heisenberg (direct)

5.0 ) 0.2 (C)

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C l -358 D. D. BETTS, R. V. DITZIAN, C. J. ELLIOTT AND M. H. LEE and XY models. These predictions together with

direct estimates of Heisenberg indices [9, 101 are included in Table T. As the values of the indices for the XY and Ising models have been chosen t o obey scaling the predicted (indirect) Heisenberg indices also obey scaling. (We have omitted those indices related to the correlations because some evidence from the three dimensional Ising model is against scaling for these indices.)

It would be desirable to have direct computation of more than two indices for the XY model as a test of the scaling hypothesis. I t would also be desirable to have experimental measurements on an XY ferromagnet or antiferromagnet. In any case the scaled values of the spin one half Ising and XY model indices plus the linearity conjecture lead to predicted indices for the spin one half Heisenberg model in rather good agreement with those directly calculated, considering the fairly large errors involved in the esti-

mate. We note, however, t h e Bowers-an-& Woolf argue for a value of y = 1 318 for the Heisenberg model for all spin. Working backwards the linearity conjecture would then require for the XY model y = 1 5/16 ; rather lower than seems permitted by the direct calculations. The same linear behaviour seems satisfied within allowed errors for the case of spin oo where according to Jasnow and Wortis to within 1 %, y = 1.23, 1.32, 1.38 and 2 v = 1.25, 1.34, 1.40 for the Ising, XY and Heisenberg models respectively.

In summary, then, we have made some indirect predictions of values of critical indices for the XY and Heisenberg models based on the hypothesis of scaling and lineer dependence of the indices on order parameter symmetry respectively. It would be desirable to test these predictions with further direct calculations and with measurements on Heisenberg and XY-like magnetic systems.

References

[I] MATSUBARA (T.) and MATSUDA (H.), Prog. Theov. [7] DOMB (C.) and HUNTER (D. L.), PYOC. Phys. Soc., Phys. (Kyoto), 1956, 16, 416. 1965, 86, 1147.

[2] WIELINGA (R. F.), LUBBERS (J.) and HUISKAMP (W. J.), [8] JASNOW (D.) and WORTIS (M.), Phys. Rev., 1968,

Physica, 1967, 37, 375. 176, 739.

[3 ] FISHER (M. E.), Rep. Prog. Phys., 1967, 30, 61 5. [9] BAKER (G. A.), GILBERT (H. E.), EVE (J.) and RUSH- [4] BETTS (D. D.), ELLIOTT (C. J.) and LEE (M. H.), ^BROOKE(G. S.), Phys. Rev., 1967, 164, 800.

Can. J. Phys., 1970, 48, and references therein. [lo] BAKER (G. A.), EVE (J.) and RUSHBROOKE (G. S.), [5] DITZIAN (R. V.) and BETTS (D. D.), Phys. Letts, Phys. Rev. (1970), in the press.

1970, 32 ^A, 152. [ll] BOWERS (R. G.) and WOOLF (M. E.), Phys. Rev., [6] FALK (H.) and BRUCH (J.), Phys. Rev., 1969,180,442. 1969, 177, 917.