PHYSICAL REUIE% B VOLUME 33, NUMBER 5 1 MARCH 1986

Magnetic excitations on two-dimensional percolating clusters

G.-J. Hu and D. L. HuberDepartment of Physics, University of Wisconsin, Madison, Wisconsin 53706

(Received 15 October 1985}

We have carried out numerical studies of the exponent characterizing the distribution of low-energy mag-

non modes on the infinite percolating cluster of the two-dimensional, bond-dilute, nearest-neighbor Heisen-

berg antiferromagnet and the XY model. In the case of the antiferromagnet we obtain agreement with thepredictions of scaling theory using the value of the transverse susceptibility exponent given by Kumar and

Harris. The agreement with scaling theory is somewhat less satisfactory for the XY model, but may im-

prove with larger arrays.

The study of excitation in systems having fractal dimen-sions has attracted a great deal of attention in recent years. '

Among such systems the infinite percolation cluster (IPC)occupies a special place, in that physical realizations of themodel are found in dilute magnetic materials with finite-range interactions. In this paper we are concerned with thelinearized magnon modes on the IPC's of two-dimensional(2D) magnets with nearest-neighbor interactions. We as-sume that the dilution is brought about by the random re-moval of bonds so that with a square lattice the percolationthreshold p, occurs when 50'lo of the bonds have been re-moved.

Of particular interest here is the limiting behavior of thedensity of states at low frequencies. The analysis of Ref. 1,as well as earlier scaling arguments, ' predict that in the caseof a Heisenberg ferromagnet the density of magnon modes,p(E), should vary as

p(E)~ E—I

~here

z= lt —(d —2)v]/(r —p+2v) . (2)

Here d is the dimension of the lattice, t is the exponentcharacterizing the percolation conductivity, and v and AS

have their usual meaning as exponents associated with thecritical percolation point.

Recently, Lewis, and Stinchcombe~ tested Eq. (1) by nu-merically determining the density of magnon modes in atwo-dimensional bond-dilute Heisenberg ferromagnet at p, .From an analysis of 15 configurations on an array of 65 x 65sites, they obtained A. =0.33+0.03. This value is in goodagreement with theoretical predictions, which range from0.33 to 0.34 depending on the value of t.

In this paper we extend the study of the distribution ofmagnon modes to the two-dimensional XY model and thetwo-dimensional Heisenberg antiferromagnet. In both caseswe assume infinitesimal anisotropy so as to have long-rangeorder. Using eigenvalue-counting techniques ' we calculatethe distribution of modes in a single randomly generatedpercolating cluster on a 50x 50 square lattice. Our results,which are obtained from an analysis of the combined datafrom 65 configurations, are compared with the predictionsof scaling theory.

For the purpose of comparison, it is convenient to consid-er both the Heisenberg ferromagnet and the XY model onan equal footing. The eigenvalue equation for the former

takes the form

Ea, = XA,)aj, (3)

where the dynamical matrix is given by

As=S XJ;,;+s Sg —ASS, ;ps,8

which is to be interpreted as a relative measure of the good-ness of fit, rather than the uncertainty in the value of ~.'As in Ref. 3, the E; are separated by an energy interval

in which S is the spin and J;;+& is the interaction betweenthe spin i and its nearest neighbor i + 5.

The analysis of the linearized magnons in the XY modelyields an eigenvalue equation similar to (3) except that E isidentified with the square of the magnon energy and thedynamical matrix is modified to read'

Ai'= Av(A;;A)i)'i

The form of the modification, involving multiplication byproducts of the diagonal elements of the dynamical matrixfor the ferromagnet, suggests that the two systems shouldbehave similarly. ~ This argument is supported by a scalinganalysis based on a hydrodynamic description of the long-wavelength modes above p, .

Since the frequency of the hydrodynamic spin waves inthe XY model is inversely proportional to the square root ofthe zz susceptibility, which does not diverge at p, due to theeasy-plane anisotropy, the modes have a dispersion relationof the form ~ - cq, where c varies as (p —p, )" ~'~'. Apply-ing the scaling analysis to the square of the magnon fre-quency, one obtains Eq. (1) with X equal to its ferromagnet-ic value, Eq. (2).

Our numerical results for the distribution of modes forthe Heisenberg ferromagnet, the XY model, and the Heisen-berg antiferromagnet are sho~n in Fig. 1, awhile the corre-sponding values of ) are listed in Table I. As mentioned,the data were obtained from 65 configurations of a 50 x 50array. The values of P were inferred by fitting the loga-rithrn of the integrated density of states to the expressionA +(1—X)lnE for 0(E «E~. The plus and minus referto the standard statistical error,

N

W 2 $ ln„' p(E)dE —(A + (1 —h. )lnE, ]~=1

33 3599 1986 The American Physical Society

3600 G.-J. HU AND D. L. HUBER 33

TABLE I. Values of A. .

20- Ferromagnet XY Antiferromagnet

t0-

0.10 (10)

0.20 (20)

0.30 (30}

0.314 + 0.001

0.317+0.001

0.316+0.001

0.280 + 0.001

0.275 + 0.001

0.271+ 0.001

—0.06 + 0.01

—0.00 + 0.01

—0.04 + 0.01

Ld 8-

(b)

'E& is the upper cutoff in the integrated density of states. Thenumber of points which were fit is shown in parentheses. The datawere obtained from 65 configurations of a 50x50 array of sites.Units are such that J=5 =1. For comparison, the maximum mag-non energy in the fully occupied ferromagnetic lattice is equal to 8.

given by

Z= [t+r+2P —2v(d —I)]/(t+r+2v) . (6)

4-UJ

In Ref. 10 heuristic arguments were presented which ledto the inequality

00

FIG. 1. Histogram of the distribution of magnon modes on the2D IPC at p, . (a) Heisenberg ferromagnet, (b) XY model, (c)Heisenberg antiferromagnet. N(E) is the number of modes in a

bin of width 0.01 in a SOX50 array, averaged over 65 configura-tions. Note the difference in vertical scales. In the case of the XYmodel, E is identified with the square of the magnon energy.

equal to 0.01 in units where J = S = 1.The data for the ferromagnet are consistent with

1=0.31-0.32, which is comparable to the estimate of Lewisand Stincheomb and close to the value for the infinite array,0.34, obtained with v= T, p= ~6, and t = 1.31. In the caseof the XY model, the discrepancy between our numericalresults for ~ and the predicted value is noticeably greater,although the former are within about 10% of the value weobtain for the ferromagnet. Moreover, the agreement ap-pears to be improving with decreasing F~. This suggeststhat calculations carried over on larger arrays, which make itpossible to have smaller values of E~ with good statistics,would yield results in better agreement with theory.

As in the XY model, the frequency of the hydrodynamicspin waves in the Heisenberg antifcrromagnet is inverselyproportional to the square root of the perpendicular suscep-tibility, p&. Unlike the XY model, however, pq diverges asthe concentration of bonds (or occupied sites) approachesp, .' Because of this divergence the spin-wave velocityvaries as (p —p, )" 't2, where r is the exponent associatedwith x. According to scaling arguments, ' the parameter ofX characterizing the density of magnon modes on the IPC is

r ~t —p —(d —2)v.

More recently, Kumar and Harris" developed static scalingarguments indicating that v is equal to the right-hand side of(7). Using Eq. (7) with t =1.31 and the other exponentstaking on values appropriate to 1=2, we obtain X=0.02.The data given in Table I, although showing more scatterthan the corresponding numbers for the ferromagnet, are inreasonable agreement with this value.

It should be noted that had we used t =~2, the value

which gives A. =T for the ferromagnet, ' we would have

found A, =O. Also, were X& nondivergent at p„we wouldhave obtained

X = [t + p —2v(d —I ) ]/(t —p+ 2v) . (8)

%e ~ould like to thank D. Kumar for helpful commentson this work. The research was supported by the NationalScience Foundation under Grant No. DMR82-03704.

With exponents appropriate to d = 2, Eq. (8) yields~= —0.32, which is in definite disagreement with the nu-merical data.

In conclusion, we have carried out a numerical study ofthe exponent A. characterizing the density of magnon modeson the IPC of a square lattice. Results were obtained forthe Heisenberg antiferromagnet and the XY model. Ourvalue of ~ for the antiferromagnet is in reasonable agree-ment with the predictions of scaling theory when we use theexponent for the transverse susceptibility given in Ref. 11.In the ease of the XY model, the agreement is not quite asgood. However, the data suggest the discrepancy withtheory ~ould be reduced if calculations were carried out onlarger arrays.

MAGNETIC EXCITATIONS ON TWO-DIMENSIONAL. . . 3601

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JETP 43, 1174 (1976)] (ferromagnets); 75, 352 (1978) t4&, 175(1978)] (antiferromagnets) .

~S. J. Lewis and R. B. Stinchcombe, Phys. Rev. Lett. 52, 1021(1984).

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91

' A. B. Harris and S. Kirkpatrick, Phys. Rev. B 16, 5242 (1977).' D. Kumar and A. B. Harris, Phys. Rev. B 32, 3251 (1985).