senhora da limpeza atende utentes na extens£o de sade

50

Progress of Theoretical Physics, Vol. 53, No. 1, January 1975

A Use of Bethe Lattice Solutions for Percolation-Constrained Phenomena

A. R. BISHOP

Department of Theoretical Physics University of Oxford, England*l

(Received July 4, 1974)

Classes of phenomena .are considered that are strongly influenced by percolation constraints in particular showing a critical percolation concentration for some value p. of a 'concentration' parameter p. Using this feature an attempt is made to generate useful approximations to behaviour in real lattices over a complete concentration range from exact solutions available in simple pseudo-lattices-here only Bethe lattices are used. Fitting Bethe solutions at P=Pc and p=1 is found to give a numerically useful interpolation for all p in a range of lattices and for several problems; detailed comparisons with ·best available data are given for (quenched) dilute Ising ferromagnets, random resistor networks and spin-wave stiffness constants. Limitations of the approach are emphasised numerically-especially the inadequate distinction between bond and site problems or treatment of critical exponents. It is also suggested that localization in cellularly disordered lattices might be amenable to a similarly motivated approach. -

§ I. Introduction

In earlier work1l (hereafter referred to as I and II) the present author introduced a highly successful ad hoc procedure for calculating the critical (Curie) temperature of an undiluted Ising ferromagnet. The philosophy of this approach (see I and II for details) is to approximate behaviour in general lattices by reinterpreting the appropriate behaviour in certain simple infinite dimensional branching media (infinite ·Cayley trees or Bethe lattices) in terms of suitably sensitive lattice constants. It was found .in I and II that the critical site percolation concentration Psc is a particularly successful lattice constant in this context when used in a specific way.

This paper will be concerned with the same basic philosophy-the interpretation of exact Bethe lattice solutions (see II) in .terms of Psc and Pbc (the critical bond percolation consentration) -but extended to more general 'critical' phenomena where Psc and Pbc play essential roles of clear physical importance. In this they contrast with the undiluted Ising ferromagnet problem of I and II where the reasons for p,c's effectiveness are not obvious. We will :find the approximations here to be far less precise than in I and II but nevertheless useful

*> Address during academic year 1974~5: Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, U.S.A.

Dow

nloaded from https://academ

ic.oup.com/ptp/article-abstract/53/1/50/1861612 by guest on 12 April 2019

A Use of Bethe Lattice Solutions for Percolation-Constrained Phenomena 51

guides over a whole range of a 'concentration' parameter p. Such approximations may be of practical interest in several problems (see § 6). Typical systems where this approach is applicable are the diluted Ising model of ferromagnetism (§ 2) and random resistor networks (§§ 3, 4). In § 5 we will conjecture that a less obvious example might possibly be provided by electron localization in cellularly disordered lattices.

The idea we will examine here is a very simple one and it may be useful to emphasise this now so that it' is not obscured by the details of particular examples; we will be interpreting Bethe lattice solutions of various percolationconstrained phenomena as functions of Pc, deduced from the relationship determining Pc on. Bethe lattices (Eq. (1) of I). The object will be to generate useful approximations over a complete 'concentration' range for more general lattices using their (assumed known) values of Pc· It will become clear that our procedure is in effect a simple scaling of the Bethe lattice solution and that its success arises from the facts that we fit the solution at two points. (p = Pc and p=1) and also guarantee a degree of the correct (e.g., thermodynamic) structure by using the exact Bethe lattice result. Since our aim is to demonstrate the quantitative (and not merely qualitative) usefulness of this approach, the bulk of the succeeding sections will be concerned with detailed quantitative comparison with available analytical and numerical data.

§ 2. The Ising ferromagnet with impurities

In I and II we considered the undiluted Ising ferromagnet. We now leave this model and turn to .a situation much discussed in the literature, e.g., Refs. 2) "-'6) where non-magnetic inwurities have been introduced-either randomly replacing a fraction 1-p, of the spins (the dilute Ising ferromagnet or site impurity system) or occupying a fraction 1-pb of the bonds and operationally preventing interaction between the spins at the ends of an occupied bond (the bond impurity system). To date most investigations have been limited to two ideal limits. In the 'annealed' system the impurities are assumed to be in complete equilibrium with the spins, whereas in the 'quenched' system they are taken to be so massive as to be completely independent of the spin fluctuations and to be fixed randomly on a fraction 1-p of sites (or bonds). Extensive numerical inve~tigations -in these -systems are now available, _principally employing series expansion methods both in concentration, e.g., Refs. 3), 6), and high temperature, e.g., Refs. 4) "-'6). Here we· will limit discussion to the variations in the critical Ising temperature Tc (p) with impurity concentration predicted by these means. It should be noted, however, that there are sophisticated questions of existence and regularity7),B) that cannot be answered by finite series data. We w:ill not be concerned with these important questions in this work.

As in I and II our approach will be to approximate the real lattice behaviour

Dow



52 A. R. Bishop

by interpreting the exact solution ('exact' in the sense discussed in II) on infinite Cayley tree lattices. We will find presently that our approximation is far less precise than in the undiluted problem of I and II. However, the use of the critical percolation concentration is now very natura.!. As discussed in II, if P<Pc the percolation probability is zero, i.e., no spins belong to infinite connected clusters, and no co-operative macroscopic transition is possible even at zero temperature in Ising or Heisenberg ferromagnets. 6>· 9> Thus Tc(P) is zero for p<pc'>Pc· It is not proven in general that Po' =Pc, although in the quenched site impurity Ising system the coefficients occurring in the concentration expansion for the susceptibility at zero temperature are precisely those of the site percolation expansion for the mean cluster size,6> which indeed diverges at P=Pc· Numerical evidence suggests more generally that Pc and p/ are close6>• 7>' 19>•11> and they are equal on Bethe lattices. Henceforth we shall assume this to be generally true as suggested elsewhere9>' 12 '>13> and our approximation will correspond to a ~imple scaling of the Bethe solution to the real lattice Pc value. The unsealed Bethe 'mean field approximation has been considered previously.2> We now 'confine our remarks to the quenched problem, although a similar analysis could be applied to the annealed model.

The Oth order mean field theory (c.f., Eq. (1) of II) is simply

TcMF(p) =pqJ=PTcMF(1) (1)

and shows no critical concentration (q is the lattice co-ordination number). This has led to the suggestion14> that only those sites (or bonds) belonging to infinite clusters (in which the co-operative transition can take place) should be taken into account, i.e., only the fraction P(p) (the percolation probability: .see II). This 'percolation mean field' (PMF) approximation is then

T'!MF(p) =P(p)qJ=P(p)YJP)MF(1). (2)

We will see (numerically) presently that T.(P)/Tc(1) is usually distinctly overestimated by (2). This might be expected since spin fluctuation effects mean that the Weiss theory overestimates Tc increasingly for smaller (effective) lattice co-ordination.

The exact Bethe lattice solution to the quenched (site or bond) impurity Ising model is known.2>· 8>• 16> Of interest to us is the critical temperature result:

P tanh(J/Tc(P)) = (q-1)-1• (3)

Equation (3) can be written

p tanh(J/Tc(P)) =tanh(J/Tc{1)), (4)

which was derived by Elliott and Heap8> as a real lattice approximation in a concentration expansion and called by them the 'Ising chain approximation' for obvious reasons. They used (4) in an attempt to estimate Tc(P)/Tc(1) by supplying (assumed known) values of Tc (1). This is not particularly successful,

Dow




however. For example, it gives Pc=tanh(J/Tc(1)):::::::K (using Eq. (4) of I). As a better approximation we suggest that (3) might be generalized by interpreting it in terms of Pc instead of Tc (1), thereby explicitly introducing the condition pc' =fie· Adopting the same mean field scaling as m Eq. (5) of II yields

q 1

(1+p;;-1) tanh-1 (Pc/P)' (5)

which is now accurate at p = Pc and approximately at p = l. Further, we deduce from (5) that

Tc(P) = ln((1-Pc)/(1+Pc)) (6) Tc(1) ln((P-Pc)/(P+Pc))

This critical curve has infinite gradient at p = Pc of the form expected in real bond and site impuri.ty problems5J,a), 1~l and is of course exact at p = 1. We will refer to (5) and (6) as the Bethe-Pc approximation. The alternative interpretation of (3) in terms qf K to give a corresponding Bethe-K approximation (the self-avoiding walk approximation1l' 17l) fails to distinguish bond and site problems at all and is always inferior to (6), though less so for bond cases. However, Harris17l has shown rigorously that this Bethe-K approximation gives a. strict upper bound for Tc(P) (strengthening the MF one). We have not been able to establish similarly rigorous results for (5) but it seems to be a generally numerically superior approximation for site cases. We will concentrate attention on the ratio Tc(P)/Tc(1) in this paper.

In Figs. 1 (a), (b) we have compared the best available numerical estimates of TcCP)/Tc(1) 5l' 6l' 18l with the approximations (1), (2) and (6) for the bond and site impurity problems on the face-centred cubic lattice. P(p) is taken here from best available Monte Carlo data, see Refs. 10), 11), which is consistent with a power law formnear Pc with exponent 0.3~0.4.10) D!mensional arguments limit this critical exponent to 0~2/3 .in 3-dimensions/9l so that the critical curve in the PMF approximation is also expected to have the correct gradient at threshold at least in real lattices (see below). The PMF is clearly an improvement on the simple MF one, but we see in Figs. 1(a), (b) that approximation (6) gives a decisive further improvement, as we suggest it will in all lattices; in Figs. 1 (a), (b) the various approximations are also compared with best estimates for the site-simple cubic and site-square lattices, .supporting this suggestion. Agreement is particularly good for the smallest threshold case (the f.c.c. lattice-bond impurity) but even here there is a slight overestimation of the exact critical curve for all p. The overestimation is more pronounced in the site case (higher Pc value) and apparently mostly for p'?::_pc, although there is considerable uncertainty in the 'exact' results at low p.5J,BJ, 18l

We can quite confidently expect this error to increase with increasing Pco • for the gradient of the critical curve at p = 1 provides a useful indicator and we

Dow



•

54 A. R. Bishop

are able to check this with fairly precise numerical (as well as some analytical) series expansion predictions. The comparison is exhibited in Table I, where we see that the gradient is consistently underestimated, especially for large Pc· The limiting gradient is of course unity for all lattices in the MF and PMF approximations (P(p)____,p+O((l-p)q), as P----'>1-). We suggest that (6) is a

(a) Face-centred cubic lattice, bond case. Simple cubic lattice,

P--+- site case.

(b) Face-centred cubic and plane square lattices. Both site cases.

(c) Bethe lattices of co-ordination numbers 3 and 4. The Bethe-p, and exact solutions are coincident (see § 2) .

Fig. 1. Comparison of MF, PMF and Bethe-p, approximations (see § 2) with exact or best estimates of the phase separation curves in the quenched diluted Ising ferromagnet model for various lattices. Solid line: MF approximation (1) or 'exact' results taken from Rapaport•>, •> or estimated from Saville.'"' Dotted line: PMF approximation (2) with P(p) taken from Frisch et aL'•> and Kirkpatrick.'•>,"> Dashed line: Bethe-p, approximation (6).

Dow



A Use of Bethe Lattice Solutions for Percolation-Constrained Phenomena 55 .,

useful interpolation approximation, especially for more highly co-ordinated 3-

dimensional lattices. In fact, although there is this increasing discrepancy in

gradient with increasing p., the simultaneous decrease in the P-range of physical

interest means that (6) may still be a numerically useful approximation (see the

square lattice in Fig. 1 (b), for example). Rushbrooke et al.4l have considered the asymptotic behaviour of T.(P)/T.(l)

in the nonphysical limit P-'> oo. They use the exact Ising model result

T.(P)/T.(l) ~ pKtanh-1V"'

where V.=tanh(J/T.(l)). Equation (6) gives

T.(P)/T.(l) ~ (P/P.)tanh-1Pe.

These last two formulae indicate something of the limitations of the approxima

tions introduced in (6) -real lattice behaviour naturally depends on other sensitive

Table I. Exact or best estimates of the gradient of T. (p) /T. (1) as P--'> 1- and oo compared with values according tb. various approximations (§2): Bethe-q (from (3)); Bethe-K (see Ref. 17)); Bethe-p. (Eq. (6)). Clearly, both of these gradients are unity in the MF and PMF approximations (Eqs. (1) and (2)).

d/dp[T.(p)IT.(1)], P--'>1- T.(p)/pT.(1), p--'>oo Lattice

I Bethe-q I Bethe-K17> I Bethe-p. 'Exact 'd> I Bethe-q I Bethe-K17 >1 Bethe·Pc 'Exact'

Square (bond) 1.33•l,b) 1.08 1.11 1.23

(site) "'1.5•> 1.08 1.11 1.34

1.57•>;

Triangular (site) 1.45dl 1.03 1.04 1.23 1.140 1.014 1.020 1.099

±.05

Tetragonal

Zircon (site) "-'1.2•) 1.08 - 1.14

s. cubic (bond) 1.06bl 1.03 1.03 1.04

(site) "'1.1 cl 1.03 1.03 1.07

b.c. cubic (site) ~1.05dl 1.01 1.02 1.04 1.028 1.007 1.008 1.020

±.02

f.c. cubic (b. on d) 1.02*> 1.01 1.01 1.01

(si~e) l.05cl,d) 1.01 1.01 1.03 1.025 1.003 1.003 1.013

±.02

a) These are limiting gradients for annealed systems taken from Refs. 5), 21) and 22). We expect intuitively that the gradients will be the same for the corresponding quenched systems••> (for instance compare Refs. 8) and 21)). This is already proven in. bond cases. •>, •>, "J

b) Ref. 8). c) Ref. 18). The data here are inconclusive (except for the f.c.c. lattice) because of the open

nature of the lattices and the relatively small number of expansion terms used. The estimates given are only freehand ones after adjusting the data to fit at p=p,. and 1. The bare data consistently gives gradie_nts that are 'too low (according to the general trends in Table I).

d) Ref. 4). These authors observe that it is very difficult to ,distinguish the critical temperature curves on f. c. c. and b. c' c. lattices from series expansion results with P"-'L They suggest a gradient of "'1.06.

Dow



56 A. R. Bishop

lattice characteristics besides simply Pc (sufficient for Bethe lattices where Vc = Pc = K-1). This is clear here and also in the underestimation of fluctuation effects especially near P< (Figs. 1). Fluctuations of an increasingly long range become crucial on approaching the critical point but these are artifically constrained in any effective medium approach such as a Bethe lattice solution (c.f., discussion in II), so that their role is underemphasized, arising purely from first cluster considerations in Bethe lattices.·' This effect will be seen again in the context of random resistor networks (§ 3). T~ere, as here, the critical exponent is inadequately accounted for and it can be shown (J.W. Essam, private communication, and Ref. 47) to be determined entirely by self-consistent first cluster properties in Bethe lattices (if finite and infinite clusters are carefully distinguished).

It is interesting to compare the PMF and .exact solutions on Bethe lattices. P(p) is known· exactly here24l and shows anomalous behaviour in that its gradient at Pc is finite (a feature shared by other branching media). On a Bethe lattice of connectivity K it can be shown (e.g., from Ref. 24)) that P(p) vanishes with unit exponent at Pc and gradient 2 (K + 1)/(K -1). The exact critical temperature curve is .required by the same thermodynamic arguments as in real lattices to have infinite gradient here. Thus there are concentrations near Pc for which tl;le PMF approximation underestimates Tc (P)/Tc (1), in apparent contrast with all real lattices. We have illustrated this behaviour in Fig. 1 (c). Of course, by its derivation, (6) is exact on Bethe lattices.

§ 3. Random resistor networks

There are other systems in which the concept of a critical percolation concentration can be expected to be important and for which exact Cayley tree solutions are available. Currently fashionable 'random resistor networks' are a good example.lo),11),26),26)

' Here a regular lattice is considered with conductances between near-neighbour sites, the conductance values being independently distributed according to some common probability distribution. If the distribution is bimodal and the two conductance values some finite value and zero, then Pc enters naturally-there will only be macroscopic current flow if an infinite path of finite conductances spans the whole system. As before we must distinguish between site and bond cases by specifying either the probability pb that a bond is occupied by the finite conductance or the probability p, that a site is 'occupied', in which case a given bond will be occupied only if both of its end sites are. The conductivity (J (p), then, is zero for P<Pc· Experiments25,l• 26l· 82l have su,pported this interpretation and served to emphasize important differences between P(p) and fJ(P)/fJ(l}, In particular the latter approaches threshold with a concave 'toe' and zero gradient, described (available evidence suggests) by a power law dependence with exponent "-'8/5 in 3-dimensions.10l• 11l This is very different from the P(p) behaviour

Dow



A Use of Betke -Lattice Solutions for Percolation-Constrained Phenomena 57

described in § 2, and has been attributed to conduction near Pe via bottlenecks

connecting (3-dimensional) finite clusters.11>• 26l' 27> Again the unit gradient of

P(p) as p~1- is replaced by 1+a, where a is lattice-dependent (see later).

The observed behaviour differs significantly in the bond and site situations. In

the former a single bond effective medium theory (entirely analogous11> to the

coherent potential _approximation) is successful for p:S1 over a substantial range

where 0" (p )/0" (1) is a linear function of p. This theory fails as P~Pe + and

the critical toe develops. By contrast, in the site problem there is an implicit

· correlation of absent bonds so that a single bond theory is inadequate everywhere,

failing even to give a correctly;11> More elaborate treatments are necessary to

remedy this.29>• 82>• 84> It is characteristic of the type of dilution; similar differences

are observed in comparisons of site and bond diluted Ising (or Heisenberg)

ferromagnets.4>• 6>' 47> However, an essentially single site effective medium theory82>

replaces the linear behaviour with a quadratic one which is again accurate out

side of a similarly small critical region.

An exact theory of the (general) random resistor network problem on

infinite Cayley tree cactii has recently become available27>• 28> (see also Ref. 47))

The bond and site cases are of course degenerate for these cactii. It is indeed

found that the exact result reduces in the high q limit to the linear effective

medium theory result (a feature in common with other Oth order mean field

theories80>). For finite q, deviations occur in a 'critical region' (see later) of

the form observed in real lattices. Outside this region (p~1-) deviations

from linearity are negligible. As in § 2, the critical fluctuations are limited in

a Bethe lattice so that the critical toe tends to be less pronounced than in real lattices and the critical exponent is different, equalying 2 in all Bethe lattices.

We now attempt to reinterpret these exact results to form a Bethe-Pe approximation for general lattices. For the moment we consider only the bond

problem. Interpreted as a function of Poe, the effective medium (EM) approximations in real and Bethe lattices can be written in a common form from Refs.

11) and 28):

where

O"(plM

()" (1) Po-PoeEM 1-P~eM '

EM { 2/q' Poe = 1/(q-1),

(real lattices)

(Bethe lattices).

(7a)

(7b)

(EM theory is exact for Be the lattices at p = 1 and Pe.28>) If the right-hand

side of (7) is interpreted in terms of real lattice values of Poe (assumed known), then it will be correct there and at p = 1 but giving only a linear interpolation

with none of the critical behaviour. However, if the exact Bethe lattice solution

is interpreted as a function of q = p;;/ + 1 for a particular lattice value of Poe, then we guarantee the same limits but also begin to develop critical characteristics

Dow



58 A. R. Bishop

Table II. Exact values of gradients of reduced conductivities in several random resistor networks as p--71, compared with values from Bethe lattice-based approximations: EM (p •• ) (from (7b)) and Bethe-p.. (Eq. (8)). In the latter case 'critical ranges' are given outside of which there is essentially linear behaviour, as described in § 3.

djdp[a(p)/a(1)], p--71- ' critical range' Lattice P•ca) EM(q) I I

in Bethe·P•• 'exact' EM(p •• ) Bethe-P••

Honeycomb 0.653 3.00 2.88 3.77 0.65-1.00 Square 0.500 2.00 2.00 2.40 0.50-0.83 Triangular 0.347 1.50 1.53 1.70 0.35~0.60 Diamond 0.388 2.00 1.63 1.85 0.39~.67 S.C. 0.247 1.50 1.33 1.40 0.25~.45 b.c.c. 0.179 1.33 1.22 1.26 0.18-0.33 f.c.c. 0.119 1.20 1.14 1.15 0.12~.23

a) 2-dimensional values are exact. 3-dimensional values are best series estimate.">,">

of the correct form. This will be termed the 'Be the-he approximation'. Although this approximation results in reasonable agreement (over the

entire concentration range), it will never be an exact procedure (except for Be the lattices) -the critical exponent is always too high so that we will find in 3-dimensional lattices a tendency to overestimate rJ(P)/rJ(1) unless p>pbc when it is underestimated. In the one 2-dimensional lattice studied experimentally (plane square) the critical toe seems to be extremely small,11l so that quite good approximations are provided by (7) (or its interpretation in terms of Pbc (real)).

In an attempt to quantify these remarks we have considered in detail the limiting gradients 1 +a, as pb-41-. The results are shown in Table II. Here the exact Bethe lattice gradient has been taken from the work of Stinchcombe28l in a form suiting our purpose:

= p~-1 a= (1+Pc)_E ·

n=2 1 +P~n-1 (8)

Monte Carlo experiments m square and simple cubic lattices11l confirm that EM theory (7a) (i.e., as a function of q) will give a accurately. The increasing discrepancy (Table II) between this and our Bethe-hc approximation occurring in lower co-ordination and two-dimensional lattices (i.e., higher Pbc) is numerically not as serious as it might seem, because the extent of the Bethe-Pbc 'critical region' (where deviations from linearity are important) is simultaneously increasing: from Ref. 28) we deduce this region to be O< pb -Pbc < (1 + p;;;,l)-1• The various critical ranges are displayed in Table_ II, and it is seen, for example, that in the extreme case of the honeycomb lattice this range dominates the entire concentration range. Thus the Bethe-hc approximation is also reasonable in 2-dimensional lattices, despite seriously misrepresenting the limiting gradient (c.£., § 2). This can be seen in Figs. 2 where the various approximations are compared with

Dow




1.0

Q8 , , '

1 ' '

'~ ,, 0,6

,, ,, BETHE~q --;;,:,'

= " ,, , Q4 ,":',' ".f " ~ r .. ,,-::/ ~ . ...,

Q2 " ,'/ ·EXACT /.'' /''~EM(q) , .. ~ ... ,

t Pt>c

0,4 0.6 0.8

Pb~

(a)

1.0

Q8

Q6

0.4

0.2

1.0

/:? EM(q) ,/</ EMY(Pt>cl / /! ,,

,// , , BETHE:-q / , /

~ / 1.../--EXACf , , , , , , ,/ ,l /

/, /,/ --_BETHE -Pbc , , , ,' ,, 0.4 t 0.6 0.8 1.0

Pt>c

(b)

Fig. 2. Comparison of various approximations (see § 3) with best available data"> for reduced conductivities in random resistor networks: Bethe-q (from Ref. 28); EM (q) and EM (p,,) (from (7)); Bethe-p,, (§ 3).

(a) Simple cubic lattice. (b) Plane square lattice.

As explained in the text these represent contrasting examples of the accuracy "'f the Bethe-p,, approximation.

available data for the bond problem on square and simple cubic lattices.

A contrasting situation arises when a Bethe-p,, interpretation is used for corresponding site problems. As mentioned earlier, the effective medium theory is never successful here but accurate low concentration (of missing sites) theories are available giving for the square lattice82l (Psc =0,587±.005) a=2.14 and for the simple cubic lattice11l (Psc = 0.310 ± .002) a= 1.52. Good experimental data is also available in these lattices for all p.

The Bethe-p .. theory gives a=2.06 and 0.58 respectively. Thus the 2-dimensional lattice behaviour is reproduced rather well-deviations from linearity occurring for p,$0.95 which is numerically consistent with the experimental data.82l (The agreement is of course only numerical; we emphasize again that there is no simple linear effective 'medium theory regime in the site problem.) However, the 3-dimensional result is rather poor, the Bethe-Psc prediction being far too linear (c.f., Refs. 10) and 11)).

The reas_on for this poor agreement probably lies in the distinction between bond and site problems stressed earlier. As we have noted, this difference is lost in Cayley tree topologies and the behaviour is rather bond-like if Pc <t becoming more site-like for p,;?:t (deduced from Ref. 28)). In fact, much of the discrepancy introduced in this way by our use of Bethe lattices can be removed by using a device due to Kirkpatrick.11l He has observed that the differences between the bond and site results are much less apparent if rJ (p )/ rJ(1) for the site case is plotted as a function of the fraction of occupied bonds (i.e., the .square of p,). This is found to result in a linear behaviour except for a small critical region both in the simple cubic11l and plane square82l lattices, with initial gradients 1.17 and 1.57 respectively. We can again use our Bethe-p, approach very easily and, taking p, (simple cubic)= 0.096 and p, (square)= 0.345, find that the suggested initial .gradients are 1.12 and 1.69 respectively. In this way very reasonable agreement is produced in both cases (because the effective values of Pc have been reduced), with a predominant overestimation in the simple cubic lattice and underestimation in the square one.

Dow



60 A. R. Bishop

§ 4. Spin-wave stiffness constant

A relationship' has recently been proposed between the long-wavelength spin-wave frequency spectrum in diluted Heisenberg ferromagnets and the site percolation probability and site random resistor network problem in corresponding arrays.10>-12> A lattice dependent quantity, the 'spin-wave stiffness constant' D (p.), provides the link. Well-defined spin-waves exist 1in diluted Heisenberg ferromagnets in the long-wavelength limit with frequencies reduced from. undiluted model values by the factor D (p.) .88> The same factor relates P. (p.) and the normalized conductivity in the random resistor network, with Ps the fraction of occupied sites on the same lattice :10>-12>

rJ(P.)/6(1) =D(p.)P.(p.). (9)

In (9) D (P.) has been normalized to D (1) and following previous sections we take D (p.c) = 0. Available data10>· 11> and mean field theories29>• 84> show that D(p.) is substantially linear in the simple cubic lattice for all p.>Psc exce,pt for a very small critical toe where data for 'rJ(P)/6(1) and P,(p,) have been tentatively interpreted10> to imply from (9) a power law behaviour with exponent "'1.1.

The exact solution for D (p) on Be the lattices has arisen in the work of Stinchcombe27>• 28> and exhibits just the characteristics noted above. Namely, an, extensive linear region (p<1) outside of which fairly weak deviations occur increasing as P----'>Pc+, at which concentration D(p) vanishes with unit exponent (for all Bethe lattices).

Use of the Bethe-Psc approximation in this case for the simple cubic lattice gives good agreement, numerically because of the common linear behaviour and the similarity of the critical exponents. Theory29> shows that the exact limiting gradient (Psc----'>1-) is 1.52, whereas the Bethe-Psc interpretation gives 1.58 taking Psc = 0.31210> or 1.57 with Psc = 0.307.81> Small deviations from linearity will be evident for p,:S0.6 and this is not inconsistent with the data. Thus, our approximation will slightly underestimate the exact result for all p, except perhaps for a very small range near Psc· However, very recent numerical work on other 3-dimensional lattices (S. Kirkpatrick and A. B. Harris, AIP Magnetism Conference, December 1974) reports much less lattice-dependence of the initial gradient than suggested by the Bethe-Prc interpretation. The latter presumably gives results for D(p,) that are rather too linear in these lattices.

It is well known that 2-dimensional Heisenberg Hamiltonians are unable- to support a spontaneous magnetization, but if D (p,) is defined by Eq. (9), comparison of limiting gradients in the square lattice [2.14 (exact) (deduced from Ref. 32) and 3.06 Bethe-Psc)] suggests that agreement will not be so good-although we should note that the critical region will be large here ( c.f., Table II). As in § 3 this is a result of the degeneracy of site and bond problems on Bethe lattices. Once again, more unified agreement is found if P, is measured as a function of

Dow




the concentration of occupied bonds.11l Incidentally, the exact values of initial gradient for D(p,) (2.14) and Psc (0.587) on the square lattice, taken together, imply one (or both) of two novel features; either D (p,) has a critical exponent less than unity or it has somewhat more structure for p, intermediate between Pse and 1 than suggested in the simple cubic lattice, for example. The critical exponent for 6' (p,)j 6' (1) is 1.~6~ 1.5032) and for P, (p,) it is certainly <i-19l and probably ;S0.311l but our knowledge is imprecise, as it is for P,(p,) itsel£.20)

Nevertheless, it is clear that the form of D(p,) on the square lattice will be worthy of further elucidation.

For the Heisenberg ferromagnet with random bond removal we can expect the corresponding spin-wave stiffness constant D(pb) to again have a characteristic linear regime f_or pb;Sl. Indeed, it is known that the exact limiting gradient is qj (q- 2) just as for 6' (h) /6' (1), Eq. (7a). Thus, the Bethe-hc interpretation underestimates the gradients in 3-dimensional lattices precisely as in Table II.

In the spirit of § 2 it might be interesting to investigate D(p) as an approximation to the phase separation curve Tc(P)/Tc(1) for the Heisenberg ferromagnet (in 3-dimensions). For instance, in a mean field approach we might expect qualitative success for the ratio Tc(P)/Tc(1) by taking Tc(P) ~long-wavelength spinwave energy. This can only be qualitatively useful; clearly for P""'-' 1 shorter wavelength modes are. less damped and more important. This is evident from the exact results for the limiting gradients in the randomly bond diluted case. The long-wavelength modes alone (i.e., D (Pil)) would give gradients in s.c., b.c.c. and f.c.c. lattices of 1.50, 1.33 and 1.20 respectively (Table II). However, recent series estimates (E. Brown, ]. W. Essam and C. M. Place, preprint, 1974) have revealed very extensive linear regimes in the true phase separation curves with gradients of 1.38 ± 0.01, 1.26 ± 0.02 and 1.16 ± 0.01 respectively, so that the critical deviations are very restricted. For this reason the Bethe-pb interpretation of D (Pb) 's gradient (Table II) agrees excellently with Tc(Pb)/Tc(1) but only because of this fortuitous combinatioh of circumstances. The shorter wavelength modes appear to have the same effect in site diluted cases; the initial gradient of the quadratic regime of T 0 (p.)/Tc(1) is depressed below that for D(p,) (S. Kirkpatrick and A. B. Harris, AlP Magnetism Conference, December 1974). The uncertainly near Pc is greater and (unlike in the Ising problems (§ 2)) the gradient of the (site or bond) Heisenberg critical curve at Pc is as yet obscure.

§ 5. Electron localization in cellularly disordered lattices

The concept of electrons being spatially localized in a sufficiently disordered system has been central to recent advances in our theoretical interpretations of such systems (see Ref. 35) for a comprehensive discussion). The simplest model exhibiting this phenomenon was originally introduced by Anderson,36l and in its most tractable form87)-42l refers to non-interacting electrons in a static lattice

Dow



62 A. R. BishoP

with constant overlap t between near-neighbour sites, all other overlaps being zero,*l and one atomic state per site. Disorder is introduced by allowing the site energies to be statistically independent and determined by a common probability distribution, whose width W measures this 'cellular' or 'diagonal' disorder.

Details of the theoretical techniques involved in treating this model need not concern us here. However, it is salient to point out that the retention of the underlying lattice structure leads to lattice statistics being important as in the other problems mentioned in this paper. Specifically, self-avoiding lattice paths enter centrally so that the connectivity K plays a dominant role in most current theories. The widely accepted prediction of these theories is that for W = 0 all states within the band are extended (i.e., not localized) but for every W =FO there are energies Ee (W) such that all states with eigenenergies in the tails of the band beyond E.(W) are localized and all other states (within the band) extended. As W increases it is believed that the pe:rcentage of localized states increases until at some critical (Anderson) disorder W. the whole band_ is localized. This is the notion of Ee (W) as a 'mobility edge function': 89l for a symmetric unperturbed band and probability distribution /Eel ~B, as W ~o + and /E./ ~o, as W ~ W.-, where B is the unperturbed half-bandwidth.

Quite recently a self-consistent localization theory has been developed for the preceding model that is exact on Bethe lattices.40l In the spirit of the present work it is appropriate to point out that the form of the mobility edge function found is similar to that anticipated more generally89l' 41l with the possible exception of certain long-tailed probability distributions.89l' 40l This is particularly evident in the parabolic form of E. (W) for small W and its sharp decline for W <We (especially for a rectangular probability distribution). It is therefore feasible that a scaling of the Bethe lattice Ee (W) results of the type employed elsewhere in this paper might yield reasonable approximations for other lattices.

The introduction of Pc does not have the same physical justification here as in our earlier problems because of the possibility of (quantum~mechanical) electron tunnelling effects, absent in the earlier classical problems. Thus using Pe is far '· more contentious. It occurs naturally in only one limit, that of a bimodal site-energy probability distribution (a primitive A-B alloy model) with the energy difference between modes tending to infinity (and thereby prohibiting A-B tunnelling, c.f., §§ 2, 3). In this situation Pre will equal the maximum concentration of say A atoms for which the whole of the A-subband is localized. In this way it can be shown40l• 42l that the exact selfconsistent Bethe lattice theory gives Pse = K- 1 ( = (q -1)-1) as it should (Eq. (2) of I. This should be compared with entirely analogous calculations for the diluted Ising ferromagnet43l). Clearly, then, interpreting the Bethe lattice theory in terms of (real lattice values of)

*> i.e., a tight-binding limit, c.f., the previous examples considered in this paper.

Dow




P.e according to this· Be the lattice relationship, will predict P.e correctly for all lattices and retain the exact Bethe lattice solution for any disorder.

In the absence of accepted quantitative computer simulation results, it is not at present possible to numerically assess the merits of a Bethe-P.e interpretation more generally--for example for the whole mobility edge function. However, it is intriguing to. note that the self-consistent Bethe lattice localization theory40l

suggests (in common with some strong correlation theories89l) that Ee~EB= (K +l)t, as W~O+, whereas B (Bethe) =2K112t. This curious result has been discussed briefly by Abou-Chacra and Thouless40l but here we will note that, since

EB/(qt) (Bethe) = (K+l)/q

( c1ompare the factor qt with Eq. (5) of II), our B'ethe-P,e philosophy (or any other lattice constant!) gives EB=qt=B (real lattice), i.e., the exact result on a real lattice. At the other end of the mobility ·edge function's range, use of the Bethe-p,e interpretation suggests42l interesting values for We in real lattices, resulting for instance in an approximate dimensional invariant character for We/B of a type typical in critical percolation contexts.11),19J Within the formalism of current th~ories it is clear87l'40l that important correlation effects (absent in Bethe lattices) act to severely reduce values of We for real lattices below values deduced by interpreting the Cayley tree results as functions of K. *l The Bethe-P.e interpretation apparently at least partially accommodates this effect because p;;/<K (in all real lattices).

We musf emphasize again, however, that no unequivocal evidence is available to test these suggestions. Suitable interpre.tation of existing numerical results41l' 44l

implies that the approximation continues to overestimate We in 2- and 3-dimensions. The work of Schonhammer and Brenig41l on a square lattice suggests this also for a range of critical disorders iWe:$W<We. [N.B. the form of the whole mobility edge is very sensitive to K for 'effective connectivities' :$2,40) which we now have for 2-dimensional lattices, e.g., K (effective, square lattice) ::::::::1.7. In the extreme limit K ~ 1, Ee (W) becomes a 'delta function' at W = 040l .] In a very recent and extensive review of electron localization theories in the Anderson and other models Thouless46l has emphasised important similarities and differences between the quantum mechanical electron localization problem and that of classical percolation theory.

§ 6. Discussion

Easily applied and broadly representative approximations for percolationinfluenced phenomena can be of considerable practical interest. For instance,

*> Essentially equivalent to the 'best' real lattice estimates of Anderson"> where these correlations were neglected. This use of K in the Bethe result is comparable with the derivation of the self-avoiding walk approximation in the Ising problem given in I.

Dow



64 A. R. Bishop

the Jahn-Teller transitions in certain mixed crystal vanadates of varying concentration such as TbpGd1_pVO, and DypY1_pVO, (O<P<l) can be adequately described by a pseudo-spin model which is sin1ilar to a diluted Ising model in a transverse field/4> so that knowledge of the transition temperature as a function of p is of direct relevance. , Short-ranged interactions (and consequently percolation effects) dominate in the Dy salt (but not the Tb salt where m~an field theory is quite successful). It will be interesting to pursue our interpretation to this problem and this is being considered. Random resistor networks are being used intensively in conjectured microscopic inhomogeneity models for certain disordered systems.45> Again, a better knowledge of the mobility edge function will be useful for example as an aid to interpreting computer estimates of critical disorders (Bishop, unpublished work), or providing a better estimate of the important concept of 'minimum metallic conductivity' introduced by Mott.36>• 46>

In this paper we have investigated the possibility of developing approximations in several contexts of this kind by a: phenomenological scaling of exact solutions (in the sense discussed in II) on Bethe lattices. We have concentrated attention on the critical percolation probability rather than other lattice constants such as connectivity. The results are encouraging within the limitations built into the approach-particularly inadequate estimates of critical exponents or distinction between bond and site problems. The latter limitation was found (§ 3) to be very evident in random resistor networks. A logical future development will be to extend the idea to more elaborate branching media (see I, for example). It would also be interesting to employ it in less obvious contexts, e.g., Ising models diluted with magnetic impurities15>. or more general random resistor networks,28> since these compare more nearly with general A-B alloy and glassy distribution problems, e.g., Refs. 8), 39), although the relevance of Pc is less clear here.

Acknowledgements

The author is indebted to Professor P.W. Anderson for much initial stimulation and to Professor R.]. Elliott and Dr. R.B. Stinchcombe for several enlightening discussions and critical comments.

He is also grateful to St. John's College, Oxford, for providing financial support during the course of this work.

References

1) A. R. Bishop, ]. Phys. C: Solid State Phys. 6 (1973), 2089 (I in the text); Prog. Theor. Phys. 52 (1974~, 1798 (II in the text).

2) H. Sato; A. Arrott and R. Kikuchi, ]. Phys. Chern. Solids 10 (1959), 19. 3) R. ]. Elliott and B. R. Heap, Proc. Roy. Soc. A 265 (1962), 264. 4) G. S, Rush brooke, R. A. Muse, R. L. Stephenson and K. Pirnie, J. Phys. C: Solid State

Phys. 5 (1972), 3371. 5) D. C. Rapaport, ]. Phys. C: Solid State Phys. 5 (1972), 1830.

Dow



A Use of Bethe Lattice Solutions for Percolation-Constrained Phen0mena 65

6) Ibid., 2813. 7) R. B. Griffiths, Phys. Rev. Letters 23 (1969), 17. 8) A. B. Harris, J. Phys. C: Solid State Phys. 7 (197 4), 1671. 9) R. J. Elliott, B. R: Heap, D. J. Morgan and G. S. Rushbrooke, Phys. Rev. Letters 5

(1960) , 366. 10) S. Kirkpatrick, Solid State Comm. 12 (1973), 1279. 11) S. Kirkpatrick, Rev. Mod. Phys. 45 (1973), 574. 12) B. J. Last, J. Phys. C: Solid State Phys. 5 (1972), 2805. 13) D. Kumar and A.. B. Harris, A. I. P. Con£. Proc. 5 (1972), 1345; Phys. Rev. B 8 (1973),

2166. 14) R. T. Harley, W. Hayes, A. M. Perry, S. R. P. Smith. R. J. Elliott and I. D. Saville,

J. Phys. C: Solid State Phys. 7 (1974), to appear. 15) S. Katsura and .F. Matsubara, Can. J. Phys. 52 (1974), 120.

F. Matsubara, Prog. Theor. Phys. 51 (1974), 378. 16) R. J. Elliott, Clarendon Lab. report (Oxford) 128/63 (1963). 17) A. B. Harris, J. Phys. C : Solid State Phys. 7 (197 4), 3082. 18) I. D. Saville, D. Phil. thesis, University of Oxford (1973). 19) V. K. S. Shante and S. Kirkpatrick, Adv. in Phys. 20 (1971), 325. 20) H. L. Frisch, J. M. Hammersley and D. J. A. Welch, Phys. Rev. 126 (1962), 949.

1 21) A. A. Lushnikov, Phys. Letters 27 A (1968), 158. 22) N. Matsudaira, Prog. Theor. Phys. 49 (1973). 1748. 23) K. Sawada and T. Osawa, Prog. Theor. Phys. 50 (1973), 1232. 24) J. W. Essam, in Phase Transitions and Critical Phenomena, Vol. 2, eds., C. Domb and

M. S. Green (Academic, 1972) . 25) S. Kirkpatrick, Phys. Rev. Letters 27 (1971), 1722. 26) B. J. Last and D .. J. Thouless, Phys. Rev. Letters 27 (1971), 1719. 27) R. B. Stinchcombe, J. Phys. C: Solid State Phys. 6 (1973), Ll. 28) R. B. Stinchcombe, J. Phys. C: Solid State Phys. 7 (1974), 179. 29) Y. Izyumov, Proc. Phys. Soc. 87 (1966), 505. 30) e.g., C. Domb, Adv. in Phys. 9 (1960), 149.

R. Brout, Phase Transitions (Benjamin, 1965). L. Schwartz and E. Siggia, Phys. Rev. B 5 (1972), 383.

31) M. F. Sykes and J. W. Essam, Phys. Rev. 133A (1964), 310. 32) B. P. Watson and P. L. Leath, Phys. Rev. B9 (1974), 4893. 33) S. F. Edwards and R. C. Jones, J. Phys. C :. Solid State Phys. 4 (1971), 2109. 34) A. B. Harris, P. L. Leath, B. Nickel and R. J. Elliott, J. Phys. C: Solid State Phys. 7

(197 4), 1693. 35) N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Oxford,

1971). 36) P. W. Anderson, Phys. Rev. 109 (1958), 1492; Proc. Natl. Acad. Sci. 69 (1972), 1097. 37) D. J. Thouless, J. Phys. C: Solid State Phys. 3 (1970), 1559. 38) E. N. Economou and M. H. Cohen, Phys. Rev. B 5 (1972), 2931. 39) A. R. Bishop, Phil. Mag. 27 (1973), 651, 1489; Phys. Letters 49A (197 4), 5; Solid State

Comm. 15 (1974), 1447. . 40) R. Abou-Chacra, P. W. Anderson and D. J. Thouless, J. Phys. C: Solid State Phys. 6

(1973)' 1734. R. A. Abou-Chacra and D. J. Thouless, J. Phys. C: Solid State Phys. 7 (1974), 65.

41) K. Schonhammer and W. Brenig; Phys. Letters 42A (1973), 447. 42) A. R. Bishop, Ph. D. thesis, University of Cambridge (1973). 43) c.f., C. Domb, J. Phys. C: Solid State Phys. 3 (1970), 256. 44) K. E. Khor and P. V. Smith, J. Phys. C: Solid State Phys. 4 (1971), 2029.

J. T. Edwards and D. J. Thouless, J. Phys. C: Solid State Phys. 5 (1972), 807. B . .J. Last and D. J. Thouless, J. Phys. C: Solid State Phys. 7 (1974), 715.

45) seeM. H. Cohen and J. Jortner, Phys. Rev. Letters 30 (1973), 696, 699. 46) D. J. Thouless, Physics Reports, to be published. 47) J. W. Essam, C. M. Place and E. H. Sondheimer, J. Phys. C: Solid State Phys. 7 (1974),

L258.

Dow



senhora da limpeza atende utentes na extens£o de sade

Documents