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Percolation in Negative Field and Lattice Animals University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 1-1-1989 Percolation in Negative Field and Lattice Animals Yigal Meir Amnon Aharony A. Brooks Harris University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Physics Commons Recommended Citation Meir, Y., Aharony, A., & Harris, A. (1989). Percolation in Negative Field and Lattice Animals. Physical Review B, 39 (1), 649-656. http://dx.doi.org/10.1103/PhysRevB.39.649 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/377 For more information, please contact [email protected]. Percolation in Negative Field and Lattice Animals Abstract We study in detail percolation in a negative ‘‘ghost’’ field, and show that the percolation model crosses over, in the presence of a negative field h, to the lattice-animal model, as predicted by the field theory. This was done by exact solutions in one dimension and on a Cayley tree, and series expansions in general dimension. We confirm the scaling picture near the percolation threshold, and study the extended scaling ansatz for all values of h in terms of the nonlinear scaling field gh. Estimates for gh are obtained as a function of h in all dimensions. We also show how information on percolation clusters in all concentrations up to the percolation threshold may be extracted by studying the critical behavior of the generalized susceptibilities χk(p,h) near their critical point pc(h) as a function of h, and obtain data on the cluster distribution function and on the ratio of perimeter bonds to cluster bonds, for large clusters for all 0≤p≤pc. The crossover function is studied in one dimension, mean-field theory and the ε expansion. Disciplines Physics This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/377 PHYSICAL REVIEW 8 VOLUME 39, NUMBER 1 1 JANUARY 1989 Percolation in negative field and lattice animals Yigal Meir and Amnon Aharony Raymond and Beverly Sackier Faculty of Exact Sciences, School of Physics and Astronomy, Tel Auiu University, Tel Aviv 69978, Israel A. Brooks Harris University of Pennsylvania, Philadelphia, Pennsyluania 19104 (Received 8 June 1988) We study in detail percolation in a negative "ghost" field, and show that the percolation model crosses over, in the presence of a negative field h, to the lattice-animal model, as predicted by the field theory. This was done by exact solutions in one dimension and on a Cayley tree, and series ex- pansions in general dimension. We confirm the scaling picture near the percolation threshold, and study the extended scaling ansatz for all values of h in terms of the nonlinear scaling field gI, . Esti- mates for gh are obtained as a function of h in all dimensions. We also show how information on percolation clusters in all concentrations up to the percolation threshold may be extracted by study- ing the critical behavior of the generalized susceptibilities gk(p, h) near their critical point p, (h) as a function of h, and obtain data on the cluster distribution function and on the ratio of perimeter bonds to cluster bonds, for large clusters for all 0&p & p, The crossover function is studied in one dimension, mean-field theory and the e expansion. I. INTRODUCTION lnB (n, p) — n— (1.5a) The statistics of clusters on diluted lattices have at- tracted much attention in the last decade. In the site- which leads to' lattice-animal model' each site is assigned a fugacity K, — and the number A„of clusters (animals) with n sites, 9( ) A (p) (1.5b) scales, for large n, as 3„—n 'K,", For p ~0 the cluster numbers B (n, p) reduce to A„p", so one expects 0(0)=0, and A (p~o)= —ln(p)+0(1). where K, is a nonuniversal, lattice-dependent constant As one might expect from universality arguments, 0(p) and 0, is the animal critical exponent. In the site- is not a continuous function of p, but rather 0(p) =0, for percolation model' each site is present with probability — all p (p, and 0(p, )=r . Indeed, this is the outcome of p and absent with probability q =1 p. Although the the field theory constructed by Harris and Lubensky two models were defined above as site models, equivalent (HL) and real-space renormalization-group studies. It definitions apply for the bond models. From universality has not been easy to numerically verify this discontinuous one expects the same critical behavior for both models, behavior. Monte Carlo methods can hardly help, since and a difference between the two kinds of models is ex- for p &p, the number of large clusters is exponentially pected only when dealing with nonuniversal quantities, small. For instance, Bauchspiess and Stauffer used the such as K, . available perimeter polynomials' to study the distribu- Near the percolation threshold p„ the number per site, tion of clusters in the whole range 0 &p &p, . They found B (n, p), of clusters with n sites, scales as that 0(p) changes continuously from 0(0) =0, to ) but they did not rule out the possibility of the — (1.2) 0(p, =r, B(n,p)=n 'f(n ~ '), ~p, p discontinuity we now believe to occur. In fact, as we we here is where 7p and lmLlp are related to the order-parameter ex- shall see, the series analysis present perhaps ponent P and the susceptibility exponent y, of percola- the best numerical evidence currently available for the tion, via discontinuous scenario. For such an analysis it is convenient to define the sus- (1.3a) ceptibilities gk (p, h ) as (1.3b) y„(p, h)= n "B(n,p)e (1.6) At the percolation threshold Eq. (1.2) reduces to n=0g T B(n, )-n (1.4) p, h here is the usual ghost field introduced in the percola- ' while below p, it was shown rigorously ' that tion problem. Using (1.5b) we find 39 649 1989 The American Physical Society 650 YIGAL MEIR, AMNON AHARONY, AND A. BROOKS HARRIS 39 (1 —5 )/5 yk(p, h) —[ A (p)+h] Comparing with (1.13) we find dg, /dh -g„ The crossover from percolation to animals is described (1.7) as follows within the field theory. The model Hamiltoni- At h=0 Eq. (1.2) implies that yk(0)=y~+(k —2)b, . an contains three independent parameters, which can be expressed in terms of h, and the bond percolation For h & 0 these quantities all diverge at p, (h), where p, (h ) p, q, variables, where the probability for a bond to be va- is the solution of 3 [p, (h)] = —h. In the limit p ~0, one q, again uses 8 (n, p) —A„p" to find the dominant singular cant, is taken to be independent of p. HL showed that in behavior: this parameter space there is a critical surface on which — there are three nontrivial fixed points describing, respec- r(k e. +1) tively, percolation, lattice animals, and the theta point of (1.8) k+I branching polymers. If we vary at h=O we intersect (1 — e ") p pK, the percolation fixed point which is unstable in both the field con- where I is the usual gamma function. Thus, Xk (p, h ) thermal (i.e., p) and directions, as usual for a diverge in this limit at p, (h) =exp(h)/K„with exponents tinuous phase transition. If we vary p for h & 0 we hit a critical line (h) from which we fiow to the animal yk ——k+1 —O„and the gap exponent, defined as the p =p, fixed point. The theta point requires to not be equal to difference between yk and yk+1, is unity for the animal q " 1 — and hence we need not consider this point here. problem. For h=O the gap exponent is given by Eq. p, — (1.3a). The q =1 p plane of this three-dimensional phase space The scaling form (1.2) for 8 (n, p) implies, with is displayed in Fig. 1. In this two-dimensional phase space the critical surface appears as a critical line t =(p, p)/p, , p, (h), where the susceptibilities yk(p, h), defined in Eq. (1.6), —y —(k —2)A gk(p, h)=t Fk( ht )— (1.9) diverge. We see that for p (p, the critical behavior is determined by the animal fixed point, with exponents in- which should hold in the vicinity of the percolation criti- dependent of the percolation exponents. This also im- is a first-order cal point. Equation (1.9) only approxima- plies that (1.11) is obeyed for all h&0 near the appropri- form' ' ' tion in terms of t and h of the exact scaling ate threshold. Some aspects of this phase space were in- — — — y (k 2)h dependently confirmed by real-space renormalization- xk(p h) g +k(gh/g (1.10) group studies. in terms of g, and gz, the nonlinear scaling fields for As was mentioned above, information on larger clus- which the recursion relations of the renormalization ters below the percolation threshold is rather hard to ex- ' simulations. will group become exactly linear. g, and g& reduce in linear tract by computer We demonstrate order to t and —h, respectively, and Eq. (1.10) is correct below how such information can be obtained by studying to corrections due to irrelevant variables. The extend- percolation in the presence of a negative field. In particu- up ' ed scaling ansatz' implies that the scaling relation (1.10) lar we discuss the perimeter ratio, the ratio of the num- holds for any h, near the appropriate threshold p, (h): ber of perimeter sites, n, to the number of cluster sites.
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