University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 5-1-1987 Potts-Model Formulation of the Random Resistor Network A. Brooks Harris University of Pennsylvania, [email protected] Thomas C. Lubensky University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Physics Commons Recommended Citation Harris, A., & Lubensky, T. C. (1987). Potts-Model Formulation of the Random Resistor Network. Physical Review B, 35 (13), 6987-6996. http://dx.doi.org/10.1103/PhysRevB.35.6987 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/307 For more information, please contact [email protected]. Potts-Model Formulation of the Random Resistor Network Abstract The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in ε=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as φ1=1+ε/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)ν+φ1, where ν is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving φ=1 to all orders in ε. The new result for φ1 removes the paradox associated with the numerical result that t>1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the ε expansion. Disciplines Physics This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/307 PHYSICAL REVIEW B VOLUME 35, NUMBER 13 1 MAY 1987 Potts-model formulation of the random resistor network A. B. Harris and T. C. Lubensky Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396 (Received 7 August 1986) The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in a=6—d, where d is the spatial dimensionality. The recur- sion relations are shown to be the same as for the xy model. Their detailed analysis (given in the ac- companying paper) gives the resistance crossover exponent as P~ —1+ /e42, and determines the criti- cal exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t =(d —2)v+P&, where v is the critical exponent for the corre- lation length at the percolation threshold. These results correct previously accepted results giving /=1 to all orders in e Th.e new result for P& removes the paradox associated with the numerical re- sult that t & 1 for d =2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the e expansion. I. INTRODUCTION proper context is given in the introduction of the accom- panying paper, and will not be repeated here. However, The properties of the randomly diluted resistor network some comments specific to the technique based on the has received considerable attention over the past several Potts model are in order. In the original analysis of years. ' In the usual model one associates a finite Kasteleyn and Fortuin' and in our succeeding work, use nonzero conductance o with occupied bonds and zero has been made of the connection between the resistance conductance with unoccupied bonds. Each bond is ran- between two terminals in a network and the generating domly occupied with probability p and unoccupied with function for spanning trees on that network. Of course, in probability 1 —p. Consider the resistance R (x,x') be- the presence of random dilution, one obtains the proper tween two terminals at the sites x and x'. If the sites x quenched average by considering the n~O limit of the and x are in different clusters, this quantity will be infi- appropriate n-replicated Hamiltonian. The most impor- nite. Even if the two sites are in the same cluster, R(x,x') tant modification of the development from that presented ' will still be described by a (conditional) probability distri- by Kasteleyn and Fortuin, is to replace the v s in their bution. For p near the percolation threshold, p„ this dis- formulation by suitable limiting operations on s and on tribution function may be characterized by an infinite set the coupling constant J appearing in the Hamiltonian. of crossover exponents Pk, k =1, 2, 3, . In mean- These changes facilitate the development of a field- field theory, i.e., for spatial dimension d & 6, Pk = 1 for all theoretic Hamiltonian upon which the appropriate limit- k. For d =6—e & 6 these exponents may be calculated in ing processes can readily be performed. two apparently very different formalisms. ' In the ac- Briefly, this paper is organized as follows. In Sec. II we companying paper" we give the details of such calcula- discuss in some detail the formalism whereby the ap- tions' based on consideration of a random xy model, fol- propriate quenched average of the arbitrary moments of ' lowing the technique introduced by Stephen. Here we the resistance between two terminals can be obtained us- describe the calculations based on a modification of the ing the replica formalism for the s-state Potts model. In connection found by Kasteleyn and Fortuin' between the particular, we show that it is important to take the limits resistor network and the s~O limit of the s-state Potts in the order, n~O first, then s~O, and finally J~ao. model. We show that the renormalization-group recur- In Sec. III this model is converted into a field theory us- sion relations obtained using the Potts model are identical ing a Hubbard-Stratonovich transformation. The recur- to those obtained and fully analyzed in the accompanying sion relation from which the crossover exponents are ob- paper for the xy model. To avoid undue repetition their tained are derived in Sec. IV and are shown to be analysis is not reproduced here. The results of this paper equivalent, for J~oo to those in the accompanying pa- were summarized previously. ' per, " where their consequences are analyzed in detail. Fi- A brief discussion placing these calculations in the nally, Sec. V contains discussion and concluding remarks. 35 6987 1987 The American Physical Society 6988 A. B. HARRIS AND T. C. LUBENSKY 35 II. FORMULATION Since we will be interested in the partition function and thermal correlation functions of this Potts model, we note In this section we derive the Potts-model Hamiltonian the following relations from which one can obtain the resistance between two sites in a random resistor network. This formulation is a (2.4a) modification in two respects of that given originally by ' *'* — Kasteleyn and Fortuin. First of all, we introduce the re- =e '*[1+(e l )6,(,) ~„)]. (2.4b) plicated Hamiltonian to treat the disorder in the quenched Now we consider some arbitrary assembly I of conduc- resistor network. our treatment the s-state Secondly, of tances. In general, this network will consist of indepen- Potts model in the limit s~0 is more suitable for setting dent clusters (denoted y), y2, . , etc.) of sites, such that a field than was the original formulation. up theory voltages and currents in one cluster are completely in- We start by discussing the Hamiltonian of a Potts dependent of those in other clusters. This situation is ob- model in which there is associated with each site x a set of viously relevant for a discussion of the randomly dilute vectors [e':r=1,2, . , s} corresponding to the s dif- — resistor network. The Potts model associated with this ferent directions from the center of the (s 1)- network can be written as dimensional simplex (or regular multihedron) to its s ver- tices. For later convenience these vectors are normalized H(l )= g H(y), (2.5) to have length (s —1)'~ and therefore obey the relations' year where the Hamiltonian for the cluster is e e =s6, —1, (2. 1) y = — o„„e"").e~" where 6 is the Kronecker delta. The interaction between H(y) J g ', (2.6) &x, x'& ey two sites x and x', denoted h „„is given by where (x,x') Hy means that the bond connecting site x (2.2) and x is in the set y. For an isolated site, i.e.