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Physical review. B, Condensed matter

Phase diagrams for the randomly diluted resistor network andXYmodel

A. Brooks Harris, Amnon Aharony A

Published Oct 1, 1989• 4 citations
Publisher

Abstract

The randomly diluted resistor network and XY model at low temperature T are studied near the d-dimensional percolation threshold using the \ensuremath{\varepsilon} expansion, where \ensuremath{\varepsilon}=6-d. The series expansion of the inverse susceptibility in powers of T for the XY model is identical to that of the appropriate resistive inverse susceptibility in powers of ${\ensuremath{\sigma}}_{0}^{\mathrm{\ensuremath{-}}1}$, where ${\ensuremath{\sigma}}_{0}$ is the conductance of a bond. However, the temperature-dependent critical concentration ${p}_{c}$(T) for the XY model has no analog in the resistor network, where ${p}_{c}$ clearly does not depend on ${\ensuremath{\sigma}}_{0}$. This distinction arises from a rather subtle difference between the Fourier component representation of the Gaussian model for the resistor network and that of the bounded potential energy associated with the XY model. We introduce a family of models which provides a smooth interpolation between these two models and show that the phase boundary for the XY model satisfies certain simple self-consistency checks involving other susceptibilities. In particular we provide the first explicit calculation of the universal crossover function to finite temperature of the dilute XY model.

DOI
10.1103/physrevb.40.7230

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The randomly diluted resistor network and XY model at low temperature T are studied near the d-dimensional percolation threshold using the \ensuremath{\varepsilon} expansion, where \ensuremath{\varepsilon}=6-d. The series expansion of the inverse susceptibility in powers of T for the XY model is identical to that of the appropriate resistive inverse susceptibility in powers of \ensuremathσ0\ensuremath−1{\ensuremath{\sigma}}_{0}^{\mathrm{\ensuremath{-}}1}\ensuremathσ0\ensuremath−1​, where \ensuremathσ0{\ensuremath{\sigma}}_{0}\ensuremathσ0​ is the conductance of a bond. However, the temperature-dependent critical concentration pc{p}_{c}pc​(T) for the XY model has no analog in the resistor network, where pc{p}_{c}pc​ clearly does not depend on \ensuremathσ0{\ensuremath{\sigma}}_{0}\ensuremathσ0​. This distinction arises from a rather subtle difference between the Fourier component representation of the Gaussian model for the resistor network and that of the bounded potential energy associated with the XY model. We introduce a family of models which provides a smooth interpolation between these two models and show that the phase boundary for the XY model satisfies certain simple self-consistency checks involving other susceptibilities. In particular we provide the first explicit calculation of the universal crossover function to finite temperature of the dilute XY model.
The randomly diluted resistor network and XY model at low temperature T are studied near the d-dimensional percolation threshold using the ɛ expansion, where ɛ=6-d. The series expansion of the inverse susceptibility in powers of T for the XY model is identical to that of the appropriate resistive inverse susceptibility in powers of σ0−1, where σ0 is the conductance of a bond. However, the temperature-dependent critical concentration pc(T) for the XY model has no analog in the resistor network, where pc clearly does not depend on σ0. This distinction arises from a rather subtle difference between the Fourier component representation of the Gaussian model for the resistor network and that of the bounded potential energy associated with the XY model. We introduce a family of models which provides a smooth interpolation between these two models and show that the phase boundary for the XY model satisfies certain simple self-consistency checks involving other susceptibilities. In particular we provide the first explicit calculation of the universal crossover function to finite temperature of the dilute XY model.
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 \ensuremath{\epsilon}=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 ${\ensuremath{\varphi}}_{1}$=1+\ensuremath{\epsilon}/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)\ensuremath{\nu}+${\ensuremath{\varphi}}_{1}$, where \ensuremath{\nu} is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving \ensuremath{\varphi}=1 to all orders in \ensuremath{\epsilon}. The new result for ${\ensuremath{\varphi}}_{1}$ 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 \ensuremath{\epsilon} expansion.
An approach by Stephen [Phys. Rev. B 17, 4444 (1978)] is used to investigate the critical properties of randomly diluted resistor networks near the percolation threshold by means of renormalized field theory. We reformulate an existing field theory by Harris and Lubensky [Phys. Rev. B 35, 6964 (1987)]. By a decomposition of the principal Feynman diagrams, we obtain diagrams which again can be interpreted as resistor networks. This interpretation provides for an alternative way of evaluating the Feynman diagrams for random resistor networks. We calculate the resistance crossover exponent phi up to second order in epsilon=6-d, where d is the spatial dimension. Our result phi=1+epsilon/42+4epsilon(2)/3087 verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts-model formulation of the random resistor network.
We consider the critical properties of the two-point resistance and its fluctuations due to microscopic noise in a randomly diluted resistor network near the percolation threshold pc{p}_{c}pc​. We introduce a n\ifmmode\times\else\texttimes\fi{}m replicated Hamiltonian in order to treat separately the configuration average over the randomly occupied bonds denoted [ ]av{]}_{\mathrm{av}}]av​ and the average over probability distribution function of the fluctuating microscopic bond conductance, denoted { ${\mathrm{}}}_{f}$. We evaluate a family of exponents {${\ensuremath{\psi}}_{l}$} (l=2,3,. . .) whose values are 1+O(\ensuremath{\epsilon}) with \ensuremath{\epsilon}=6-d where d is the spatial dimensionality. Each ${\ensuremath{\psi}}_{l}$ governs the critical behavior of the lth cumulant of the resistance between the sites x,x' conditionally averaged subject to the sites being in the same cluster such that CR{C}_{R}CR​\ifmmode\bar\else\textasciimacron\fi{} (l)^{(l)}(l)(x,x')\ensuremath{\sim}\ensuremath{\Vert}x- for p near pc{p}_{c}pc​, where ${\ensuremath{\nu}}_{p}$ is the correlation-length exponent for percolation. Furthermore, ${\ensuremath{\psi}}_{2}$=1+\ensuremath{\epsilon}/105 determines the dependence of the variance of the resistance in a finite network on size L as . .AE
We consider the critical properties of the two-point resistance and its fluctuations due to microscopic noise in a randomly diluted resistor network near the percolation threshold pc. We introduce a n×m replicated Hamiltonian in order to treat separately the configuration average over the randomly occupied bonds denoted [ ]av and the average over probability distribution function of the fluctuating microscopic bond conductance, denoted { }f. We evaluate a family of exponents {ψl} (l=2,3,. . .) whose values are 1+O(ε) with ε=6-d where d is the spatial dimensionality. Each ψl governs the critical behavior of the lth cumulant of the resistance between the sites x,x’ conditionally averaged subject to the sites being in the same cluster such that C¯R (l)(x,x’)~‖x-x'‖ψl/vp for p near pc, where νp is the correlation-length exponent for percolation. Furthermore, ψ2=1+ε/105 determines the dependence of the variance of the resistance in a finite network on size L as Lψ2/vp.
We present numerical data and scaling theories for the critical behavior of random resistor networks near the percolation threshold. We determine the critical exponents of a suitably defined resistance correlation function by a Pad\'e analysis of low-concentration expansions as a function of dimensionality. We verify that d=6d=6d=6 is the critical dimensionality for the onset of mean-field behavior. We use the coherent-potential approximation to construct a mean-field scaling function for the critical region.