Phase Diagrams for the Randomly Diluted Resistor Network and XY Model

A. Brooks Harris, Amnon Aharony A

Published Oct 1, 1989• 0 citations

Abstract

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.

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Phase Diagrams for the Randomly Diluted Resistor Network and XY Model

A. Brooks Harris, Amnon Aharony A

Published Oct 1, 1989• 0 citations

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Abstract

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Phase diagrams for the randomly diluted resistor network andXYmodel

A. Brooks Harris, Amnon Aharony A

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.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.

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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 ${p}_{c}$ . 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 [ ${]}_{\mathrm{av}}$ 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 ${C}_{R}$ \ifmmode\bar\else\textasciimacron\fi{} $^{(l)}$ (x,x')\ensuremath{\sim}\ensuremath{\Vert}x- for p near ${p}_{c}$ , 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 . .AEWe 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 ${p}_{c}$. 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 [ ${]}_{\mathrm{av}}$ 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 ${C}_{R}$\ifmmode\bar\else\textasciimacron\fi{} $^{(l)}$(x,x')\ensuremath{\sim}\ensuremath{\Vert}x- for p near ${p}_{c}$, 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

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{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.

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

{p}_{c}

. 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 [

{]}_{\mathrm{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

{C}_{R}

\ifmmode\bar\else\textasciimacron\fi{}

^{(l)}

(x,x')\ensuremath{\sim}\ensuremath{\Vert}x- for p near

{p}_{c}

, 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

d=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.