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 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.

Oct 1, 19890 citations

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.

Physical review. B, Condensed matterOct 1, 19894 citations

Critical properties of the two-dimensional XY model

Douglas M. Lublin

Jan 1, 19763 citations

Dynamics of the anisotropic two-dimensional XY model

L.S. Lima, Antonio S. T. Pires

The European Physical Journal BJul 4, 200924 citations

{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.