## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

10-1-1989

#### Publication Source

Physical Review B

#### Volume

40

#### Issue

10

#### Start Page

7230

#### Last Page

7238

#### DOI

10.1103/PhysRevB.40.7230

#### 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 σ_{0}^{−1}, where σ_{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*clearly does not depend on σ

_{c}_{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.

#### Recommended Citation

Harris, A.,
&
Aharony, A.
(1989).
Phase Diagrams for the Randomly Diluted Resistor Network and *XY* Model.
*Physical Review B,*
*40*
(10),
7230-7238.
http://dx.doi.org/10.1103/PhysRevB.40.7230

**Date Posted:** 12 August 2015

This document has been peer reviewed.