Correlation length-exponent relation for the two-dimensional random Ising model

TL;DR

This study investigates the correlation length behavior in the 2D random Ising model on a diagonal strip, revealing universal ratios and properties of surface correlations at criticality.

Contribution

It introduces an iterative star-triangle transformation method to accurately determine correlation lengths and demonstrates universality and self-averaging in the 2D dilute Ising model.

Findings

The ratio of correlation length to strip width approaches 2/π for large widths.

Surface correlation functions are self-averaging and conformally covariant at criticality.

Decay exponent of surface correlations is η_parallel=1.

Abstract

We consider the two-dimensional (2d) random Ising model on a diagonal strip of the square lattice, where the bonds take two values, $J_1>J_2$, with equal probability. Using an iterative method, based on a successive application of the star-triangle transformation, we have determined at the bulk critical temperature the correlation length along the strip, $\xi_L$, for different widths of the strip, $L \le 21$. The ratio of the two lengths, $\xi_L/L=A$, is found to approach the universal value, $A=2/\pi$ for large $L$, independent of the dilution parameter, $J_1/J_2$. With our method we have demonstrated with high numerical precision, that the surface correlation function of the 2d dilute Ising model is self-averaging, in the critical point conformally coovariant and the corresponding decay exponent is $\eta_{\parallel}=1$.