Critical Behaviour of the Randomly Spin--Diluted 2-d Ising Model --- A Grand Ensemble Approach

TL;DR

This paper investigates the critical behavior of the 2D randomly spin-diluted Ising model using a novel grand ensemble approach combined with phenomenological renormalization, revealing continuous variation of critical exponents with impurity density.

Contribution

It introduces a new method combining grand ensemble and renormalization group techniques to analyze disordered systems, providing detailed phase diagrams and critical exponents for the diluted Ising model.

Findings

Critical exponents vary continuously with impurity density.

Weak universality observed with some exponents independent of impurity density.

Results align with recent Monte Carlo simulations.

Abstract

The critical behaviour of the randomly spin-diluted Ising model in two space dimensions is investigated by a new method which combines a grand ensemble approach to disordered systems proposed by Morita with the phenomenological renormalization group scheme of Nightingale. Accurate approximations for the phase diagram and for the connectivity length exponent of the percolation transition are obtained. Our results suggest that the thermal phase transition of the disordered system might be different from that of the pure system: we observe a continuous variation of critical exponents with the density $\rho$ of magnetic impurities, respecting, however, weak universality in the sense that $\eta$ and $\gamma/\nu$ do {\it not\/} depend on $\rho$ while $\gamma$ and $\nu$ separately do. Our results are in qualitative and quantitative agreement with a recent Monte--Carlo study.