Geometrical clusters in two-dimensional random-field Ising models

TL;DR

This study investigates the geometrical clusters in the 2D random-field Ising model, revealing a critical point where cluster size diverges and showing that critical correlations exhibit conformal invariance.

Contribution

It demonstrates the relation between the divergence of the correlation length and tricritical percolation, and confirms conformal invariance of critical geometrical correlations.

Findings

Finite cluster size for $\Delta > \Delta_c$

Divergent cluster size at $\Delta \\le \\Delta_c$

Conformal invariance of critical correlations

Abstract

We consider geometrical clusters (i.e. domains of parallel spins) in the square lattice random field Ising model by varying the strength of the Gaussian random field, $\Delta$. In agreement with the conclusion of previous investigation (Phys. Rev. E{\bf 63}, 066109 (2001)), the geometrical correlation length, i.e. the average size of the clusters, $\xi$, is finite for $\Delta > \Delta_c \approx 1.65$ and divergent for $\Delta \le \Delta_c$. The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand the divergence of the correlation length, $\xi(\Delta) \sim (\Delta - \Delta_c)^{-\nu}$, with $\nu \approx 2.$ is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.