Correlation functions in the two-dimensional random-field Ising model

TL;DR

This study uses transfer-matrix methods to analyze the probability distributions of spin-spin correlation functions in the 2D random-field Ising model, revealing temperature and field effects on distribution shapes and scaling behaviors.

Contribution

It provides analytical and numerical insights into the distribution of correlation functions and their scaling properties near criticality in the 2D RFIM.

Findings

Distributions are singly-peaked at moderate T, asymmetric.

At low T, distributions transition to a double-delta structure.

Estimated scaling exponent y=0.875 ± 0.025 aligns with theoretical predictions.

Abstract

Transfer-matrix methods are used to study the probability distributions of spin-spin correlation functions $G$ in the two-dimensional random-field Ising model, on long strips of width $L = 3 - 15$ sites, for binary field distributions at generic distance $R$, temperature $T$ and field intensity $h_0$. For moderately high $T$, and $h_0$ of the order of magnitude used in most experiments, the distributions are singly-peaked, though rather asymmetric. For low temperatures the single-peaked shape deteriorates, crossing over towards a double-$\delta$ ground-state structure. A connection is obtained between the probability distribution for correlation functions and the underlying distribution of accumulated field fluctuations. Analytical expressions are in good agreement with numerical results for $R/L \gtrsim 1$, low $T$, $h_0$ not too small, and near G=1. From a finite-size {\it ansatz} at $T=T_c (h_0=0)$, $h_0 \to 0$, averaged correlation functions are predicted to scale with $L^y h_0$, $y =7/8$. From numerical data we estimate y=0.875 \pm 0.025$, in excellent agreement with theory. In the same region, the RMS relative width $W$ of the probability distributions varies for fixed $R/L=1$ as $W \sim h_0^{\kappa} f(L h_0^u)$ with $\kappa \simeq 0.45$, $u \simeq 0.8$ ; $f(x)$ appears to saturate when $x \to \infty$, thus implying $W \sim h_0^{\kappa}$ in $d=2$.