Modified Scaling Relation for the Random-Field Ising Model

TL;DR

This paper studies the critical behavior of the 3D random-field Ising model at low temperatures, proposing a modified scaling relation for the specific heat exponent and numerically estimating critical exponents.

Contribution

It introduces a modified scaling relation for the specific heat exponent at zero temperature and verifies it through numerical simulations of the 3D random-field Ising model.

Findings

Modified Rushbrooke equation at T=0: α + 2β + γ = 1

Numerical estimates of critical exponents: ν ≈ 1.0, β ≈ 0.05, γ̄ ≈ 2.9, γ ≈ 1.5, α ≈ -0.55

Validation of the modified scaling relation through finite size scaling analysis.

Abstract

We investigate the low-temperature critical behavior of the three dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature $T \to 0$ the usual scaling relations have to be modified as far as the exponent $\alpha$ of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to $\alpha + 2 \beta + \gamma = 1$, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three dimensional random field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite size scaling analysis we calculate the critical exponents $\nu \approx 1.0$, $\beta \approx 0.05$, $\bar{\gamma} \approx 2.9$ $\gamma \approx 1.5$ and $\alpha \approx -0.55$.