Self-averaging in the random 2D Ising ferromagnet

TL;DR

This paper investigates sample-to-sample fluctuations in a 2D disordered Ising model, showing that internal energy fluctuations are Gaussian and scale with system size, while the specific heat is self-averaging and becomes sharply peaked as system size grows.

Contribution

The study provides explicit expressions for the distribution of internal energy and specific heat fluctuations in the disordered 2D Ising model, highlighting their different self-averaging behaviors.

Findings

Internal energy fluctuations are Gaussian and scale as L ln ln L.

Specific heat is self-averaging and tends to a delta function.

Results are directly relevant for experimental and simulation measurements.

Abstract

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size $L$ like $\sim L \ln\ln(L)$. In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a $\delta$-peak in the thermodynamic limit $L \to \infty$. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.