Lack of self-averaging of the critical internal energy in a weakly-disordered Baxter model

TL;DR

This study examines the non-self-averaging behavior of the critical internal energy in a weakly-disordered Baxter model, combining analytical methods and extensive simulations to reveal persistent fluctuations at large system sizes.

Contribution

It provides new insights into the lack of self-averaging of the critical internal energy in a disordered Baxter model, supported by both analytical and numerical evidence.

Findings

Relative variance of internal energy approaches a finite constant as system size increases.

Fluctuations of the critical internal energy remain relevant regardless of disorder sign.

Numerical results confirm earlier predictions about non-self-averaging in disordered Ising models.

Abstract

We investigate the first two moments of the critical internal energy $E$ in a weakly disordered two-dimensional Baxter eight-vertex model as a function of the system size $L$, evaluated at the pseudo-critical point. Disorder is introduced via an equivalent representation of the pure eight-vertex model in terms of two ferromagnetic Ising models coupled by a four-spin interaction of strength $g_0$, where the Ising couplings consist of a uniform ferromagnetic part $J>0$ supplemented by weak Gaussian spatial disorder. In the critical regime, the model is formulated in terms of interacting Grassmann-Majorana spinor fields with quartic interactions and analyzed, for small positive $g_0$, using a combination of replica and renormalization-group methods. We also run extensive numerical simulations measuring the critical internal energy. Our results show that its relative variance increases with $L$ and approaches a finite constant as $L \to \infty$ for both $\pm g_0$. Hence, fluctuations remain relevant independently of the sign of $g_0$ (and thus of the specific-heat exponent), implying a lack of self-averaging of both the critical internal energy and the free energy. Consequently, reliable estimates of these quantities require averaging over many disorder realizations. In addition, we numerically confirm earlier predictions concerning the absence of self-averaging of the critical internal energy in the disordered Ising model.