Critical behaviour of three-dimensional Ising ferromagnets at imperfect surfaces: Bounds on the surface critical exponent $\beta_1$

TL;DR

This paper derives bounds on the surface critical exponent in 3D Ising ferromagnets with surface imperfections, explaining the observed robustness of the exponent through inequalities and Monte Carlo simulation insights.

Contribution

It establishes theoretical bounds on the surface critical exponent considering surface disorder and terraces, linking inequalities to empirical observations.

Findings

Bounds on the surface critical exponent $eta_1$ are derived.

Surface imperfections do not alter $eta_1$ if they are below a certain threshold.

Monte Carlo simulations support the robustness of $eta_1^{ord}$ against imperfections.

Abstract

The critical behaviour of three-dimensional semi-infinite Ising ferromagnets at planar surfaces with (i) random surface-bond disorder or (ii) a terrace of monatomic height and macroscopic size is considered. The Griffiths-Kelly-Sherman correlation inequalities are shown to impose constraints on the order-parameter density at the surface, which yield upper and lower bounds for the surface critical exponent $\beta_1$. If the surface bonds do not exceed the threshold for supercritical enhancement of the pure system, these bounds force $\beta_1$ to take the value $\beta_1^{ord}$ of the latter system's ordinary transition. This explains the robustness of $\beta_1^{ord}$ to such surface imperfections observed in recent Monte Carlo simulations.