Magnetization Profile in the d=2 Semi-Infinite Ising Model and Crossover between Ordinary and Normal Transition

TL;DR

This paper studies how the surface magnetization in the two-dimensional semi-infinite Ising model varies with distance and surface magnetic field, revealing a crossover between different boundary conditions and confirming findings with simulations.

Contribution

It provides a detailed analysis of the magnetization profile and crossover phenomena in 2D, extending previous 3D results and connecting short-distance behavior to surface field effects.

Findings

Magnetization increases sharply near the boundary for small surface fields.

Short-distance power law is modified by a logarithm in 2D.

Monte Carlo simulations support the theoretical analysis.

Abstract

We theoretically investigate the spatial dependence of the order parameter of the two-dimensional semi-infinite Ising model with a free surface at or above the bulk critical temperature. Special attention is paid to the influence of a surface magnetic field $h_1$ and the crossover between the fixed points at h_1=0 and h_1=infinity. The sharp increase of the magnetization m(z) close to the boundary generated by a small h_1, which was found previously by the present authors in the three-dimensional model, is also seen in two dimensions. There, however, the universal short-distance power law is modified by a logarithm. By means of a phenomenological scaling analysis, the short-distance behavior can be related to the logarithmic dependence of the surface magnetization on h_1. Our results, which are corroborated by Monte Carlo simulations, provide a deeper understanding of the existing exact results concerning the local magnetization and relate the short-distance phenomena in two dimensions to those in higher dimensionality.