Ordinary, extraordinary, and normal surface transitions: extraordinary-normal equivalence and simple explanation of $|T-T_c|^{2-\alpha}$ singularities

TL;DR

This paper demonstrates that surface magnetization and energy density at various surface transitions in semi-infinite Ising systems share the same critical singularities as the bulk free energy, providing a simple, exact explanation for these phenomena.

Contribution

The authors establish the equivalence of extraordinary and normal surface transitions and derive their singularities using exact arguments and mappings in Ising models.

Findings

Surface quantities have singularities with the same critical exponent as bulk free energy.

The extraordinary-normal transition equivalence is proven through exact model mappings.

Surface energy density and magnetization are proportional to differences in bulk free energies.

Abstract

With simple, exact arguments we show that the surface magnetization $m_1$ at the extraordinary and normal transitions and the surface energy density $\epsilon_1$ at the ordinary, extraordinary, and normal transitions of semi-infinite $d$-dimensional Ising systems have leading thermal singularities $B_\pm |t|^{2-\alpha}$, with the same critical exponent and amplitude ratio as the bulk free energy $f_b(t,0)$. The derivation is carried out in three steps: (i) By tracing out the surface spins, the semi-infinite Ising model with supercritical surface enhancement $g$ and vanishing surface magnetic field $h_1$ is mapped exactly onto a semi-infinite Ising model with subcritical surface enhancement, a nonzero surface field, and irrelevant additional surface interactions. This establishes the equivalence of the extraordinary ($h_1=0, g>0$) and normal ($h_1\neq 0, g<0$) transitions. (ii) The magnetization $m_1$ at the interface of an infinite system with uniform temperature $t$ and a nonzero magnetic field $h$ in the half-space $z>0$ only is shown to be proportional to $f_b(t,0)-f_b(t,h)$. (iii) The energy density $\epsilon_1$ at the interface of an infinite system with temperatures $t_+$ and $t$ in the half-spaces $z>0$ and $z<0$ and no magnetic field is shown to be proportional to $f_b(t,0)-f_b(t_+,0)$.