Ordinary, extraordinary, and normal surface transitions: Extraordinary-normal equivalence and simple explanation of |T−Tc|2−α singularities

Abstract

With simple, exact arguments we show that the surface magnetization ${m}_{1}$ at the extraordinary and normal transitions and the surface energy density ${\ensuremath{\epsilon}}_{1}$ at the ordinary, extraordinary, and normal transitions of semi-infinite $d$-dimensional Ising systems have leading thermal singularities $B\ifmmode\pm\else\textpm\fi{}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, 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 (${h}_{1}\ensuremath{\ne}0, g<0$) 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 ${f}_{b}(t,0)\ensuremath{-}{f}_{b}(t,h)$. (iii) The energy density ${\ensuremath{\epsilon}}_{1}$ 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 ${f}_{b}(t,0)\ensuremath{-}{f}_{b}({t}_{+},0)$.