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

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 $B\pm {\mathrm{}\left|t\right|\mathrm{}}^{2-\alpha }$, with the same critical exponent and amplitude ratio as the bulk free energy $f_b\mathrm{}(t,0\mathrm{})\mathrm{}$. 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 ($h_1=0, g>\mathrm{}0\mathrm{}$) and normal ($h_1\ne 0, g<\mathrm{}0\mathrm{}$) 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 $z>\mathrm{}0\mathrm{}$ only is shown to be proportional to $f_b\mathrm{}(t,0\mathrm{})-f_b\mathrm{}(t,h)\mathrm{}$. (iii) The energy density ${\epsilon }_1$ at the interface of an infinite system with temperatures $t_{+}$ and $t$ in the half-spaces $z>\mathrm{}0\mathrm{}$ and $z<\mathrm{}0\mathrm{}$ and no magnetic field is shown to be proportional to $f_b\mathrm{}(t,0\mathrm{})-f_b(t_{+},0)$.