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

Theodore W. Burkhardt; H. W. Diehl

doi:10.1103/PhysRevB.50.3894

paper

Ordinary, extraordinary, and normal surface transitions: extraordinary-normal equivalence and simple explanation of $∣ T - T_{c} ∣^{2 - α}$ singularities

arXiv:cond-mat/9402077 · doi:10.1103/PhysRevB.50.3894

Abstract

With simple, exact arguments we show that the surface magnetization $m_{1}$ at the extraordinary and normal transitions and the surface energy density $ε_{1}$ at the ordinary, extraordinary, and normal transitions of semi-infinite $d$ -dimensional Ising systems have leading thermal singularities $B_{\pm} ∣ t ∣^{2 - α}$ , 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 $ε_{1}$ 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 $f_{b} (t, 0) - f_{b} (t_{+}, 0)$ .

10 pages (RevTex 3.0) including one 1 figure, generated via a pictex source file

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

Abstract

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Ordinary, extraordinary, and normal surface transitions: extraordinary-normal equivalence and simple explanation of $∣ T - T_{c} ∣^{2 - α}$ singularities