ORDINARY, EXTRAORDINARY, AND NORMAL SURFACE TRANSITIONS - EXTRAORDINARY-NORMAL EQUIVALENCE AND SIMPLE EXPLANATION OF VERTICAL-BAR-T-TC-VERTICAL-BAR-2-ALPHA SINGULARITIES

With simple, exact arguments we show that the surface magnetization m1 at the extraordinary and normal transitions and the surface energy density epsilon1 at the ordinary, extraordinary, and normal transitions of semi-infinite d-dimensional Ising systems have leading thermal singularities B+/- Absolute value of 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 h1 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 (h1 = 0, g > 0) and normal (h1 not-equal 0, g < 0) transitions. (ii) The magnetization m1 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 epsilon1 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).