Massive Field-Theory Approach to Surface Critical Behavior in Three-Dimensional Systems

Abstract

The massive field-theory approach for studying critical behavior in fixed space dimensions $d<4$ is extended to systems with surfaces.This enables one to study surface critical behavior directly in dimensions $d<4$ without having to resort to the $ε$ expansion. The approach is elaborated for the representative case of the semi-infinite $|\bboxϕ|^4$ $n$-vector model with a boundary term ${1/2} c_0\int_{\partial V}\bboxϕ^2$ in the action. To make the theory uv finite in bulk dimensions $3\le d<4$, a renormalization of the surface enhancement $c_0$ is required in addition to the standard mass renormalization. Adequate normalization conditions for the renormalized theory are given. This theory involves two mass parameter: the usual bulk `mass' (inverse correlation length) $m$, and the renormalized surface enhancement $c$. Thus the surface renormalization factors depend on the renormalized coupling constant $u$ and the ratio $c/m$. The special and ordinary surface transitions correspond to the limits $m\to 0$ with $c/m\to 0$ and $c/m\to\infty$, respectively. It is shown that the surface-enhancement renormalization turns into an additive renormalization in the limit $c/m\to\infty$. The renormalization factors and exponent functions with $c/m=0$ and $c/m=\infty$ that are needed to determine the surface critical exponents of the special and ordinary transitions are calculated to two-loop order. The associated series expansions are analyzed by Padé-Borel summation techniques. The resulting numerical estimates for the surface critical exponents are in good agreement with recent Monte Carlo simulations. This also holds for the surface crossover exponent $Φ$.