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 the surface critical behavior directly in dimensions d < 4 without having to resort to the e expansion. The approach is elaborated for the representative case of the semi-infinite | Φ| 4 n-vector model with a boundary term case1 2 c 0 ∫ ∂V Φ 2 in the action. To make the theory UV finite in bulk dimensions 3 ⩽ 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 parameters: 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 → 0 with c m → 0 and c m → ∞ , respectively. It is shown that the surface-enhancement renormalization turns into an additive renormalization in the limit c m → ∞ . The renormalization factors and exponent functions with c m = 0 and c m = ∞ that are needed to determine the surface critical exponents of the special and ordinary transitions are calculated to two-loop order at d = 3. 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 data. This is also true for the surface crossover exponent Φ, for which we obtain Φ(n = 0) ⋍ 0.52 and Φ(n = 1) ⋍ 0.54, values considerably lower than the previous ϵ-expansion estimates.