Physics – Condensed Matter – Statistical Mechanics
Scientific paper
1998-04-07
Nucl.Phys. B528 (1998) 595-647
Physics
Condensed Matter
Statistical Mechanics
Revtex, 40 pages, 3 figures, and 8 pictograms (included in equations)
Scientific paper
10.1016/S0550-3213(98)00489-1
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 $\epsilon$ expansion. The approach is elaborated for the representative case of the semi-infinite $|\bbox{\phi}|^4$ $n$-vector model with a boundary term ${1/2} c_0\int_{\partial V}\bbox{\phi}^2$ 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 Pad\'e-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 $\Phi$.
Diehl H. W.
Shpot M.
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