Universality, irrelevant surface operators, and corrections to scaling in systems with free surfaces and defect planes

Abstract

The effects of irrelevant surface perturbations on the local critical behavior of ($4\ensuremath{-}\ensuremath{\epsilon}$)-dimensional $n$-vector models with a free surface or an internal defect plane are investigated with particular attention paid to the corrections to scaling induced by the surface term $\frac{1}{2}c\ensuremath{\int}{\mathrm{surface}}^{}{\ensuremath{\varphi}}^{2}$ near the ordinary transition. A systematic expansion in ${c}^{\ensuremath{-}1}$ is developed. In the case of a semi-infinite system, ${c}^{\ensuremath{-}1}$ couples to the square of the normal derivative of $\ensuremath{\varphi}$. The associated crossover exponent is ${\ensuremath{\Phi}}^{o}=\ensuremath{-}\ensuremath{\nu}$ to all orders in $\ensuremath{\epsilon}$. In the case of an infinite-space model with a defect plane, ${c}^{\ensuremath{-}1}$ couples to the squares of the normal derivatives on either side of the plane and to the product of the normal derivative on one side with the normal derivative on the other side. The latter operator represents perpendicular interactions across the defect plane, while the other two describe parallel interactions between spins on the same side of the plane. The associated crossover exponents are ${\ensuremath{\Phi}}_{\ensuremath{\parallel}}^{o}=\ensuremath{-}\ensuremath{\nu}$ and ${\ensuremath{\Phi}}_{\ensuremath{\perp}}^{o}={\ensuremath{\gamma}}_{11}^{o}$. A reanalysis of Bray and Moore's derivation of the (incorrect) relation ${\ensuremath{\Phi}}^{o}={\ensuremath{\gamma}}_{11}^{o}=\ensuremath{\nu}\ensuremath{-}1$ is presented which reveals where their arguments go wrong.