In a system with long-ranged correlations, the behavior of correlation functions is sensitive to the presence of a boundary. We show that surface deformations strongly modify this behavior as compared to a flat surface. The modified near surface correlations can be measured by scattering probes. To determine these correlations, we develop a perturbative calculation in the deformations in height from a flat surface. Detailed results are given for a regularly patterned surface, as well as for a self-affinely rough surface with roughness exponent ζ. By combining this perturbative calculation in height deformations with the field-theoretic renormalization-group approach, we also estimate the values of critical exponents governing the behavior of the decay of correlation functions near a self-affinely rough surface. We find that for the interacting theory, a large enough ζ can lead to a different surface critical behavior. We also provide scaling relations between roughness induced critical exponents for thermodynamic surface quantities.
Hanke, Andreas and Kardar, Mehran, "Correlation functions near modulated and rough surfaces" (2002). Physics and Astronomy Faculty Publications and Presentations. 369.
Physical Review E