Correlation functions near modulated and rough surfaces

Abstract

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 novel surface critical behavior.We also provide scaling relations between roughness induced critical exponents for thermodynamic surface quantities.

Critical phenomena at perfect and non-perfect surfaces

“…These results in two dimensions have been interpreted in terms of a scale-dependent distribution of opening angles of the fractal boundary [259]. The same analogy has been evoked in the discussion of the surface critical behaviour of self-affine surfaces in [256,257].…”

“…Using a perturbative calculation in the deformations in height from a flat surface they showed that the leading power law decay of the correlations is the same as for a flat surface, but with a modulated amplitude reflecting the shape of the surface [256,257].…”

“…Whereas the different kinds of roughnesses discussed so far have been shown to be (or are supposed to be) irrelevant for the surface critical behaviour at the ordinary transition, this is not always the case. In a recent interesting work, Hanke and Kardar [256,257] showed that self-affine rough surfaces may give rise to novel surface critical behaviour with surface critical exponents different from those of the perfect case. Similar results have previously been obtained in two-dimensional systems with fractal boundaries [258,259] yielding new multifractal boundary exponents.…”

Fluctuation-Induced Effects in Smectic Films with Nontrivial Boundary Conditions

“…General expression for correlation function G ðuÞ r; r 0 ð Þ ¼ũ uðrÞũ uðr 0 Þ h i can be obtained from the generating functional [17,18] ZfJg ¼ exp using auxiliary field J(r). With the help of Eq.…”

“…102=[1572]E. S. Pikina long-range correlations[18] (see Appendix C). The leading contribution to the two-dimensional Fourier transform of the correlation G…”