  Bulk-boundary correspondence is the emergence of features at the boundary of
a material that are dependent on and yet distinct from the properties of the
bulk of the material. The diverse applications of this idea in topological
insulators as well as high energy physics prove its universality. However,
whether a form of bulk-boundary correspondence holds also in soft matter such
as gels, polymers, lipids and other biomaterials is thus far unknown.
Aerosil-dispersed liquid crystal gels (LC+aerosil) provide a good testing
ground to explore the relation between the controlled variations of the aerosil
density within the liquid crystal host bulk and the surface topography of the
sample. Here we report on one of the earliest if not the first direct
observation of such a correspondence where the controlled strength of random
disorder created by aerosil dispersion in the bulk liquid crystal is correlated
with the fractal dimension of the surface. We obtained the surface topography
of our gel samples with different quenched random disorder strengths by using
atomic force microscope techniques, and computed the fractal dimension for each
sample. We found that an increase of the aerosil gel density in the bulk
corresponds to an increase in the fractal dimension at the surface. From our
results emerges a new method to acquire the bulk properties of soft matter such
as density, randomness and phase merely from the fractal dimension of the
surface.
