  Surface probes such as scanning tunneling microscopy (STM) have detected
complex patterns at the nanoscale, indicative of electronic inhomogeneity, in a
variety of high temperature superconductors. In cuprates, the pattern formation
is associated with the pseudogap phase, a precursor to the high temperature
superconducting state. Symmetry breaking (i.e. from C4 to C2) in the form of
electronic nematicity has recently been implicated as a unifying theme of the
pseudogap phase,[1] however the fundamental physics governing the nanoscale
pattern formation has not yet been identified. Here we use universal cluster
properties extracted from STM studies of cuprate superconductors in order to
identify the fundamental physics controlling the complex pattern formation. We
find that the pattern formation is set by a delicate balance between disorder
and interactions, leading to a fractal nature of the cluster pattern. The
method introduced here may be extended to a variety of surface probes, enabling
the direct measurement of the dimension of the phenomenon being studied, in
order to determine whether the phenomenon arises from the bulk of the material,
or whether it is confined to the surface.
