Connecting Complex Electronic Pattern Formation to Critical Exponents

Scanning probes reveal complex, inhomogeneous patterns on the surface of many condensed matter systems. In some cases, form self-similar, fractal geometric clusters. this paper, we advance theory criticality as it pertains to those clusters (defined connected sets nearest-neighbor aligned spins) in context Ising models. We show how data from can be used distinguish whether electronic observed at a material are confined surface, or originate bulk. Whereas thermodynamic critical exponents derived behavior Fortuin-Kasteleyn (FK) clusters, similarly defined for find that these not only distinct numerically and uncorrelated percolation exponents, but they separately satisfy scaling relations fixed points discussed text. furthermore two-dimensional (2D) cross-sections three-dimensional (3D) model display bulk phase transition temperature. particular, when considered 2D slice 3D system, pair connectivity function familiar displays more robust than spin-spin correlation function, calculate corresponding exponent. discuss implications two length scales also exponent clean case. These results extend universality classes, facilitate broad application cluster analysis techniques maximize information extracted scanning image probe