Scale invariance, fractal dynamics, and critical exponents at the phase transition

Abstract

We propose that, at criticality, equilibrium dynamics effectively evolves on a fractal subspace rather than throughout the full Euclidean space. Starting from Fisher's formulation of the order-parameter correlation function, we interpret the anomalous critical exponent $\eta$ geometrically through a correlation fractal dimension associated with this effective subspace. Using fractional-calculus tools, we derive a form of the correlation function that recovers critical behavior below the upper critical dimension and obtain an explicit relation between $\eta$ and a fractal dimension $d_R$ linked to the Riesz fractional derivative. We also examine the Rushbrooke scaling relation and investigate critical behavior in non-integer-dimensional and disordered systems. In the disordered case, controlled by a parameter $\sigma$, we test the range over which the proposed geometric interpretation remains valid. The results connect scaling laws, critical exponents, correlations, and fractal geometry, suggesting that the anomalous behavior observed at criticality can be understood as a consequence of dynamics constrained to an effective fractal subspace.