Fractional diffusion without disorder in two dimensions

Abstract

We analyse how simple local constraints in two dimensions lead a defect to exhibit robust, non-transient, and tunable, subdiffusion. We uncover a rich dynamical phenomenology realised in ice- and dimer-type models. On the microscopic scale the path of a single defect exhibits anomalously long retractions, amounting to dynamical caging in a continuous-time random-walk framework, culminating in an effective fractional diffusion equation. Mapping to a height field yields an effective random walk subject to an emergent (entropic) logarithmic potential, whose strength is tunable, related to the exponent of algebraic ground-state correlations. The defect's path, viewed as non-equilibrium growth process, yields a frontier of fractal dimension of $5/4$, the value for a loop-erased random walk, rather than $4/3$ for simple and self-avoiding random walks. Such frustration/constraint-induced subdiffusion is expected to be relevant to platforms such as artificial spin ice and quantum simulators aiming to realize discrete link models and emergent gauge theories.