Critical Dynamics in Short-Range Quadratic Hamiltonians

We investigate critical transport and the dynamical exponent through the spreading of an initially localized particle in quadratic Hamiltonians with short-range hopping in lattice dimension d_{l}. We consider critical dynamics that emerges when the Thouless time, i.e., the saturation time of the mean-squared displacement, approaches the typical Heisenberg time. We establish a relation, z=d_{l}/d_{s}, linking the critical dynamical exponent z to d_{l} and to the spectral fractal dimension d_{s}. This result has notable implications: it says that superdiffusive transport in d_{l}≥2 and diffusive transport in d_{l}≥3 cannot be critical in the sense defined above. Our findings clarify previous results on disordered and quasiperiodic models and, through Fibonacci potential models in two and three dimensions, provide nontrivial examples of critical dynamics in systems with d_{l}≠1 and d_{s}≠1.