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 dₗ. 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=dₗ/dₛ, linking the critical dynamical exponent z to dₗ and to the spectral fractal dimension dₛ. This result has notable implications: it says that superdiffusive transport in dₗ≥ 2 and diffusive transport in dₗ≥ 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 non-trivial examples of critical dynamics in systems with dₗ≠1 and dₛ≠1.