Analytical description of intermediate-time (sub)diffusive dynamics

Derive a complete analytical description of the intermediate-time (sub)diffusive scaling of the wave-packet width for the one-dimensional disordered long-range correlated hopping model with complex phase parameter θ, defined by H = ∑_n ε_n |n><n| + ∑_{n≠m} J_0 e^{iθ sign(n−m)}/|n−m|^a |n><m| with ε_n i.i.d. Gaussian and a>0. Specifically, determine the analytical form of the scaling of the disorder-averaged variance [σ_2(t)]^2 (and, more generally, σ_q(t)) between the early-time ballistic regime and the long-time saturation across the relevant parameter regimes of a and θ, complementing the established results for the ballistic and saturation limits.

Background

The paper establishes a static phase diagram for a one-dimensional model with fully correlated long-range hopping that breaks time-reversal symmetry via a phase θ, and demonstrates robustness of algebraic localization up to |θ|=πa/2. Dynamically, the authors analyze wave-packet spreading and determine the short-time ballistic scaling and long-time saturation behavior analytically.

Numerically, they observe that the intermediate-time dynamics of the variance is subdiffusive for θ=0 and diffusive for any finite θ, but a full analytical theory for this regime is not provided. Establishing such a theory would complete the dynamical characterization between the ballistic onset and saturation.