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Hierarchy of timescales in a disordered spin-$1/2$ XX ladder

Understanding the timescales associated with relaxation to equilibrium in closed quantum many-body systems is one of the central focuses in the study of their non-equilibrium dynamics. At late times, these relaxation processes exhibit universal behavior, emerging from the inherent randomness of chaotic Hamiltonians. In this work, we investigate a disordered spin-$1/2$ XX ladder - an experimentally realizable model known for its diffusive dynamics - to explore the connection between transport properties and spectral measures derived solely from the system's energy levels via these relaxation timescales. We begin by analyzing the spectral form factor, which yields the time when the system begins to follow the random matrix theory (RMT) statistics, known as the RMT time. We then determine the Thouless times - the average times for a local excitation to diffuse across the entire finite system - through the linear-response theory for both spin and energy transport. Our numerical results confirm that the RMT time scales quadratically with system size and upper bounds the Thouless times. Interestingly, we also find that, unlike other non-integrable models, spin diffusion proceeds faster than energy diffusion.