Time scale for equilibration/irreversibility in the free fermion chain

Determine explicit quantitative bounds or scaling laws for the "sufficiently large" observation time T in Theorem 3 for the L-site periodic free fermion chain with Hamiltonian H = ∑_{x=1}^L (e^{iθ} c_x^† c_{x+1} + e^{-iθ} c_{x+1}^† c_x), coarse-grained into m intervals with relative density tolerance δ, such that for any initial N-particle state the probability ((ψ(t)|P_neq|ψ(t))) that the measured coarse-grained densities deviate from p0 by at least p0δ is exponentially small for all typical times t ∈ [0,T] \ A (with l(A)/T exponentially small). Ascertain how T depends on N, L, m, δ, and the initial state |ψ(0)⟩, providing explicit upper/lower bounds or asymptotic scaling.

Background

The paper proves macroscopic irreversibility for a free fermion chain by showing that coarse-grained density measurements become (with overwhelming probability) uniform at sufficiently large and typical times. This is formalized in Theorem 3, which guarantees that for any initial N-particle state and sufficiently large observation window [0,T], there exists an exponentially small set of atypical times outside of which the coarse-grained densities are close to the uniform value.

However, while the qualitative result is established, the authors state that their theory does not provide any information about the actual time scale needed to reach the regime where the high-probability uniformity holds. They emphasize the importance of controlling this time scale and note the absence of general results, though they remark that in certain Slater-determinant forms one might derive estimates, and they reference related work for time-scale results in non-interacting systems.