  In this work, we investigate the active dynamics and ergodicity breaking of a
nonequilibrium fractional Langevin equation (FLE) with a power-law memory
kernel of the form $K(t)\sim t^{-(2-2H)}$, where $1/2<H<1$ represents the Hurst
exponent. The system is subjected to two distinct noises: a thermal noise
satisfying the fluctuation-dissipation theorem and an active noise
characterized by an active Ornstein-Uhlenbeck process with a propulsion memory
time $\tau_\mathrm{A}$. We provide analytic solutions for the underdamped
active fractional Langevin equation, performing both analytical and
computational investigations of dynamic observables such as velocity
autocorrelation, the two-time position correlation, ensemble- and time-averaged
mean-squared displacements (MSDs), and ergodicity-breaking parameters. Our
results reveal that the interplay between the active noise and long-time
viscoelastic memory effect leads to unusual and complex nonequilibrium dynamics
in the active FLE systems. Furthermore, the active FLE displays a new type of
discrepancy between ensemble- and time-averaged observables. The active
component of the system exhibits ultraweak ergodicity breaking where both
ensemble- and time-averaged MSDs have the same functional form with unequal
amplitudes. However, the combined dynamics of the active and thermal components
of the active FLE system are eventually ergodic in the infinite-time limit.
Intriguingly, the system has a long-standing ergodicity-breaking state before
recovering the ergodicity. This apparent ergodicity-breaking state becomes
exceptionally long-lived as $H\to1$, making it difficult to observe ergodicity
within practical measurement times. Our findings provide insight into related
problems, such as the transport dynamics for self-propelled particles in
crowded or polymeric media.
