  Entanglement in many-body systems may display interesting signatures of
quantum phase transitions and similar properties are starting to be encountered
in the analysis of work fluctuations. Here, we consider the fermionic
superfluid-to-insulator transition (SIT) and relate its entanglement properties
with its work distribution statistics. The SIT is modeled by the attractive
fermionic Hubbard model in the presence of randomly distributed impurities. The
work distribution is calculated across two quench protocols, both triggering
the SIT. In the first, the concentration of impurities is increased; in the
second, the impurities' disorder strength is varied. Our results indicate that,
the critical state that induces minimization of the entanglement also maximizes
the average work. We demonstrate that, for this state, density fluctuations
vanish at all orders, hence all central moments of the work probability
distribution are exactly zero at criticality. For systems undergoing a
precursor to the transition (short chains with finite impurity potential)
numerical results confirm these predictions, with higher moments further from
the ideal result. For both protocols, at criticality, the system absorbs the
most energy with almost no penalty in terms of fluctuations: ultimately this
feature could be used to implement a quantum critical battery. The effects of
temperature on these signatures of critical behaviour are also investigated and
shown to favor work extraction for high enough temperatures.
