Superfluid-Insulator Transition unambiguously detected by entanglement in one-dimensional disordered superfluids

We use entanglement to track the superfluid-insulator transition (SIT) in disordered fermionic superfluids described by the one-dimensional Hubbard model. Entanglement is found to have remarkable signatures of the SIT driven by i) the disorder strength $V$, ii) the concentration of impurities $C$ and iii) the particle density $n$. Our results reveal the absence of a critical potential intensity on the SIT driven by $V$, i.e. any small $V$ suffices to decrease considerably the degree of entanglement: it drops $\sim 50\%$ for $V=-0.25t$. We also find that entanglement is non-monotonic with the concentration $C$, approaching to zero for a certain critical value $C_C$. This critical concentration is found to be related to a special type of localization, here named as fully-localized state, which can be also reached for a particular density $n_C$. Our results show that the SIT driven by $n$ or $C$ has distinct nature whether it leads to the full localization or to the ordinary one: it is a first-order quantum phase transition when leading to full localization, and a smoother transition when reaching ordinary localization. In contrast, the SIT driven by $V$ is always a smoother transition independently on the type of localization reached.