Many-body mobility edges in a one-dimensional model of interacting fermions

We analyze many body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold value of the potential $h < h_c$, the non-interacting system has single particle mobility edges at $\pm E_c$ while for $h > h_c$ all the single particle states are localized. We demonstrate that even in the presence of single particle mobility edges, the interacting system can have MBL. Our numerical calculation of participation ratio in the Fock space and Shannon entropy shows that both for $h < h_c$ (quarter filled) and $h>h_c$ ($h\sim h_c$ and half filled), many body states in the middle of the spectrum are delocalized while the low energy states with $E < E_1$ and the high energy states with $E> E_2$ are localized. Variance of entanglement entropy (EE) also shows divergence at $E_{1,2}$ indicating a transition from MBL to delocalized regime. We also studied eigenstate thermalisation hypothesis (ETH) and found that the low energy many body states, which show area law scaling for EE do not obey ETH. The crossings from volume to area law scaling for EE and from thermal to non-thermal behaviour occurs deep inside the localised regime. For $h \gg h_c$, all the many body states remain localized for weak to intermediate strength of interaction and the system shows infinite temperature MBL phase.