Incommensurate many-body localization in the presence of long-range hopping and single-particle mobility edge

Single-particle lattice model can possess an intermediate phase with coexisting extended and localized states separated by a single-particle mobility edge (SPME). To investigate the role of SPME in the presence of short-range interactions, we specifically study the $t_1$-$t_2$ model, namely the Aubry-Andre (AA) model with next-nearest-neighbor (NNN) hopping. For weak interactions ($Ull t_1$), the system preserves its single-particle properties, resulting in the distinction between the AA and the $t_1$-$t_2$ model. However, for intermediate interactions ($Usim t_1$), two models exhibit qualitatively similar thermal-MBL transition, which agrees with the experimental observations. Additionally, for strong interactions ($Ugg t_1$), the two models are disparate because the NNN hopping breaks down the Hilbert space fragmentation in the AA model. We anticipate that the phenomena for weak and intermediate interactions are generic for models with SPME, while the breakdown in the strong-interaction regime is limited to the $t_1$-$t_2$ model.