  We study many-body localization (MBL) in the quasiperiodic $t_1$-$t_2$ model,
focusing on the role of next-nearest-neighbor (NNN) hopping $t_2$, which
introduces a single-particle mobility edge. The calculated phase diagram can be
divided into three distinct regimes, depending on the strength of the
short-range interaction $U$. For weak interactions ($U\ll t_1$), this model is
always nonthermal. For intermediate interactions ($U\sim t_1$), the thermal-MBL
phase transition in this model is qualitatively the same as that of the
Aubry-Andre (AA) model, which is consistent with existing experimental
observations. For strong interactions $(U\gg t_1)$, the NNN hopping produces
qualitatively new physics because it breaks down the Hilbert space
fragmentation present in the AA model. The NNN hopping is thus irrelevant when
the interaction is intermediate but relevant for strong interactions.
