Carrier Density Dependent Quantum Hall States Sequence of Holes in WSe$_2$*

The high intrinsic hole mobility and existence of Ohmic contacts to the valence band have enabled the magnetotransport study of holes in WSe$_2$. Using multiple monolayer and bilayer WSe$_2$ samples encapsulated in hexagonal boron nitride, we report an unusual carrier density-dependent quantum Hall states (QHSs) sequence in hole-doped WSe$_2$. At a fixed carrier density, the samples exhibit either a predominantly even or a predominantly odd QHSs sequence, which is insensitive to an applied transverse electric field. As the density is reduced from $9\times10^{12}$ cm$^{-2}$ to $2\times10^{12}$ cm$^{-2}$, we observe a transition from even to odd, and back to even QHS filling factors. Furthermore, magnetotransport measurements in a tilted magnetic field reveal that the QHSs sequence is insensitive to the in-plane magnetic field, indicating that the electron spin is locked perpendicular to the WSe$_2$ plane. These findings suggest that the Landau level Zeeman splitting depends linearly on the perpendicular magnetic field via a carrier density dependent $g$-factor, as a result of electron-electron interaction.