  We use the higher-order tensor renormalization group method to study the
two-dimensional generalized XY model that admits integer and half-integer
vortices. This model is the deformation of the classical XY model and has a
rich phase structure consisting of nematic, ferromagnetic, and disordered
phases and three transition lines belonging to the
Berezinskii-Kosterlitz-Thouless (BKT) and Ising class. We explore the model for
a wide range of temperatures, $T$, and the deformation parameter, $\Delta$, and
compute specific heat along with integer and half-integer magnetic
susceptibility, finding both BKT-like and Ising-like transitions and the region
where they meet.
