  We present a second-order extension of the first-order Lohe hermitian
sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed
model incorporates an inertial effect as a second-order extension. The inertia
term can generate an oscillatory behavior of particle trajectory in a small
time interval(initial layer) which causes a technical difficulty for the
application of monotonicity-based arguments. For emergent estimates, we employ
two-point correlation function which is defined as an inner product between
positions of particles. For a homogeneous ensemble with the same frequency
matrix, we provide two sufficient frameworks in terms of system parameters and
initial data to show that two-point correlation functions tend to the unity
which is exactly the same as the complete aggregation. In contrast, for a
heterogeneous ensemble with distinct frequency matrices, we provide a
sufficient framework in terms of system parameters and initial data, which
makes two-point correlation functions close to unity by increasing the
principal coupling strength.
