  We study the quantum phase transitions in the two-dimensional spin-orbit
models in terms of fidelity susceptibility and reduced fidelity susceptibility.
An order-to-order phase transition is identified by fidelity susceptibility in
the two-dimensional Heisenberg XXZ model with Dzyaloshinsky-Moriya interaction
on a square lattice. The finite size scaling of fidelity susceptibility shows a
power-law divergence at criticality, which indicates the quantum phase
transition is of second order. Two distinct types of quantum phase transitions
are witnessed by fidelity susceptibility in Kitaev-Heisenberg model on a
hexagonal lattice. We exploit the symmetry of two-dimensional quantum compass
model, and obtain a simple analytic expression of reduced fidelity
susceptibility. Compared with the derivative of ground-state energy, the
fidelity susceptibility is a bit more sensitive to phase transition. The
violation of power-law behavior for the scaling of reduced fidelity
susceptibility at criticality suggests that the quantum phase transition
belongs to a first-order transition. We conclude that fidelity susceptibility
and reduced fidelity susceptibility show great advantage to characterize
diverse quantum phase transitions in spin-orbit models.
