In addition to phase transitions that take place at finite temperatures, where usually a classical description of the critical behavior is adequate, phase transitions at very low temperatures require a quantum approach. In the extreme case, one can define quantum phase transitions in the ground state as a result of a change of parameters that induce a qualitative change of the state of the system. Two related models are considered as examples such as the Ising model in a transverse field and the Kitaev model for a spinless triplet superconductor. The critical behavior is analyzed in terms of the deviation from a critical coupling value, instead of a critical temperature. As temperature increases, there is a crossover from a quantum regime to a classical regime. The zero-temperature Hertz dynamical theory is discussed, and a brief discussion of a quantum information theory quantity, the fidelity, is presented as an alternative and complementary way to characterize a transition. A short discussion of the time dependence of an induced phase transition by changing some parameter is presented in diabatic and adiabatic quenches, namely, the Landau-Zener crossings and the Kibble-Zurek mechanism.