  With the Lipkin-Meshkov-Glick (LMG) model as an illustration, we construct a
thermodynamic cycle composed of two isothermal processes and two isomagnetic
field processes and study the thermodynamic performance of this cycle
accompanied by the quantum phase transition (QPT). We find that for a finite
particle system working below the critical temperature, the efficiency of the
cycle is capable of approaching the Carnot limit when the external magnetic
field \lambda_{1} corresponding to one of the isomagnetic processes reaches the
crosspoint of the ground states' energy level, which can become critical point
of the QPT in large N limit. Our analysis proves that the system's energy level
crossings at low temperature limits can lead to significant efficiency
improvement of the quantum heat engine. In the case of the thermodynamics
limit, analytical partition function is obtained to study the efficiency of the
cycle at high and low temperature limits. At low temperatures, when the
magnetic fields of the isothermal processes are located on both sides of the
critical point of the QPT, the cycle obtains the maximum efficiency and the
Carnot efficiency can be achieved. This observation demonstrate that the QPT of
the LMG model below critical temperature is beneficial to the thermodynamic
cycle's operation.
