First order quantum phase transitions of the XX spin-1/2 chain in a uniform transverse field

Quantum phase transitional behavior of a finite periodic XX spin-12 chain with nearest neighbor interaction in a uniform transverse field is studied based on the simple exact solutions. It is found that there are [N/2][N/2] level-crossing points in the ground state, where N   is the periodic number of the system and [x][x] stands for the integer part of x, when the interaction strength and magnitude of the magnetic field satisfy certain conditions. The quantum phase transitional behavior in the thermodynamic is also studied.