Field Driven Pairing State Phase Transition in d_x^2-y^2+id_xy-Wave Superconductors

Within the framework of the Ginzburg-Landau theory for $d_{x^2-y^2}+id_{xy}$-wave superconductors, we discuss the pairing state phase transition in the absence of the Zeeman coupling between the Cooper pair orbital angular momentum and the magnetic field. We find that above a temperature $T_{\ast}$, the pairing state in a magnetic field is pure $d_{x^{2}-y^{2}}$-wave. However, below $T_{\ast}$, the pairing state is $d_{x^{2}-y^{2}}+id_{xy}$-wave at low fields, and it becomes pure $d_{x^{2}-y^{2}}$-wave at higher fields. Between these pairing states there exists a field driven phase transition . The transition field increases with decreasing temperature. In the field-temperature phase diagram, the phase transition line is obtained theoretically by a combined use of a variational method and the Virial theorem. The analytical result is found to be in good agreement with numerical simulation results of the Gingzburg-Landau equations. The validity of the variational method is discussed. The difference to the case with the Zeeman coupling is discussed, which may be utilized to the detection of the Zeeman coupling.