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The Ginzburg-Landau (GL) effective theory is a useful tool to study asuperconductivity or superfluidity near the critical temperature, and usuallythe expansion up to the 4th order in terms of order parameters is sufficientfor the description of the second-order phase transition. In this paper, wediscuss the GL equation for the neutron $^{3}P_{2}$ superfluidity relevant forinterior of neutron stars. We derive the GL expansion up to the 8th order inthe condensates and find that this order is necessary for the system to havethe unique ground state, unlike the ordinary cases. Starting from the $LS$potential, which provides the dominant attraction between two neutrons at thehigh density, we derive the GL equation in the path-integral formalism, wherethe auxiliary field method and the Nambu-Gor'kov representation are used. Wepresent the detailed description for the trace calculation necessary in thederivation of the GL equation. As numerical results, we show the phase diagramof the neutron $^{3}P_{2}$ superfluidity on the plane spanned by thetemperature and magnetic field, and find that the 8th order terms lead to afirst-order phase transition, whose existence was predicted in theBogoliubov-de Gennes equation but has not been found thus far within theframework of the GL expansion up to the 6th order.The first-order phasetransition will affect the interior structures inside the neutron stars.

Reexamining Ginzburg-Landau theory for neutron $^3P_2$ superfluidity in neutron stars