We study the thermodynamics and critical behavior of neutron P23 superfluids in the inner cores of neutron stars. P23 superfluids offer a rich phase diagram including uniaxial/biaxial nematic phases, the ferromagnetic phase, and the cyclic phase. Using the Bogoliubov-de Gennes equation as in superfluid Fermi liquid theory, we show that a strong (weak) magnetic field drives the first-order (second-order) transition from the dihedral-two biaxial nematic phase to the dihedral-four biaxial nematic phase at low (high) temperatures and their phase boundaries are divided by the critical end point (CEP). We demonstrate that the set of critical exponents at the CEP satisfies the Rushbrooke, Griffiths, and Widom equalities, indicating a different universality class. At the CEP, the P23 superfluid exhibits critical behavior with nontrivial critical exponents, indicating an alternative universality class. Furthermore, we find that the Ginzburg-Landau (GL) equation up to the eighth-order expansion satisfies three equalities and properly captures the physics of the CEP. This implies that the GL theory can provide a tractable way for understanding critical phenomena which may be realized in the dense core of realistic magnetars.
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