*Published Paper*

**Inserted:** 17 dec 2020

**Last Updated:** 28 dec 2022

**Journal:** Acta Appl Math

**Volume:** 182

**Number:** 7

**Pages:** 26

**Year:** 2022

**Doi:** https://doi.org/10.1007/s10440-022-00544-4

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**Abstract:**

In the $p$-wave spin-triplet pairing model of superfluid Helium-3, at each point of the region occupied by the system, the order parameter field is described by a $3\times 3$ complex matrix $A$ encoding the orientation of the spin and orbital angular momentum of the Cooper pairs of Helium-3 atoms. The transition of liquid Helium-3 to a superfluid state is associated with a spontaneous breaking of the overall symmetry group of the system. In the Ginzburg-Landau regime (i.e., in regions near to the critical phase-transition temperature), the free-energy density of superfluid Helium-3 is expanded into powers of the components of $A$ and of its gradient $\nabla A$, and can be decomposed in the sum of the bulk part and the gradient part. The free-energy density must be invariant under the action of the overall symmetry group. We address the question of invariance for a general free-energy density in the Ginzburg--Landau energy functional and determine all linearly independent quartic terms in the expansion of the gradient free-energy density. It is known that the superfluid phases of Helium-3 near the critical temperature correspond to the minima of the bulk free energy and that the absolute minimum corresponds to a stable equilibrium phase. In zero magnetic field, there are two distinct superfluid phases, A and B, which exhibit an absolute minimum of the bulk free energy in different regions of the phase diagram. Explicit expressions for the generalized gradient energy densities are provided for both the A and B phases. Finally, a unified approach to A and B phases is proposed, which involves an auxiliary control parameter. In this framework, the extremal properties of A and B phases are recovered and a transition between the two phases is observed in dependence of pressure.

**Keywords:**
phase transition, Helium-3, Superfluidity, Superfluid phases A and B, Ginzburg-Landau energy

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