Macroscopic behavior of the P1 ( β ) and P2 (distorted β ) phases of superfluid He3 : Soundlike excitations

Abstract

We describe the macroscopic behavior of superfluid $^{3}\mathrm{He}$ in the ${P}_{1}$ ($\ensuremath{\beta}$) phase in an anisotropic aerogel. It turns out that the ${P}_{1}$ phase shares many features with superfluid $^{3}\mathrm{He}\text{\ensuremath{-}}{\mathrm{A}}_{1}$. It exists only in an external magnetic field and has only one spin orientation in its superfluid condensate. In the ${P}_{1}$ phase, the direction of the external magnetic field, the preferred direction in orbit space $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{m}}$ and the average preferred direction of the silica strands ${\ensuremath{\zeta}}_{i}$, are all parallel. While the preferred direction in orbit space and the average preferred direction of the silica strands are even under time reversal, the preferred direction $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{w}}$ in spin space is odd under time reversal. As new macroscopic variables compared with superfluid $^{3}\mathrm{He}$ we have for the superfluid ${P}_{1}$ phase in an aerogel the strain field associated with the aerogel network. As a result of these additional macroscopic variables we find new static and dynamic cross-coupling terms, which come as reversible (zero entropy production) as well as as irreversible contributions. As an outstanding feature of the ${P}_{1}$ phase we find that second sound and, to a lesser extent, fourth sound assume spin-wave character, a feature that should be testable experimentally. This result closely parallels that for the ${A}_{1}$ phase of bulk superfluid $^{3}\mathrm{He}$ in spite of the fact that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{l}}$ vector of the ${A}_{1}$ phase, which is odd under time reversal, does not exist in the ${P}_{1}$ phase. We also discuss briefly the macroscopic behavior of the distorted $\ensuremath{\beta}$ (${P}_{2}$) phase, which shares several features with the distorted $A$ phase of bulk superfluid $^{3}\mathrm{He}$, in particular with respect to its properties in spin space.