Signatures of Crystalline Phases and Domain Walls in Superfluid $^3$He Thin Films*

Thin films of superfluid $^3$He may spontaneously break translation symmetry in the plane of the film.\footnote{Phys. Rev. Lett. 98,045301 (2007).} Near a critical film thickness, $D_{c_1}\approx 13\,\xi_0$, a one-dimensional ``stripe phase'' develops as a periodic array of domain walls separating degenerate, but inequivalent B-phases, $(\Delta_{||},\Delta_{||},+\Delta_{\perp})$ and $(\Delta_{||},\Delta_{||},-\Delta_{\perp})$. These defects have a unique spectrum of topological excitations bound to the domain wall. We present results for the order parameter and Fermionic spectrum, and their observable signatures, for a single domain wall and for the stripe phase. The combination of particle-hole asymmetry and broken translational symmetry of the order parameter leads to a weak modulation of the density, $\delta n \sim \ln\left(E_f/k_B T_c\right)\left(k_B T_c/E_f\right)^2\,\bar{n}$, where $\bar{n}$ is the mean particle density. This leads to a modulation of the van der Walls attraction, and thus a small, static modulation of the film thickness. We report theoretical results for the density modulation, film thickness profile and optical reflectivity for the crystalline phases of superfluid $^3$He.