The stripe phase as a dynamical instability of the $J=2^{+}$, $m_J=0$ Higgs field in confined superfluid $^3$He

Superfluid $^3$He exhibits a variety of topological and broken symmetry phases. In a thin film, a ``stripe'' phase that spontaneously breaks translational symmetry has been predicted on the basis of weak-coupling quasiclassical theory,\footnote{A.B. Vorontsov and J.A. Sauls, PRL {\bf 98}, 045301 (2007).} and within strong-coupling Ginzburg-Landau (GL) theory.\footnote{J.J. Wiman and J.A. Sauls, JLTP {\bf 184}, 1054 (2016).} Starting from a Lagrangian formulation for the dynamics of the order parameter for $^3$He, we show that a superfluid film with translational symmetry is dynamically unstable for a range of film thicknesses of order several coherence lengths. The time-dependent GL Lagrangian describes the space-time Bosonic fluctuations around a stationary state of the GL functional. For a translationally invariant B-phase film the amplitude of the Bosonic mode dispersing from the $J=2^{+},m_J=0$ Higgs mode softens at a finite wavevector, $Q\simeq 0.3/\xi_0$, then develops a pole in the upper half of the complex frequency plane signalling a dynamical instability with exponential growth towards a new ground state with spontaneously broken translation symmetry. We discuss the dynamical instability and its relation to the predicted stripe phase of thin films of superfluid $^3$He.