Crystalline Order in Superfluid ${}^3\mathrm{He}$ Films

We predict an inhomogeneous phase of superfluid ${}^3\mathrm{He}$ films in which translational symmetry is spontaneously broken in the plane of the film. This phase is energetically favored over a range of film thicknesses, $D_{c_2}(T)<D<D_{c_1}(T)$, separating distinct homogeneous superfluid phases. The instability at the critical film thickness, $D_{c_2}\approx 9\xi (T)$, is a single-mode instability generating striped phase order in the film. Numerical calculations of the order parameter and free energy indicate a second-order instability to a periodic lattice of degenerate $B$-like phases separated by domain walls at $D_{c_1}\approx 13\xi (T)$.

Figure 1

Left: $Q_z$ vs $T$ phase diagram showing reentrance for homogeneous phases. Right: instability onsets for higher $Q_z(T)$ (orange line with boxes) for an inhomogeneous phase with in-plane wave vector $Q_x(T)$ (red dashed curve). Note that ${\xi }_0=\hslash v_F/2\pi T_c=77\mathrm{nm}$ at $p=0\mathrm{bar}$.

Figure 2

Order parameter (in units of $2\pi T_c$) for the stripe phase at $T=0.5T_c$ along the film for $z\approx 2.5{\xi }_0$. Left: $D_{c2}<D=9.3{\xi }_0\lesssim D_c$. Right: $D_c<D=10{\xi }_0<D_{c1}$.

Figure 3

(a) Surface reflection (trajectory 1) with $p_z\to -p_z$ leads to strong pairbreaking for the homogenous $B$ phase. (b) Trajectory 2 is a the strong pairbreaking trajectory because of the sign change ${\Delta }_{\perp }\to -{\Delta }_{\perp }$.

Figure 4

$T-D$-phase diagram for superfluid ${}^3\mathrm{He}$ films in the weak-coupling limit ($p=0\mathrm{bar}$). The phase with crystalline order separates translationally invariant $A(P)$ and $B$ phases.