Triangular Pair Density Wave in Confined Superfluid ${}^3\mathrm{He}$

Recent advances in experiment and theory suggest that superfluid ${}^3\mathrm{He}$ under planar confinement may form a pair density wave (PDW) whereby superfluid and crystalline orders coexist. While a natural candidate for this phase is a unidirectional stripe phase predicted by Vorontsov and Sauls in 2007, recent nuclear magnetic resonance measurements of the superfluid order parameter rather suggest a two-dimensional PDW with noncollinear wave vectors, of possibly square or hexagonal symmetry. In this Letter, we present a general mechanism by which a PDW with the symmetry of a triangular lattice can be stabilized, based on a superfluid generalization of Landau’s theory of the liquid-solid transition. A soft-mode instability at a finite wave vector within the translationally invariant planar-distorted B phase triggers a transition from uniform superfluid to PDW that is first order due to a cubic term generally present in the PDW free-energy functional. This cubic term also lifts the degeneracy of possible PDW states in favor of those for which wave vectors add to zero in triangles, which in two dimensions uniquely selects the triangular lattice.

Figure 1

(a) Eigenvalues ${\lambda }^{(j)}(\mathit{q})$ in units of the normal-state density of states $N(0)$ vs momentum $q=|\mathit{q}|$ for real (black) and imaginary (blue) normal modes in the PDB phase; their square root (b),(c) is proportional to collective mode frequencies. Parameters are chosen as $D=300\mathrm{nm}$, $P=10\mathrm{bar}$, and $T=0.914\mathrm{mK}$; other parameters in the PDB phase give qualitatively similar results.

Figure 2

Black lines: eigenvalues ${\lambda }^{(j)}(\mathit{q})$ vs momentum $q=|\mathit{q}|$ for real normal modes in the vicinity of the softening instability at $T=T_1^{*}$: (a) before softening ($T=1.318\mathrm{mK}$, $r>0$); (b) at softening ($T=T_1^{*}=1.336\mathrm{mK}$, $r=0$); (c) after softening ($T=1.380\mathrm{mK}$, $r<0$). Here, $r$ is the tuning parameter for the instability, appearing in the approximate form ${\lambda }^{(j_{*})}(\mathit{q})\approx r+\kappa ({\mathit{q}}^2-Q^2{)}^2$ of the critical mode eigenvalue (dashed red line). Parameters are chosen as $D=300\mathrm{nm}$ and $P=10$ bar, and we find $Q\approx \pi /(\sqrt{3}D)$ as in Ref. [23].

Figure 3

(a) Phase diagram of a $D=300\mathrm{nm}$ slab, with mode-softening temperatures $T_1^{*}$ and $T_2^{*}$ flanking the would-be A-PDB transition line $T_{\mathrm{AB}}$. The instability on the PDB side ($T_1^{*}$) is preempted by a first-order transition at $T_{\mathrm{PDW}}$, which (b) is approximately 4% lower than $T_1^{*}$ relative to the width of the instability region.

Figure 4

(a) Irreducible $z$-dependent amplitudes of the PDW order parameter; these multiply invariants of the triangular spin-orbital point group that involve ${\mathit{r}}_{\parallel }$-dependent basis functions $\mathrm{\Psi }({\mathit{r}}_{\parallel })$ for the (b) $A_1$, (c) $E_1$, and (d) $E_2$ irreducible representations of $D_6$.