Stabilized Pair Density Wave via Nanoscale Confinement of Superfluid ${}^3\mathrm{He}$

Superfluid ${}^3\mathrm{He}$ under nanoscale confinement has generated significant interest due to the rich spectrum of phases with complex order parameters that may be stabilized. Experiments have uncovered a variety of interesting phenomena, but a complete picture of superfluid ${}^3\mathrm{He}$ under confinement has remained elusive. Here, we present phase diagrams of superfluid ${}^3\mathrm{He}$ under varying degrees of uniaxial confinement, over a wide range of pressures, which elucidate the progressive stability of both the $A$ phase, as well as a growing region of stable pair density wave state.

Figure 1

(a) Schematic momentum-space representations of the quasiparticle gaps for the relevant superfluid states under planar confinement: from left to right, the isotropic $B$ phase, the planar distorted $B$ phase, and the $A$ phase. (b) Confinement along the $z$ axis. When Cooper pairs scatter off a wall, the $z$ component ${\mathrm{\Delta }}_{\perp }$ of the order parameter changes sign, which for the planar-distorted $B$ phase causes pair breaking. This can be minimized by the formation of domain walls between regions of planar-distorted $B$ phase with alternating signs of ${\mathrm{\Delta }}_{\perp }$. A regular arrangement of these domain walls leads to broken translational symmetry and the PDW state.

Figure 2

Experimental system and superfluid Helmholtz resonator frequency data. (a) Helmholtz resonator, fabricated out of single-crystal quartz etched to define the nanoscale confinement, then deposited with aluminum inside the etched area. (b) Under an applied voltage the electrodes are slightly displaced towards each other, pushing the superfluid ${}^3\mathrm{He}$ out of the channel and exciting the Helmholtz mode. This is detected using a capacitance bridge that outputs a current that is digitized. (c) Three Helmholtz resonators, with varying confinement, are wired in parallel and measured simultaneously using a chirped pulse scheme described in the text, while warming the sample cell via remagnetization of the copper nuclear stage [20]. Due to the measurement configuration the resonances appear Fano-like, yet are easily resolved, as shown in the inset—corresponding to the cross section marked with the dashed line in the main panel.

Figure 3

Enlargement around phase transitions for 22 bar data. (a) Cooling run with phase transitions in the 1067 nm device (upper trace) and 805 nm device (middle trace) marked with blue dashed lines. (b) Same as in (a) but for a warming run and phase transitions marked with red dashed lines. Note that (a) and (b) are on the same scales and the dashed lines cross both panels to emphasize that there is no significant temperature hysteresis between warming and cooling in our experiments. The hysteresis in shape of the transitions can be understood from the evolution of latent heat at first order phase transitions, as observed in the shape hysteresis between the $A$ and $B$ phases in bulk superfluid ${}^3\mathrm{He}$ [32, 33]. Observation of two first-order phase transitions demonstrates not only that the PDW state at intermediate temperatures is stable [34], but the lack of hysteresis points to an intermediate phase between the $A$ and planar distorted $B$ phases that eliminates supercooling [14, 35].

Figure 4

Pressure ($P$)-temperature ($T$) phase diagrams of superfluid ${}^3\mathrm{He}$, on warming. (a) Bulk phase diagram shown for reference, with the $A$ phase (pink) and $B$ phase (blue). Phase diagrams for (b) 1067 nm, (c) 805 nm, and (d) 636 nm confinements are shown with $A$ phase (pink) and planar-distorted $B$ phase (blue). Under confinement, a new phase appears, which we have colored grey. As the confinement increases from left to right, the width of the grey region grows, as does the stability of the $A$ phase. Ginzburg-Landau calculations, including strong coupling corrections [34, 38] and the effect of confinement [37] with diffuse boundary conditions [illustrated schematically in the inset of (b)], are included as the black curves—without any adjustable parameters. The PDW must lie between the $A$ phase and the planar distorted $B$ phase [8, 34]; hence the grey region is concluded to be the stable PDW state.