Fast and Slow Sound Excitations in Nematic Aerogel in superfluid 3He

Abstract

Nematic aerogel (nAG) supports so-called polar phase in liquid 3He. The experiments by [Dmitriev et al, JETP Lett. 112, 780 (2020)] showed that the onset of polar phase inside the nAG is accompanied by emergence of a sound wave with frequency quickly growing with cooling down from transition temperature and reaching a plateau. To describe this behavior, we start by calculating the elastic properties of the dry nematic AG that appear to depend only on Young's modulus of the parent material (e.g. mullite), the volume fraction of the solid phase and the aspect ratio of the representative volume of nAG. The elastic constants are then used to solve elasto-hydrodynamic equations for various sound vibrations of nAG filled with 3He. The (isotropic) first sound and anisotropic second sound in the polar phase are strongly hybridized with fourth sound and standard elastic modes in nAG. The hybrid second and the transverse fourth sound start with zero velocity at the transition, similar to pure 3He, and quickly grow with lowering temperature until they hit the sample finite size cutoff.

Figures (5)

• Figure 1: (Color online) (a) Representative volume of nematic aerogel used to estimate its elastic constants viewed as an equivalent orthorhombic cell with fused nodes. The strands are fused together over the typical distance $c \gg a,b$ , where the parameters $a$ and $b$ are typical separation between the strands ($\sim$60 nm in Ref. dmit20.) $P$ and $F$ mark the shear and the axial forces resulting in shear strain $e_{23}\equiv e_5$ and axial strain (not shown). Under the load, the fused strands bend and rotate about the nodes. The resulting bending torques on the strands (elastic Euler-Bernoulli "beams") are shown by curved arrows. (b) The STEM picture of nematic aerogel (courtesy V.V. Dmitriev, A.A.Soldatov, and A.N. Yudin.) The size bar is 0.6 $\mu$m.
• Figure 2: (Color online) Schematic of sound wave propagating along the strands of nematic aerogel filled with the polar phase of $^3$He that shows the Dirac nodal line at the equator plane perpendicular to the nematic director $\vec{m}$. The irrotational superfluid motion with velocity $\textbf{v}^s$ follows the wave vector $\hat{k}=(0,0,1)$. The normal velocity has one longitudinal and two transversal possible orientations ($L, T1, T2$), The modes $T1$ and $T2$ are degenerate since the nematic aerogel is the transversely isotropic system.
• Figure 3: (Color online) Schematic of sound wave propagating perpendicular to the strands of nematic aerogel filled with the polar phase of $^3$He. The superfluid velocity $\textbf{v}^s$ is collinear with the wave vector $\hat{k}=(1,0,0)$. The normal velocity has one longitudinal and two transversal possible orientations ($L, T1, T2$), The modes $T1$ and $T2$ are not degenerate: the mode $T1$ involves the normal density along the strands, $T2$ is perpendicular to the strands and they are different stemming from anisotropy of the superconducting gap.
• Figure 4: (Color online) Schematic of the hybrid sound velocities versus temperature in extended nematic aerogel. The hybrid second sound $U_{2a}$ and $S_{2a}$ emerge at the transition into polar phase with zero velocity. Both modes exhibit sharply increasing velocity until they hit the size cutoff that grows linearly with the linear size of aerogel sample (see text.) The shear modes due to elastic response of AG skeleton exist above $T_{ca}$ and weakly depend on temperature (gray band). The observed main VW mode (solid line) exists above $T_{ca}$ and exhibits an avoided crossing with the additional mode that starts at $T_{ca}$. The velocity of additional mode (upper branch below$T_{ca}$) increases until it approaches the nAG sample size cutoff corresponding to about 1600 Hz dmit20.
• Figure 5: (Color online) Comparison of the frequency of hybrid second sound $U_{2a}$ with data dmit20. The frequency rises sharply from its emergence at $T_{ca}$. It quickly hits a plateau that is likely due the velocity of this mode reaching the size cutoff value $U_c \sim 10$ m/s. The size cutoff increases linearly with $L$ (see text). Inset shows the aerogel sample with size $L \approx 3$ mm attached to the vibrating wire dmit20.