Sound-like Oscillations in the Polar Phase of Superfluid 3He in a Nematic Aerogel

The acoustic spectrum of the polar phase of superfluid 3He in nematic aerogel has been studied. An equation determining the dispersion of sound oscillations of the system has been derived. Solutions of the system have been obtained for low-frequency oscillations, whose propagation velocity is much lower than the speed of first sound in pure 3He, for directions along and across the anisotropy axis of the aerogel. In the same approximation, a solution for purely shear oscillations of the system has been found under special boundary conditions corresponding to an incompressible liquid surrounding the system. The temperature dependence of the found frequencies has been compared with the existing experimental data.


INTRODUCTION
The polar phase of superfluid 3 He belongs to topological superfluid phases and has a number of unique properties. The order parameter of this phase is a complex 3 × 3 matrix , where d μ and m j are the components of the unit vectors in the spin and orbital spaces, respectively [1]. The spectrum of quasiparticles in the polar phase has a topologically stable Dirac line of zeros on the equator of the Fermi surface in the plane perpendicular to the vector m. As a result, superfluid properties of the system are described by a uniaxial superfluid density tensor with the maximum principal value along the m direction. The topological properties of this phase are also manifested in the existence of a specific excitation branch at the phase interface, which is due to the principle of correspondence and belongs to Majorana surface states [2]. The polar phase can be observed experimentally only in the presence of nematic aerogel with a high degree of anisotropy, i.e., aerogel where strands are well codirected (nafen, mullite aerogel) [3,4]. In the presence of strands of the aerogel in the system, interesting topological defects-half-quantum vortices [5,6], which were observed in the experiment reported in [7]-can be stabilized. In addition to the mentioned topological properties of the polar phase, the superfluid response of the system is also of interest. Superfluid properties of the system can be observed in experiments on oscillating of the aerogel inside the volume filled with superfluid 3 He [8,9]. A feature of the problem is the effective interaction between the superfluid component and the framework of the aerogel. As a result, the oscillating spectrum of the system is determined not only by the superfluid properties of the polar phase but also by the elastic properties of the aerogel. Since the elastic properties of the nematic aerogel are obviously anisotropic, interesting mixed vibrational modes, which are impossible in the isotropic case, can be expected in the system. Two oscillating modes were detected in the mentioned experiments on oscillations of the nematic aerogel in superfluid 3 He [8,9]. One of the modes corresponds to mechanical vibrations of the system and is observed in the entire temperature range. The temperature dependence the frequency of the first mode is due both to the temperature dependence of the viscosity of 3 He surrounding the aerogel and to change in the flow-lines around the aerogel at the time of the superfluid transition in the aerogel. An attempt to describe the temperature dependence of the frequency of the first mode taking into account the potential flow of superfluid currents through the aerogel under the condition that both the liquid and aerogel are incompressible was performed earlier [10]. The second oscillating mode appears only at the time of the superfluid transition in the aerogel. The features of the observed temperature dependence of the frequency for the second mode is its fast increase near the transition point with the subsequent approximate saturation at about 1.6 kHz. Such a low frequency of oscillations excludes the possibility of locking of the first sound of 3 He in this mode. The frequency is saturated in a quite narrow temperature range of several hundredths of the transition temperature T c . This feature distinguishes the observed mode from the "slow" mode excited in experiments on the propagation of sound in silica aerogels [11]. The aims of this work are

CONDENSED MATTER
to study the acoustic spectrum of the polar phase and nematic aerogel and to identify the experimentally observed second vibrational mode. The accurate derivation of the system of hydrodynamic equations necessary to solve this problem is a separate interesting item, but it is beyond the scope of this work. In this work, the acoustic spectrum of the system is determined using the system of linear hydrodynamic equations that can be obtained heuristically as in [12]. The system is solved for the actual case of low-frequency oscillations whose propagation velocity is much lower than the speed of first sound in pure superfluid 3 He. Effective boundary conditions describing the case where the system is surrounded by an incompressible superfluid liquid are discussed qualitatively. It is noteworthy that the system of linear hydrodynamic equations for the polar phase in the nematic aerogel was derived in [13], but the results obtained in [13] cannot be used in this work because of inaccuracies in it.

LINEARIZED HYDRODYNAMIC EQUATIONS FOR THE POLAR PHASE IN THE NEMATIC AEROGEL
The system of hydrodynamic equations for the considered system consists of five equations describing the conservation laws of the mass of the aerogel, mass of 3 He, the total momentum of the composite system, entropy (damping is ignored), and the potentiality of superfluid motion. The density of the aerogel ρ a , the density of 3 He ρ l , entropy per unit volume s, superfluid velocity v s , and local displacement vector of particles of the aerogel u are used as independent hydrodynamic variables. The last quantity can be used as an independent quantity under the no-slip condition between the aerogel framework and the normal component of 3 He, which is satisfied at low frequencies of motion. In this case, the velocity of the normal component and the total current density are determined by the expression j i = (ρ a δ ij + (ρ n ) ij ) + (ρ s ) ij (v s ) j , where (ρ s ) ij and (ρ n ) ij ( ) are the tensors of the superfluid and normal components of 3 He, respectively. The spin vector d and orbital vector m, which are hydrodynamic variables specific to the polar phase, are ignored. The former is not involved in motion because it is uncoupled with other hydrodynamic variables in the absence of the spin-orbit coupling. The inclusion of the latter exceeds the accuracy of the linear approximation and long-wavelength limit used below because the energy of the texture of the orbital vector includes higher derivatives of the displacement vector than the elastic energy of the aerogel. Thus, the linearized equations for the system under consideration have the form where , , and δρ a = ρ aare small deviations of the entropy, density of 3 He, and density of the aerogel from the equilibrium values, respectively; and , , and are the changes in the chemical potential of the liquid per particle, pressure, and stress tensor, respectively. The presence of the stress tensor of the aerogel in the law of conservation of the total momentum of the system is an additional feature of the considered system. If the pressure is defined as a change in the total energy of the composite system under the variation of its volume, this stress tensor should refer to the elastic energy independent of a change in the volume of the system. Since the change in the volume is determined by the divergence of the displacement vector, to exclude the mentioned terms, should be set. To close the system, it is necessary to express , , and in terms of small deviations of the introduced hydrodynamic variables. To do this, the energy per unit volume is expanded in relative deviations from the equilibrium value, disregarding the kinetic energy, up to the second order terms: (6) where the subscripts a and l mark quantities for the aerogel and 3 He, respectively; ; the z axis is directed along the anisotropy axis of the aerogel; ; and the tensors and are given by the expressions c a1 , c l1 , etc., are the phenomenological coefficients with a dimension of velocity including the speeds of the first and second sounds in superfluid 3 He c l1 and c l2 , respectively; the speeds of the ith sound in the aerogel c ai determined by its elastic coefficients; the velocities c as , c ls , and c us associated with the thermal expansion effect in the aerogel and 3 He; and the coefficients c al and c ul describing the interaction between the aerogel and 3 He that are the most interesting for the problem under consideration. Since the number of variables is large, conditions of thermodynamic stability of the system are not considered below; i.e., the consideration below concerns the region of the parameters where the above quadratic form is positive definite. Several remarks are in order concerning values of the phenomenological coefficients in the expansion of the energy. The density of 3 He in the systems under consideration is on the order of the density of the aerogel; i.e., ρ a ~ ρ l . The speed of first sound in 3 He is much higher than the speeds of sound in the aerogel; i.e., c l1 ~ 300 m/s ≫ c ai . The speed of second sound in superfluid Fermi systems is also very low. The elastic characteristics of nematic aerogels have not yet been systematically measured. However, some relations between the phenomenological coefficients c ai can be assumed: the largest coefficient is c a5 because it determines the elasticity of the system along the z axis coinciding with the direction of the strands of the aerogel. It can be accepted that velocities associated with the thermal expansion effect are much lower than the speed of first sound and all phenomenological velocities in the aerogel. Finally, the velocities describing the interaction of the aerogel with 3 He should be much lower than the speeds of first sound because of a low concentration of impurities. Summarizing the aforesaid, we can confirm that the speed of first sound in the system under consideration is much higher than all other velocities.
First, using the above quadratic form, we obtain exact expressions for small variations of the chemical potential of the liquid, pressure, and stress tensor. These expressions can then be simplified using the above quantitative relations between the phenomenological coefficients. By definition,   1) and (2), Eq. (9) can be simplified to the form (11) where (12) In the linear approximation, the variation of the pressure is given by the formula A similar algebra for , , and gives the following expressions for the variation of the pressure and stress tensor: where the second and third terms on the right-hand side of Eq. (14) include the coefficient of the first term in Eq. (5)    δϕ ω δϕ = ω δϕ δ + δϕ − δϕ

LOW-FREQUENCY MODES
The exact solution of Eq. (32) is too lengthy. We consider only several simple limits corresponding to recent experiments [8]. Resonances observed in oscillating experiments with the nematic aerogel in 3 He have frequencies of about 1 kHz. Since the characteristic wavelengths of excited waves are determined by the sizes of the aerogel about mm, the first sound in 3 He inside the aerogel is not excited in the discussed experiments. For this reason, we solve Eq. (32) only for low-frequency oscillating modes of the system, i.e., taking into account the small parameter in this equation. Furthermore, since the speed of sound in 3 He is about 300 m/s and the superfluid density is measured with an accuracy limited by the condition , the more stringent condition is satisfied in the entire available temperature range. Thus, in the zeroth order in the mentioned small parameters, the first term and the ratios of all velocities to the speed of first sound c l1 in the denominators of the expressions for , , and can be neglected. In this approximation, the ⊥ × ω ρ ρ − ρ − ρ .
where . The first oscillating mode obviously corresponds to transverse oscillations of the aerogel and the normal component of 3 He, which propagate along the z axis. This solution exists in both the normal and superfluid phases and is due exclusively to the elasticity of the aerogel. The second vibrational mode is an analog of second sound in the superfluid system. Indeed, in the limit of zero velocities associated with the aerogel and zero density of aerogel, the conventional expression for the frequency of second sound in the superfluid system is obtained: In is noteworthy that the propagation velocity of oscillations of this mode is now determined only by the elasticity of the aerogel along the z axis because of the interaction with the aerogel.
We obtain the following results for oscillations propagating in the directions perpendicular to the z axis: The spectrum of eigenmodes of the system depends on the boundary conditions on the surface of the aerogel. In the considered experimental situation where the rectangular parallelepiped aerogel is in a cell and is surrounded by 3 He, it is additionally necessary to solve hydrodynamic equations for the corresponding superfluid phase outside the aerogel with the condition of absence of current at infinity. This problem is qualitatively discussed in the next section.

SHEAR OSCILLATIONS
We now approximately consider oscillations where all components of the displacement vector and the wave vector are nonzero. It is substantial that the aerogel is surrounded by superfluid 3 He. Simple estimates can demonstrate that shear oscillations can be excited in this case in the system. The spatial part of the solution inside the aerogel is sought in the form Oscillations of the aerogel can excite waves of the first and second sounds. Since the speed of second sound is much lower than the elastic velocities of the aerogel, second sound will be emitted to the environment, making a contribution to the damping of eigenmodes of the system. On the contrary, the speed of first sound is much higher than speeds of sound in the aerogel; as a result, waves of the first sound are damped far from ⊥ ρ ω + ρ + ρ