Second Sound Mode Research Articles

Dipole superfluid hydrodynamics

We construct a theory of hydrodynamic transport for systems with conserved dipole moment, U(1) charge, energy, and momentum. These models have been considered in the context of fractons, since their elementary and isolated charges are immobile by symmetry, and have two known translation-invariant gapless phases: a “p-wave dipole superfluid” phase where the dipole symmetry is spontaneously broken and a “s-wave dipole superfluid” phase where both the U(1) and dipole symmetries are spontaneously broken. We argue on grounds of symmetry and thermodynamics that there is no transitionally-invariant gapless fluid with unbroken dipole symmetry. In this work, we primarily focus on the hydrodynamic description of p-wave dipole superfluids, including leading dissipative corrections. That theory has, in a sense, a dynamical scaling exponent z = 2, and its spectrum of fluctuations includes novel subdiffusive modes ω ∼ −ik4 in the shear sector and magnon-like sound mode ω ∼ ±k2 − ik2. By coupling the fluid to background fields, we find response functions of the various symmetry currents. We also present a preliminary generalization of our work to s-wave dipole superfluids, which resemble z = 1 fluids and feature sound waves and diffusive shear modes, as in an ordinary fluid. However, the spectrum also contains a magnon-like second-sound mode ω ∼ ±k2 ± k4 − ik4 with subdiffusive attenuation.

Non-equilibrium effective field theory and second sound

We investigate the phenomenon of second sound in various states of matter from the perspective of non-equilibrium effective field theory (EFT). In particular, for each state of matter considered, we find that at least two (though sometimes multiple) qualitatively different EFTs exist at finite temperature such that there is always at least one EFT with a propagating second-sound wave and at least one with no such second-sound wave. To aid in the construction of these EFTs, we use the method of cosets developed for non-equilibrium systems. It turns out that the difference between the EFTs with and without second-sound modes can be understood as arising from different choices of a new kind of inverse Higgs constraint. Finally, we demonstrate that it is possible to bypass the need for new inverse Higgs constraints by formulating EFTs on a new kind of manifold that is like the usual fluid worldvolume, but with reduced gauge symmetries.

Hydrodynamics of a superfluid smectic

We determine the hydrodynamic modes of the superfluid analog of a smectic-A liquid crystal phase, i.e., a state in which both gauge invariance and translational invariance along a single direction are spontaneously broken. Such a superfluid smectic provides an idealized description of the incommensurate supersolid state realized in Bose–Einstein condensates with strong dipolar interactions as well as of the stripe phase in Bose gases with spin–orbit coupling. We show that the presence of a finite normal fluid density in the ground state of these systems gives rise to a well-defined second-sound type mode even at zero temperature. It replaces the diffusive permeation mode of a normal smectic phase and is directly connected with the classic description of supersolids by Andreev and Lifshitz in terms of a propagating defect mode. An analytic expression is derived for the two sound velocities that appear in the longitudinal excitation spectrum. It only depends on the low-energy parameters associated with the two independent broken symmetries, which are the effective layer compression modulus and the superfluid fraction.

Chiral Second-Sound Collective Modes at the Edge of 2D Systems with a Nontrivial Berry Curvature.

We study the thermal transport in two-dimensional systems with a nontrivial Berry curvature texture. The physical realizations are many; for the sake of definiteness, we consider undoped graphene gapped by the presence of an aligned hexagonal-boron-nitride substrate. The same phenomenology applies, i.e., to surface states of 3D topological insulators in the presence of a uniform magnetization. We find that chiral valley-polarized second-sound collective modes propagate along the edges of the system. The localization length of the edge modes has a topological origin stemming from the anomalous velocity term in the quasiparticle current. At low temperature, the single-particle contribution to the transverse thermal conductance is exponentially suppressed, and only second-sound modes carry heat along the boundary. A sharp change in the behavior of the thermal Hall conductance, extracted from nonlocal measurements of the temperature along the edge, marks the onset of ballistic heat transport due to second-sound edge modes.

Second-sound studies of coflow and counterflow of superfluid 4He in channels

We report a comprehensive study of turbulent superfluid 4He flow through a channel of square cross section. We study for the first time two distinct flow configurations with the same apparatus: coflow (normal and superfluid components move in the same direction), and counterflow (normal and superfluid components move in opposite directions). We realise also a variation of counterflow with the same relative velocity, but where the superfluid component moves while there is no net flow of the normal component through the channel, i.e., pure superflow. We use the second-sound attenuation technique to measure the density of quantised vortex lines in the temperature range 1.2 K ≲ T ≲ Tλ ≈ 2.18 K and for flow velocities from about 1 mm/s up to almost 1 m/s in fully developed turbulence. We find that both the steady-state and temporal decay of the turbulence significantly differ in the three flow configurations, yielding an interesting insight into two-fluid hydrodynamics. In both pure superflow and counterflow, the same scaling of vortex line density with counterflow velocity is observed, L∝Vcf2, with a pronounced temperature dependence; in coflow instead, the vortex line density scales with velocity as L ∝ V3/2 and is temperature independent; we provide theoretical explanations for these observations. Further, we develop a new promising technique to use different second-sound resonant modes to probe the spatial distribution of quantised vortices in the direction perpendicular to the flow. Preliminary measurements indicate that coflow is less homogeneous than counterflow/superflow, with a denser concentration of vortices between the centre of the channel and its walls.

Silent flocks: constraints on signal propagation across biological groups.

Experiments find coherent information transfer through biological groups on length and time scales distinctly below those on which asymptotically correct hydrodynamic theories apply. We present here a new continuum theory of collective motion coupling the velocity and density fields of Toner and Tu to the inertial spin field recently introduced to describe information propagation in natural flocks of birds. The long-wavelength limit of the new equations reproduces the Toner-Tu theory, while at shorter wavelengths (or, equivalently, smaller damping), spin fluctuations dominate over density fluctuations, and second-sound propagation of the kind observed in real flocks emerges. We study the dispersion relation of the new theory and find that when the speed of second sound is large, a gap in momentum space sharply separates first- from second-sound modes. This gap implies the existence of silent flocks, namely, of medium-sized systems across which information cannot propagate in a linear and underdamped way, either under the form of orientational fluctuations or under that of density fluctuations, making it hard for the group to achieve coordination.

First and second sound of a unitary Fermi gas in highly elongated harmonic traps

Using a variational approach, we present the full solutions of the simplified one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which was recently derived by G. Bertaina and coworkers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of the temperature and the form of superfluid density. We show that the density fluctuations in second-sound modes, due to their coupling to first-sound modes, are large enough to be measured in current experimental setups such as that exploited by M. K. Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second-sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far has remained elusive.

Probing the critical exponent of the superfluid fraction in a strongly interacting Fermi gas

We theoretically investigate the critical behavior of a second-sound mode in a harmonically trapped ultracold atomic Fermi gas with resonant interactions. Near the superfluid phase transition with critical temperature ${T}_{c}$, the frequency or the sound velocity of the second-sound mode crucially depends on the critical exponent $\ensuremath{\beta}$ of the superfluid fraction. In an isotropic harmonic trap, we predict that the mode frequency diverges like ${(1\ensuremath{-}T/{T}_{c})}^{\ensuremath{\beta}\ensuremath{-}1/2}$ when $\ensuremath{\beta}<1/2$. In a highly elongated trap, the speed of the second sound reduces by a factor of $1/\sqrt{2\ensuremath{\beta}+1}$ from that in a homogeneous three-dimensional superfluid. Our prediction could readily be tested by measurements of second-sound wave propagation in a setup, such as that exploited by Sidorenkov et al. [Nature (London) 498, 78 (2013)] for resonantly interacting lithium-6 atoms, once the experimental precision is improved.

Light scattering in a phonon gas

Light-scattering spectrum in dielectric crystals is studied with extended thermodynamics (ET) for a phonon gas, covering from hydrodynamic to ballistic (collisionless) regimes. The ET equation is solved to obtain the power spectrum of the energy density in a phonon-gas mixture, which consists of interacting phonon gases of longitudinal and transverse acoustic (LA and TA) phonon modes. In the hydrodynamic regime, where phonon collisions take place frequently, it is found that the light-scattering spectrum consists of two components, which can be interpreted as the two normal modes formed by the two second-sound modes defined in each of the LA and TA phonon gases. Out of the two normal-mode spectra, the low-frequency component arises from thermal fluctuations and gives rise to a narrow quasielastic spectrum, which corresponds to the well-known thermal Rayleigh scattering due to thermal diffusion, i.e., due to overdamped second sound. With sufficient momentum-conserving phonon collisions (normal phonon scattering), the narrow quasielastic component is demonstrated to develop, from the diffusive central peak, into a pair of shifted inelastic peaks due to propagation of underdamped second sound. The other spectrum component gives rise to a much broader quasielastic scattering, whose wing extends out to the Brillouin-scattering lines of the LA and TA phonons (first sounds). The broader quasielastic spectrum has a linewidth equal to the phonon-collision rate, which suggests that this component originates from a nonequilibrium process in the phonon gas. As the ballistic regime is approached, the line shapes and linewidths of the two normal-mode spectra approach each other, and the two components finally coincide in the limit of the ballistic regime, which is in good agreement with the reported behavior for these spectra that were experimentally observed in many crystals. In the ballistic limit, the light-scattering mechanism in the present model is found to become formally equivalent to the previously proposed microscopic framework, i.e., the second-order difference Raman scattering (two-phonon difference light scattering) on a same phonon dispersion. The derived spectral formula is fitted to the spectra previously observed in the experiments for rutile $({\text{TiO}}_{2})$ and strontium titanate $({\text{SrTiO}}_{3})$. The fits are quite successful in wide ranges of frequency and temperature, i.e., regardless of degree of nonequilibrium, owing to the ET analysis. The relaxation times for the normal and resistive phonon collisions (${\ensuremath{\tau}}_{\text{N}}$ and ${\ensuremath{\tau}}_{\text{R}}$) are determined through the analysis. The temperature dependences of ${\ensuremath{\tau}}_{\text{N}}$ and ${\ensuremath{\tau}}_{\text{R}}$ indicate that the origin of the broad shifted peaks (the ``broad doublet''), which were observed in ${\text{SrTiO}}_{3}$ at around 30 K, is likely due to underdamped second sound at least in a narrow temperature range around 30 K.

Lattice-Boltzmann modeling of phonon hydrodynamics

Based on the phonon Boltzmann equation, a lattice-Boltzmann model for phonon hydrodynamics is developed. Both transverse and longitudinal polarized phonons that interact through normal and umklapp processes are considered in the model. The collision term is approximated by the relaxation time model where normal and umklapp processes tend to relax distributions of phonons to their corresponding equilibrium distribution functions-the displaced Planck distribution and the Planck distribution, respectively. A macroscopic phonon thermal wave equation (PTWE), valid for the second-sound mode, is derived through the technique of Chapman-Enskog expansion. Compared to the dual-phase-lag (DPL) -based thermal wave equation, the PTWE has an additional fourth-ordered spatial derivative term. The fundamental difference between the two models is discussed through examining a propagating thermal pulse in a single-phased medium and the transient and steady-state transport phenomena on a two-layered structure subjected to different temperatures at boundaries. Results show that transport phenomena are significantly different between the two models. The behavior exhibited by the DPL model, as thermal wave behavior goes over to diffusive behavior, tau_{T}-->tau_{q} is incompatible with any microscopic phonon propagating mode. Unlike the DPL model, in which tau_{T} only has an effect on the transient phenomena, in the PTWE model tau_{T} shows effects on phenomena at both transient and steady state. With the intrinsic compatibility to the microscopic state, discontinuous quantities, such as a jump of temperature at a boundary or at an interface, can be calculated naturally and straightforwardly with the present lattice-Boltzmann method.

Dynamical models and the phase ordering kinetics of thes=1spinor condensate

The $s=1$ spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We investigate the formation of magnetic domains at finite temperature and magnetic field in two dimensions in an optical trap. We study the general ground state phase diagram of a spin-1 system and focus on a phase that has a magnetic Ising order parameter and numerically determine the nature of the finite-temperature superfluid and magnetic phase transitions. We then study three different dynamical models: model A, which has no conserved quantities, model F, which has a conserved second-sound mode, and the Gross-Pitaevskii (GP) equation, which has a conserved density and magnetization. We find the dynamic critical exponent to be the same for models A and F $(z=2)$ but different for GP $(z\ensuremath{\approx}3)$. Externally imposed magnetization conservation in models A and F yields the value $z\ensuremath{\approx}3$, which demonstrates that the only conserved density relevant to domain formation is the magnetization density.

Emission of the second sound with an expanding3He-concentrated droplet and phase-separation kinetics in a superfluid3He−4Hemixture

We study the growth kinetics of a droplet of the ${}^{3}\mathrm{He}$-concentrated phase in a superfluid ${}^{3}{\mathrm{H}\mathrm{e}\ensuremath{-}}^{4}\mathrm{He}$ supersaturated mixture. The growth equation, which generalizes the Rayleigh-Plesset equation for a radial expansion of bubbles in the normal fluids, is derived under the assumption of an arbitrary boundary condition for the normal velocity. The total intensity of the first- and second-sound emissions for a droplet expanding in the superfluid mixture is calculated. The emission of the second-sound mode is found to be predominant due to the smallness of the second-sound velocity compared with the velocity of the first sound. In contrast to demixing normal mixtures the diffusion and heat conduction processes play a minor role in the phase-separation kinetics of the supersaturated ${}^{3}{\mathrm{H}\mathrm{e}\ensuremath{-}}^{4}\mathrm{He}$ superfluid mixtures.

Gradient expansions in kinetic theory of phonons

For simple models of the phonon transport in rigid insulators, it is demonstrated that the extended diffusional mode transforms into a second-sound mode after its coupling to a nonhydrodynamic mode at some critical value of the wave vector. This criticality shows up as a branching point of the extension of the diffusional mode within the Chapman-Enskog method, found explicitly for these models. The solution is used to test validity of several nonpolynomial approximate methods to capture this criticality.

Dynamics and rheology of diblock copolymers quenched into microphase-separated states.

We propose a theoretical interpretation of observed anomalous rheological behavior of diblock copolymers quenched into microphase-separated states having locally lamellar morphology with disorder. The proposed mechanism of the anomaly is associated with the second-sound modes occurring in a locally lamellar structure that are overdamped in a wide wave-vector region due to large shear viscosity. The resulting complex shear modulus is proportional to (i\ensuremath{\omega}${)}^{1/2}$, which contains no adjustable parameter and is consistent with the experimental result by Bates et al. [J. Chem. Phys. 92, 6255 (1990)].

Static and dynamic properties of incommensurate smectic-AIC liquid crystals.

We study the elasticity, topological defects, and hydrodynamics of the recently discovered incommensurate smectic (${A}_{\mathrm{IC}}$) phase, characterized by two collinear mass density waves of incommensurate spatial frequency. The low-energy long-wavelength excitations of the system can be described by a displacement field u(x) and a ``phason'' field w(x) associated, respectively, with collective and relative motion of the two constituent density waves. We formulate the elastic free energy in terms of these two variables and find that when w=0, its functional dependence on u is identical to that of a conventional smectic liquid crystal, while when u=0, its functional dependence on w is the same as that for the angle variable in a slightly anisotropic XY model. An arbitrariness in the definition of u and w allows a choice that eliminates all relevant couplings between them in the long-wavelength elastic energy. The topological defects of the system are dislocations with nonzero u and w components. We introduce a two-dimensional Burgers lattice for these dislocations, and compute the interaction between them. This has two parts: one arising from the u field that is short ranged and identical to the interaction between dislocations in an ordinary smectic liquid crystal, and one arising from the w field that is long ranged and identical to the logarithmic interaction between vortices in an XY model. The hydrodynamic modes of the ${A}_{\mathrm{IC}}$ include first- and second-sound modes whose direction-dependent velocities are identical to those in ordinary smectics. The sound attenuations have a different direction dependence, however. The breakdown of hydrodynamics found in conventional smectic liquid crystals, with three of the five viscosities diverging as 1/\ensuremath{\omega} at small frequencies \ensuremath{\omega}, occurs in these systems as well and is identical in all its details. In addition, there is a diffusive phason mode, not found in ordinary smectic liquid crystals, that leads to anomalously slow mechanical response analogous to that predicted in quasicrystals, but on a far more experimentally accessible time scale.

Two-phase sound in stratified3He-4He mixtures

The hydrodynamic boundary conditions at the normal liquid-superfluid interface in stratified3He-4He mixtures are formulated and applied to the propagation of low frequency two-phase sound. Coupling between the first- and second-sound modes occurs at low temperature because of the large increase of the Fermi-liquid transport coefficients. Starting with a normal homogeneous mixture, a coupling in a narrow range of temperature near the phase separation point is also predicted.

Absolute measurement of the critical behavior of the smectic elastic constant of bilayer and monolayer smectic-Aliquid crystals on approaching the transition to the nematic phase

A new technique which provides an absolute measurement of the smectic elastic constant $B$ is presented. This technique measures second sound on samples whose top surface is an air---liquid-crystal interface. The free surface of the sample is electrically driven and the resulting surface displacements are measured with an optical heterodyne technique. The smectic elastic constant is obtained from the spectrum of the scattered light which is determined by the bulk second-sound modes. Two bilayer and two monolayer smectic-$A$ liquid crystals were studied. Near the smectic-$A$---to---nematic phase transition the data can be described by a simple power law $B={B}_{1}{(T\ensuremath{-}{T}_{\mathrm{NA}})}^{\ensuremath{\varphi}}$. However, the resulting critical exponents $\ensuremath{\varphi}$ are not universal. In addition, the ${B}_{1}$ values for the bilayer smectics are considerably smaller than the values for the monolayer smectics.

Mutual friction and vortex motion near the superfluid transition

The mutual friction parametersB and B′ for a moving vortex are calculated near the superfluid transition. They are proportional to the kinetic coefficient associated with the order parameter and, asT →λ, diverge as (Tλ − T)−1/3, in agreement with experiment. The nonlinear couplings between the order parameter Ψ and the entropym, both the reversible one and the one in the free energy, are found to be crucial in the mutual friction near the λ point. These couplings were neglected in a previous paper by the author. First, the reversible coupling in the dynamic equations makes the chemical potential deviation long-ranged and causes the dissipation to take place only near the vortex core. Second,B′ can diverge asT → Tλ only in the presence of the coupling of the formm|Ψ|2 in the free energy. Thus theE model of Halperin et al., where the latter coupling is absent, cannot explain the critical anomaly ofB′. The helical mode of a single vertex line is also investigated and its dispersion relation is found to be quite different from that at low temperatures. This mode has the same time scale as that of the second-sound mode when the wave vectors are of the order of the inverse correlation length, thus obeying the usual dynamic scaling law. The time correlation functions of the displacement fluctuations of a vortex line are also obtained. The force acting on a moving vortex is calculated and is found to be equal to the classical Magnus force.

Microscopic modes in a Fermi superfluid. II. Dispersion relations

This is the second of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this paper we obtain approximate solutions to the linearized kinetic equations for the bogolon spin density and total density for the case of long-wavelength disturbances after long times when a fluctuating superfluid velocity is present. In so doing, we obtain microscopic expressions for the amplitude and dispersion relations for the spin diffusion mode, the two shear modes, and the four longitudinal modes (two first-sound modes and two second-sound modes).

Early stages of spinodal decomposition in superfluidHe3-He4mixtures

A generalization of the linearized analysis of Cahn, Hilliard, and Cook is applied to a simple model of phase separation dynamics in $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures at early times. An instability region is found in which the structure function for concentration fluctuations exhibits maximum growth at a nonzero wave vector. The coupling of concentration fluctuations to the superfluid second-sound mode produces an additional flickering component in the scattering. Equilibrium experimental data from both sides of the coexistence curve are used to estimate the size of these effects. The instability region is found to be quite asymmetrically placed inside the two-phase region.