aa r X i v : . [ c ond - m a t . o t h e r] M a r Massless surface wave
I. Todoshchenko
Low Temperature Laboratory, Deptartment of Applied Physics,Aalto University University, 00076 AALTO, Finland ∗ (Dated: March 29, 2018)An interface between two media is topologically stable two-dimensional object where 3D-symmetrybreaks which allows for existence of many exotic excitations. A direct way to explore surface exci-tations is to investigate their interaction with the surface waves, such as very well known capillary-gravity waves and crystallization waves. Helium remains liquid down to absolute zero where bulkexcitations are frozen out and do not mask the interaction of the waves with the surface states. Herewe show the possibility of the new, massless wave which can propagate along the surface betweentwo different superfluids phases of He. The displacement of the surface in this wave occurs dueto the transition of helium atoms from one phase to another, so that there is no flow of particlesas densities of phases are equal. We calculate the dispersion of the wave in which the inertia isprovided by spin supercurrents, and the restoring force is magnetic field gradient. We calculate thedissipation of the wave and show the preferable conditions to observe it.
PACS numbers: 67.30.hj 67.30.hp 68.05.Cf
Watching the waves on a surface of ocean is probablyone of the oldest physical observation. Similar waves canpropagate along the surface between any two immisciblefluids, and particularly between liquid and its vapour, seeFig. 1a. The inertia of these waves is due to the motionof the liquid while the restoring force is due to gravity(long waves) and surface tension (short waves). Unlikethe most of known waves, dispersion relation of surfacewave is strongly nonlinear, ρω = ρgq + αq , (1)so that there is no phonon-like spectra even at q → ρ is the density of the liquid, ω is the angular fre-quency, g is the gravitational acceleration, α is the sur-face tension, and q is the wavevector. Capillary-gravitywaves have been observed on surface of He and He andhave been utilized for high-precesion measurements onthe surface tension. In this way critical exponents nearcritical liquid-gas point have been measured [1].For helium liquid-solid interface there opens a possibil-ity for quite exotic wave, a crystallization wave. At lowenough temperatures, where liquid phase is superfluidand thus provides extremely fast mass and heat trans-port, the interface between solid and liquid becomes mo-bile enough to support wave of crystallization and melt-ing. The dispersion relation is similar to that of waves onfree liquid surface, and the only difference is that for thesame amplitude of the wave, liquid phase carries smallermass flux (see Fig. 1b) which is proportional to densitydifference ∆ ρ between the solid and the liquid,∆ ρ ρ ω = ∆ ρgq + αq . (2) ∗ Electronic address: [email protected].fi
Crystallization waves have been predicted by Andreevand Parshin in 1978 [2] and discovered by Keshishev etal . two years later in He below 0.5 K [3]. By measuringcrystallization waves at surfaces of different orientationsthe singularity of the surface tension at the basal c-facetorientation has been observed [4].In this Letter we show the possibility of the new kindof surface wave which does not have any associated massflux. Imagine an interface between two immiscible liq-uids which both consist of the same atoms or molecules.Such interface will support, in addition to usual capillary-gravity wave described above, another type of wave whichis associated with the transition of particles near the sur-face from one phase to another one. Basically, such waveis an analog of the crystallization wave as one of thephases locally “grows” into another phase. The hydro-dynamical motion in both liquids is thus only due to thedifference in mass densities of the phases.Generally, it is possible that both phases have the sameor nearly the same mass densities. Then such a wave willnot have any hydrodynamical inertia and its frequencywill be infinitely high at any finite wavevector q unlesssome other kind of inertia is involved. The simplest ex-ample of the discussed system is liquid in contact withits vapour close to the critical liquid-vapour point wheredensities of the both phases equalize. Moreover, at thecritical point surface tension also vanishes, and thus thereis no restoring force unless some other force stabilizingthe surface is introduced.Although it seems impossible to satisfy all the condi-tions for the discussed massless wave, nevertheless, thereis an example of such system, a surface between two su-perfluid phases of helium-3, stabilized with the magneticfield [5, 6]. These two phases, being referred to as “A”and “B” phases, have the same mass densities, and thedescribed phase wave on the A-B interface does not re-quire mass transport in the bulk phases, see Fig. 1c.However, the A-phase has larger magnetic susceptibil- FIG. 1: Three types of surface waves. a) Usual capillary-gravity wave: oscillations of the surface are accompanied withthe motion of the liquid in the bulk. b) Crystallization wave:oscillations of the surface are accompanied with the muchslower motion of the liquid because density of liquid and solidare similar. c) Phase wave on the surface between two su-perfluid phases of He: no mass transport is needed becausedensities of both superfluid phases are equal. ity and thus prefer large magnetic field. By applyingvertical magnetic field gradient ∇ H one can stabilize theA-phase in high field region on top of the B-phase. Thefield gradient is therefore an analog of gravity and playsa role of the restoring force of the wave.As the magnetization of the A-phase is larger thanthe magnetization of the B-phase, the motion of the A-Binterface is accompanied with the spin currents in bothphases. As it was noted by Andreev, spin currents havea kinetic energy [7], and thus play the role of inertia. We therefore have a wave on the surface between twoliquids in which there is no mechanical motion at all: He particles transit from one phase to another stayingat rest, and both kinetic and potential energy of the waveare provided by the spin degree of freedom.As it was shown by Leggett [8], the spin dynamics inthe longitudinal direction can be described in the termsof the angle θ of rotation of the order parameter vector ~d around the field direction. The corresponding equationsfor the A-phase ( z >
0) and for the B-phase ( z <
0) are[7]:¨ θ a − c a ∂ θ a ∂z +Ω a θ a = 0 , ¨ θ b − c b ∂ θ b ∂z +Ω b θ b = 0 , (3)Here Ω a , Ω b , c a , and c b are longitudinal NMR frequen-cies and spin waves velocities of A- and B- phases, cor-respondingly. We consider a slow wave, ω ≪ Ω a , Ω b , forwhich the solution of (3) decaying at z = ±∞ is θ a,b ( z, t ) = θ a, b ( t ) exp (cid:0) ∓ Ω a,b c a,b z (cid:1) . (4)Following Andreev [7], we write the energy density of thespin waves as ε a,b = χ a,b γ [ c a,b ( ∂θ a,b ∂z ) + Ω a,b θ a,b ] , (5)where χ a and χ b are magnetic susceptibilities of thephases, and γ = 2 µ/ ~ is the gyromagnetic ratio. Af-ter integration over z we find the total kinetic energy ofthe wave, E kin = 12 χ a c a Ω a γ θ a + 12 χ b c b Ω b γ θ b . (6)The fluxes of the z component of the spin at the boundaryare given by j a,b = − χ a,b c a,b γ ∂θ∂z = ± χ a,b c a,b Ω a,b γ θ a, b . (7)The fluxes are connected by the boundary condition j a − j b = Hγ ( χ a − χ b ) ˙ ζ, (8)where ζ ( x, t ) = ζ ( t ) exp ( iqx ) is the displacement of theboundary from its equilibrium z = 0 position. Using (8)and minimizing kinetic energy (6) we obtain amplitudesof the spin currents, θ a = θ b = γH ( χ a − χ b ) χ a c a Ω a + χ b c b Ω b ˙ ζ. (9)Substituting amplitudes (9) to (6) we find the kineticenergy of the wave, E kin = (1 / M ˙ ζ with the effective“mass” of spin currents M = ( χ a − χ b ) H χ a c a Ω a + χ b c b Ω b . (10)The potential energy of the wave is due to the field gra-dient and due to the small surface tension α , E pot =(1 / χ a − χ b ) H ∇ H + αq ] ζ , and the dispersion rela-tion of the wave is ω = ( χ a − χ b ) H ∇ H + αq M . (11)Note that in the limit of long waves frequency doesnot depend on the wave vector and is determined by onlymagnetic parameters of superfluid phases and by the ge-ometry of the magnetic field. At low temperatures andlow pressure the A-B interface is stabilized in the field of H AB ≈ .
34 T [9], and the surface tension is α ≈ / m [10]. If we assume the field gradient of ∇ H ∼
10 T / m,then the characteristic wavelength at which the two termsin Eq. (11) equalize is λ c ∼ π r α ( χ a − χ b ) H ∇ H ∼ . (12)Longer waves are inherently magnetic, and their fre-quency ω ∼ p c Ω ∇ H/H does not depend on wavevectorbut can be tuned by the magnetic field gradient.The wave can be exited by applying an oscillating mag-netic field h = h exp ( iωt ) parallel to the constant sta-bilizing field H . This kind of experiment has been donein Lancaster by Bartkowiak et al. , who have measuredthe heat produced by the oscillating A-B interface in theultra low temperature limit at zero pressure [11]. The os-cillating field h causes the equilibrium vertical position ofthe interface to oscillate with the amplitude δζ = h/ ∇ H .The equation of motion of the interface can be writtenas M ¨ ζ + Γ ˙ ζ = − H ∇ H ( χ a − χ b )( ζ − δζ exp ( iωt )) == − M ω ( ζ − h/ ∇ H exp ( iωt )) (13)where Γ is the friction coefficient. The frequency depen-dence of the power dissipated in the wave per unit areais given by P = Γ | ˙ ζ | = Γ ω | ζ | == Γ2 ω ω ( ω − ω ) + ( ω Γ /M ) h ∇ H . (14)The quality factor of the wave Q = ω M/ Γ is propor-tional to the square root of the effective mass of the wave M which is very small because of the smallness of nuclearsusceptibilities, χ a = 3 . · − , χ b = 1 . · − at zerobar [12]. The longitudinal NMR frequencies in the lowtemperature limit at high pressure have been measuredto be Ω a | hp = 2 π ·
100 kHz, Ω b | hp = 2 π ·
250 kHz [13–16].As it was shown by Leggett, the longitudinal reso-nance frequency is scaled as Ω ∝ p N (0) aT c where N (0)is the density of states at the Fermi surface, and a isthe ratio of the relative specific heat jump to its BCSvalue (1.43) [17]. Using N (0) | hp = 1 . · J − m − , N (0) | bar = 0 . · J − m − [12], a | hp = 1 . a | bar =1, T c | hp = 2 .
49 mK, and T c | bar = 0 .
93 mK [18–20] wefind Ω a | bar ≃ π ·
25 kHz, and Ω b | bar ≃ π ·
65 kHz. Forthe estimation of the velocity c of spin waves one can usethe relation c = l (Ω / π ) where l ≈ µ m is the dipolehealing length. Finally, we find the mass of the wave athigh pressures and at zero bar, M | bar ≈ · − kg / m and M | hp ≈ · − kg / m .Another kind of the inertial mass M ∗ of the interfaceoriginates from the time variation of the order parame-ter near the moving interface. This mass has been firstconsidered by Yip and Leggett who have given a rough es-timate M ∗ ∼ − kg / m [21, 22]. Roughly, this mass issmaller than the magnetic mass by the factor ξ /l , where ξ = 18 ...
88 nm is the coherence length.There also exists usual material mass of the wave be-cause of the density difference ∆ ρ AB ∼ − kg / m be-tween phases [22]. The applied magnetic field increasesthe density difference because the A-phase has large sus-ceptibility. The magnetostriction effect can be estimatedas the additional effective pressure δP = (1 / χH which causes compression of the liquid by an amount δρ/ρ = βδP ; where β = 5 . · − / Pa is the compress-ibility [23]. The additional density difference due to mag-netostriction is thus ∆ ρ AB ( H ) = (1 / χ A − χ B ) H βρ ∼ − kg / m which is negligible compared to zero fielddifference ∆ ρ AB . The contribution to the mass of thewave from density difference of the phases is given by m = ∆ ρ AB / ( ρq ) ∼ − kg / m which is nine orders ofmagnitude smaller than the spin current mass M .With the estimated above mass M ∼ − kg / m andwith the field gradient of few Tesla per meter we can es-timate the resonant frequency of the wave ω to be of theorder of 2 π · T [21, 24]. It has been measuredat temperatures close to the A-B transition at high pres-sures in Los Alamos [25], Γ | . T c ,hp = 0 .
07 kg / (m s).This value is in very good agreement with theoretical es-timation of Kopnin, Γ ≈ π N (0) T / (30 v F ∆ A ) ( v F isFermi velocity, ∆ A is the BSC energy gap) [24]. The lowtemperature value of the friction coefficient can be foundfrom the Lancaster experiment [11]. According to theEq. (14), in the case of strong damping Γ /M ≫ ω , thedissipation has a plateau at high frequencies, P pl = 12Γ ω M h ∇ H = h H ( χ a − χ b ) , (15)which depends only on the amplitude h of the oscillat-ing field and on the friction coefficient Γ. Indeed, suchplateau has been observed in Lancaster at frequencieslarger than 10 Hz with the level P pl = 2 · − W / m independent on the field gradient [11]. The amplitudeof the oscillating field used in Lancaster experiment was h = 0 .
64 mT which gives Γ | . T c , = 3 · − kg / (m s).This is about three times larger than the theoretical es-timate.Due to the smallness of the mass of the wave, the char-acteristic attenuation rate of the wave is very fast evenat ultra low temperatures, Γ | . T c , /M ≈ · s − which should be compared to ω ≈ . · s − for thestrongest field gradient ∇ H = 2 T / m used in Lancaster experiments. To observe resonance of the A-B wave oneshould increase the field gradient by at least order ofmagnitude to shift the resonant frequency ω above thedissipation rate.The excess by factor of three of the dissipation mea-sured in Lancaster over the theoretical value might be theindication of the contribution of the surface states whichshould dominate at zero temperature limit. The A-B in-terface between two fermionic superfluids is probably therichest surface in nature and promises to support varietyof surface states such as Majorana fermions and anyons(particles which are neither bosons nor fermions) [26].The proposed magnetic massless surface wave could openaccess to these exotic surface states which may cause ad-ditional dissipation and contribute to the mass of thewave. Acknowledgments
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