NMR-like effect on Anisotropic Magnetic Moment of Surface Bound States in Topological Superfluid 3 He-B
aa r X i v : . [ c ond - m a t . o t h e r] A p r NMR-like effect on Anisotropic Magnetic Moment of Surface Bound States inTopological Superfluid He-B
M. ˇCloveˇcko, E. Gaˇzo, M. Skyba, and P. Skyba ∗ Centre of Low Temperature Physics, Institute of Experimental Physics SAS, Watsonova 47, 04001 Koˇsice, Slovakia. (Dated: April 18, 2019)We present experimental observation of a new phenomenon, that we interpret as NMR-like effecton anisotropic magnetic moment of the surface Andreev bound states in topological superfluid He-B at zero temperature limit. We show that an anisotropic magnetic moment formed near thehorizontal surface of a mechanical resonator due to symmetry violation of the superfluid He-B orderparameter by the resonator’s surface may lead to anomalous damping of the resonator motion inmagnetic field. In difference to classical NMR technique, here NMR was excited using own harmonicmotion of the mechanical resonator, and nuclear magnetic resonance was detected as a maximumin damping when resonator’s angular frequency satisfied the Larmor resonance condition.
PACS numbers: 67.30.-n, 67.30.H-, 67.30.hj, 67.30.er, 67.80.D-, 67.80.dk
INTRODUCTION
Superfluid phases of helium-3 provide one of the mostcomplex and purest physical system to which we haveaccess to. This unique system is also serving as a modelsystem for high energy physics, cosmology and quan-tum field theories. In fact, the phase transition of Heinto a superfluid state violates simultaneously three sym-metries: the orbital, the spin and the gauge symme-try (SO L (3) × SO S (3) × U(1)). Either A or B superfluidphase of He created in zero magnetic field resembles thephysical features comparable with those described by theStandard model or by the Dirac vacuum, respectively [1].Application of magnetic field breaks the spin symmetryand this leads to formation of A phase in narrow regionjust below superfluid transition temperature [2]. Further,embedding of the anisotropic impurity into He in forme.g. nematically ordered aerogel violates the orbital sym-metry, which is manifested by formation of a polar phaseof superfluid He [3, 4]. Finally, the orbital symmetry ofthe superfluid condensate is also violated near the sur-face of any object of the size of the coherence length ξ ( ξ ∼ nm ) being immersed in superfluid He-B. Pres-ence of the surface enforces only the superfluid compo-nent that consists of the Cooper pairs having their orbitalmomenta oriented in direction to the surface normal andsuppresses all others. This results in the distortion of theenergy gap in direction parallel to the surface normal onthe distance of a few coherence lengths from the surface.The gap distortion leads to a strong anisotropy in spinsusceptibility of the superfluid surface layer of He [5],as well as to the motional anisotropy of fermionic excita-tions trapped in surface Andreev bound states (SABS).It is worth to note that the dispersion relation of someof these excitations resembles the features of Majoranafermions, the fermions, which are their own antiparticles[6–9].This article deals with experimental observations of anew phenomenon, that we interpret as NMR-like effect originating from the surface paramagnetic layer in su-perfluid He-B at zero temperature limit. However, incontrast to traditional NMR techniques, the magneticresonance was excited using a mechanical resonator os-cillating in magnetic field and detected as an additional,magnetic field dependent mechanical damping of the res-onator’s motion.In order to be able to study physical properties of thesurface states using mechanical resonators, the superfluid He-B should be cooled to zero temperature limit. Whensuperfluid He-B is cooled below 250 µ K, a flux of the vol-ume excitations interacting with a mechanical resonatorfalls with temperature as φ V ∼ D ( p F ) T exp( − ∆ /k B T )due to presence of the energy gap ∆ in the spectrumof excitations [10]. Here, D ( p F ) denotes the density ofstates at the Fermi level, p F is the Fermi momentum and k B is the Boltzmann constant. On the other hand, thegap distortion in vicinity of the surface modifies the dis-persion relation for the excitations trapped in SABS to”A-like” phase and corresponding flux of the surface ex-citations φ S varies non-exponentially ( φ S ∼ D S ( p F ) T ),where D S ( p F ) is the density of states near surface whichdepends on the surface quality. Therefore, one may ex-pect that in superfluid He-B at higher temperatures φ S < φ V (in a hydrodynamic regime), while at ultralow temperatures (in a ballistic regime) a state when φ S > φ V can be achieved. This temperature transitioncan be detected using e.g. mechanical resonators as adecrease of their sensitivity to the collisions with volumeexcitations with temperature drop. It is obvious that thistransition temperature depends on the resonator’s mass,the area and quality of the resonator’s surface which de-termines the density of the surface states [5, 11–13].There are variety of mechanical resonators being usedas experimental tools to probe the physics of topological He-B at zero temperature limit [14–17]. As mechanicalresonators we utilize tuning forks. Currently, these piezo-electric devices are very popular experimental tools usedin superfluid He physics [18]. They are almost magnetic
Upper NMR coil and vwLower NMR coil and vw Tuning forks B Ag heat exchangersInner cellOuter cell
FIG. 1: (Color online) A schematic sketch of the double walledexperimental cell mounted on our nuclear stage. The orien-tation of magnetic field B is shown as well. field insensitive, simple to install, easy to excite with ex-tremely low dissipation of the order of a few fW or evenless and displacement ∼ I F is proportional to thefork velocity I F = Av , where A is the proportionalityconstant readily determined from experiment [19, 20]. EXPERIMENTAL DETAILS
We performed experiments in a double walled exper-imental cell (see Fig. 1) mounted on a diffusion-weldedcopper nuclear stage [21]. While upper tower served forNMR measurements (not mentioned here), in the lowerpart of the experimental cell, tuning forks of differentsizes and one NbTi vibrating wire were mounted. TheNbTi vibrating wire served as a thermometer of He-B inballistic regime i.e. in temperature range below 250 µ K.After cooling the fridge down to ∼ A constants for the individual forks [18, 19]. Both forks be-haved as high Q-value resonators having Q-value of theorder of 10 . The physical characteristics of the large andsmall tuning forks are as follow: the large fork resonancefrequency (in vacuum) f L = 32725.88 Hz, the width ∆ f i = 36.3 mHz and dimensions L = 3 .
12 mm, W = 0 .
25 mm, T = 0 .
402 mm give the mass m L =2.0 × − kg and valueof A L = 6.26 × − A.s.m − , while the small fork reso-nance frequency f S = 32712.968 Hz, the width ∆ f i =32.79 mHz and dimensions L = 1 .
625 mm, W = 0 . T = 0 . S =1.05 × − kg and thevalue of A S = 1.04 × − A.s.m − .Then, we filled the experimental cell with He at pres-sure of 0.1 bar. Subsequent demagnetizations of the cop- per nuclear stage allowed us to cool the superfluid He-Bin the inner cell down to 175 µ K as determined from thedamping of the NbTi vibrating wire.We also performed measurements of tuning forks insmall magnetic fields. After demagnetization, when tem-perature of the superfluid He-B was stable, we set themagnetic field ( B ) to 2.5 mT, and measured a collectionof the resonance curves at various excitations at this field.Thereafter, we reduced the magnetic field B slowly by0.25 mT and repeated the measurements of the resonancecharacteristics as a function of excitation. We reproducedthis measurement procedure while reducing the magneticfield B down to 0.25 mT. Fork velocity (mm/s)
Small forkLarge fork ∆ f ( H z ) FIG. 2: (Color online) Dependence of the width ∆ f for large(black) and small (red) tuning forks as a function of fork ve-locity and images of the surface profiles of tuning forks usedin experiment obtained from AFM scans. EXPERIMENTAL RESULTS AND ANALYSIS
Figure 2 shows the width ∆ f as a function of the forkvelocity measured for the large and small tuning forksin superfluid He-B at temperature of 175 µ K and pres-sure of 0.1 bar. As one can see, there is a remarkabledifference between them. The small tuning fork clearlydemonstrates Andreev reflection process [10, 22, 23]: asthe fork velocity is rising more and more volume excita-tions undergo the process of Andreev reflection. Duringthis scattering process they exchange a tiny momentumwith the fork of the order of (∆ /E F ) p F , where E F and p F are the Fermi energy and the Fermi momentum, re-spectively. As a result, the fork damping decreases untila critical velocity is reached. At this velocity the forkbegins to break the Cooper pairs and its damping risesagain. However, this dependence for the large tuningfork is opposite: the process of the Andreev reflection issuppressed, and the damping of the large fork increaseswith its velocity at the beginning. The different width∆ f - velocity dependence for the large fork presented inFig. 2 suggests a presence of some processes leading tothe suppression of the Andreev reflection and/or anotherdissipation mechanism than the Andreev reflection. M a g n e t i c f i e l d ( m T ) N o r m a li z ed f o r k v e l o c i t y ( m / ( s e c . V o l t )) Velocity minima E xc i t a t i on ( m V ) R M S FIG. 3: (Color online) Dependence of the normalized tun-ing fork velocity (fork velocity over excitation voltage) as afunction of the fork velocity and applied magnetic field B .Dependence clearly shows a presence of the velocity minimaat magnetic fields that satisfy the Larmor resonance condition ω L = γ ( B + B rem ). We assume that at temperature ∼ µ K, the densityof excitations near the surface of large fork satisfies thecondition φ S > φ V . However, we suppose that “non-standard” behavior of the large tuning fork is caused bythe different quality of its surface compared with that ofthe small fork. While the surface of the large fork is cor-rugated, the surface of the small fork is much smoother(see Fig. 2). Fork motion in superfluid He-B is associ-ated with creation of the back-flow i.e. the flow of thesuperfluid component around tuning fork’s body on thescale of the slip length [24, 25]. The back-flow shifts theenergy of excitations by p F · v , where v is the super-fluid velocity (in linear approximation is the same as thefork velocity). As a consequence, the excitations havingenergy less than ∆ + p F · v are scattered via Andreevprocess. This simple model assumes that direction of theback-flow is correlated with the direction of the fork ve-locity i.e. the back-flow flows in opposite direction to thetuning fork motion. However, assuming that the scale ofthe surface roughness of the large fork is larger than theslip length (see Fig. 2), the oscillating surface of the largefork makes the velocity field of the superfluid back-flowrandom. That is, the back-flow is not correlated with di-rection of the tuning fork motion. This means that thereare excitations reflected via Andreev process there dueto the back-flow flowing in different directions to that ofthe tuning fork velocity. Such reflected excitations arepractically “invisible” to the fork. On the other hand, the small fork is sensitive to the Andreev reflection sup-posing that the scale of its surface roughness is less orcomparable to the slip length. We presume that abovepresented mechanism stays behind the suppression of theAndreev reflection in case of the large fork. However, toconfirm this hypothesis additional work has to be done.Regarding to the rise of the large fork damping at lowvelocities, the origin of this phenomenon is unclear yet,and it is not a subject of this article.Another unexpected results were observed while mea-suring the damping of the large tuning fork motion insuperfluid He-B at temperature of 175 µ K in magneticfield. We surprisingly found that the damping of thelarge fork motion is magnetic field dependent and showsa maximum.
800 1000 12001.41.451.51.551.61.651.
Time (m inutes) N o r m a li z ed v e l o c i t y ( m / ( s e c . V o l t )) FIG. 4: (Color online) Time dependencies of the normal-ized tuning fork velocity (fork velocity over excitation volt-age) measured at different magnetic fields as showed. Thepoints represent the data measured for various excitations atparticular field. Figure clearly shows a presence of the ve-locity minima at magnetic field corresponding to the Larmorresonance condition ω L = γ ( B + B rem ). The dashed lineillustrates a small thermal background caused by a parasiticheat leak. Figure 3 shows the results of above mentioned mea-surements in a form of the dependence of the normalizedtuning fork velocity as a function of the fork velocity andmagnetic field. This dependence clearly shows a presenceof the minima in the fork velocity at the same value ofthe magnetic field. Presented dependencies are maskedby a tiny thermal background due to a small warm-upcaused by a parasitic heat leak into nuclear stage (seeFig. 4). Time evolution of the thermal background wasmodeled using the polynomial dependence a · t + b · t + c ,where a, b, c are fitting parameters and t is the time.We determined these parameters for particular excita-tion by fitting the time dependence via points measuredat 2.5, 2.25, 1.0, 0.75 and 0.25 mT (see illustrative thered dashed line in Fig. 4). When we subtracted-off thethermal background, the resulting dependencies are pre-sented in Fig. 5. Figure 5 shows two dependencies of thetuning fork velocity as a function of excitation and mag-netic field B . These two dependencies measured duringtwo subsequent demagnetizations demonstrate their re-producibility. -(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20) M(cid:21)(cid:22)(cid:23)(cid:24)t(cid:25)(cid:26) (cid:27)(cid:28)(cid:29)(cid:30)(cid:31) ! T ) V e l o c i t y d r o" m / s e c ) %&’ ()*+,./123456789:;<=>?@ABCDE V e l o c i t y d r FGHI m / s e c ) JKLNOPQR STUVW XY T ) Z[ c i t a t i on ( m V ) \]^_‘ c i t a t i on ( m V ) abc FIG. 5: (Color online) Two dependencies of the velocity dropof the tuning fork expressed as a function of the magnetic field B and excitation voltage measured during two subsequentdemagnetizations. Both dependencies show a clear minimumin velocity as function of the magnetic field at value thatcorresponds to the Larmor resonance frequency ω F = γ ( B + B rem ). We note that value of B rem is ∼ -0.5 mT. Figure 5 manifests a new and intriguing phenomenon:a presence of the velocity minima (i.e. an additionaldamping) at magnetic fields which satisfy the Larmor res-onance condition ω = γ ( B + B rem ) = γB , where ω is theangular frequency of the tuning fork ( ω ≃ π · γ is the He gyromagnetic ratio ( γ = − π · . × rad/s.mT ), B and B rem is the magnetic field ap-plied and remnant magnetic field from demagnetizationmagnet, respectively. We interpret this phenomenon as def ghijklmnpqrs ∆ uvwxyz{|} ~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133) (cid:134)(cid:135)(cid:136)(cid:137)(cid:138) (cid:139)(cid:140) T ) (cid:141)(cid:142) c i t a t i on ( m V ) (cid:143)(cid:144)(cid:145) FIG. 6: (Color online) Dependence of the tuning fork Q-valuesas a function of magnetic field B and excitation voltage. Thedeep minimum corresponds to nuclear magnetic resonance ofthe magnetic layer formed on tuning fork surface. NMR-like effect on the anisotropic magnetic moment M formed in vicinity of the top horizontal surface of thetuning fork. Formation of the anisotropic magnetic mo-ment M is a consequence of the symmetry violation ofthe He-B order parameter by the fork surface, simul-taneously modifying the excitation spectrum. Based ondifferent behaviors of the tuning forks showed in Fig. 2we presume that the damping of the large tuning fork mo-tion is mostly caused by the excitations trapped in thesurface states, as the rest of superfluid He-B in volumebehaves like a vacuum.Figure 6 shows the same NMR effect, however, as adrop of the tuning fork Q-value in dependence on theexcitation and magnetic field.In order to explain the measured dependencies we pro-pose a simple phenomenological model as follows. Thefork’s motion in zero magnetic field can be described bythe equation d αdt + Γ dαdt + ω α = F m sin( ωt ) , (1)where Γ is the damping coefficient characterizing thefork’s interaction with surrounding superfluid He-B, andwhich also includes its own intrinsic damping, ω is thefork resonance frequency in vacuum, ω is the angular fre-quency of the external force, α is the deflection angle ofthe tuning fork arm from equilibrium and F m is the forceamplitude normalized by the mass m and by the length l of the tuning fork arm. We assume that fork’s deflectionsare small enough and therefore interaction of the tuningfork with superfluid He-B acts in linear regime i.e. weneglect processes of the Andreev reflection [10, 23].By applying external magnetic field B = (0 , , B + B rem ) = (0 , , B ), the magnetic moment M is formed on (cid:146)(cid:147) xy z (cid:148)(cid:149) (cid:150) B M (cid:151)(cid:152) (cid:153)(cid:154)(cid:155)(cid:156)(cid:157) m (cid:158)(cid:159)(cid:160) ¡¢£⁄¥ xy z ƒ§¤ '“«‹›fifl ROTATING FRAMELABORATORY FRAME ROTATING FRAME
M MF B F m L Z a m m FIG. 7: (Color online) Schematic view on individual vectorsfor two positions of the tuning fork. In stationary position(I.) Zeeman energy is minimized and magnetic torque is zero.When fork is deflected by external force F m (II.), the rise ofZeeman energy due to anisotropy of the magnetic moment M is associated with emergence of the force F B which actsagainst external force F m and causes additional, magneticfield dependent damping. When NMR condition is satisfied(III.) i.e. when B eff = B rf , magnetic moment M precessesin z − y plane around B rf in rotating frame of the reference. the fork’s horizontal surfaces. The magnetic moment M of the surface layer includes the strong spin anisotropy ofthe superfluid He layer together with magnetic momentsof solid He atoms covering the fork’s surface [5, 26–29].However, based on measurements presented in [26], wepresume that magnetic moments of solid He atoms be-have as a paramagnet and, on a time scale of the fork os-cillation period, its Zeeman energy is always minimized.Therefore, magnetic property of the surface layer of su-perfluid He-B is responsible for the anisotropy of themagnetic moment M . According to [5], the anisotropicspin susceptibility of the surface layer of superfluid He-Bat T → χ zz = ¯ h γ k F π ∆ , (2)where p F = ¯ hk F . This susceptibility is as large as thenormal state susceptibility χ N multiplied by the width1 /κ = ¯ h v F / ∆ of the bound states. Here, v F is the Fermivelocity.In general, due to surface diffusivity the orientation of M can be tilted from the field direction B . However, we shall assume for simplicity that M = (0 , , M ). Whenfork oscillates in external magnetic field B , the normalto its horizontal surface m is deflected from the direc-tion of the external magnetic field B (see Fig. 7). Thismeans that anisotropic magnetic moment M of the sur-face layer undergoes the same deflections (oscillations).We assume that during fork oscillations the magneticmoment of solid He layer follows the direction of mag-netic field B minimizing its Zeeman energy. Therefore,in the reference frame connected to the anisotropic mag-netic moment M , this moment M experiences a lin-early polarized alternating magnetic field B rf of am-plitude B rf = B sin( α ) ≃ Bα oscillating with angularfrequency ω of the tuning fork. While magnetic fieldmagnitude in the direction of magnetic moment M is B M = B cos( α ) ≃ B . Thus, a typical experimental NMRconfiguration is set up. However, here the “virtual” ex-citation rf-field B rf acting on anisotropic magnetic mo-ment M is generated by harmonic mechanical motionof the tuning fork. We suppose that magnetic torque M × B acting on the anisotropic magnetic moment M is equivalent to the mechanical torque L × F B , where L = ( L X , , L Z ) is the vector pointed in direction of M having the magnitude equal to the length of the oscillat-ing fork prong l . The force F B emerges from the rising ofthe Zeeman energy ( − M . B ) due to deflection of M fromthe field direction, acts against excitation force F m andcauses additional field-dependent damping of the tuningfork motion. This force can be expressed as F B = 1 γl d M dt × L . (3)In order to obtain time dependence of F B one has todetermine dynamics of the magnetic moment M whichis governed by the Bloch’s equation (in rotating referenceframe) δ M δt = γ ( M × B eff ) + M − M T i . (4)Here M = ( M x , M y , M z ), B eff = ( − B rf , , B − ω/γ ),and the second term on the right side describes the pro-cesses of the energy dissipation being characterized by therelaxation time constants T i with i = 1 for z -componentand i = 2 for xy -component of the magnetic moment M .Assuming that magnetic relaxation processes act solelyin the magnetic layer near fork surface and using the ge-ometry of the problem, the amplitude of the force F B acting against excitation force F m can be expressed as F B = 1 γl dM Y dt = 1 γl [ χ D cos( ωt ) + χ A sin( ωt )] B dαdt , (5)where M Y is the y-component of magnetic moment M in the laboratory frame, ω is the tuning fork angularfrequency, χ A denotes the absorption component of the (cid:176)–†‡·(cid:181)¶•‚„”» …‰(cid:190)¿(cid:192)`´ˆ˜ ¯˘˙¨(cid:201)˚¸(cid:204)˝˛ˇ—(cid:209)(cid:210)(cid:211)(cid:212)(cid:213)(cid:214)(cid:215)(cid:216)(cid:217)(cid:218)(cid:219)(cid:220) (cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)Ł ØŒº(cid:236)(cid:237) (cid:238)(cid:239)(cid:240)æ(cid:242) (cid:243)(cid:244)ı(cid:246)(cid:247) łøœß(cid:252) (cid:253)(cid:254)(cid:255)3(cid:0) FIG. 8: (Color online) Example of the calculated resonancecharacteristics of the tuning fork using equation (8) for mag-netic field 2.25 mT and excitation 175 mV. magnetic susceptibility of the layer expressed in the form χ A = χ L ω B T ω B − ω ) T (6)and χ D denotes the dispersion component in the form χ D = χ L ω B ( ω B − ω ) T ω B − ω ) T . (7)Here ω B = γB and M = χ L B , where χ L stands for themagnetic susceptibility of the He-layer. Adding the term(5) into equation (1), one gets nonlinear differential equa-tion describing the tuning fork motion with anisotropicmagnetic layer on its horizontal surface in external mag-netic field in form d αdt + Γ dαdt + 1 γl m dM Y dt + ω α = F m sin( ωt ) . (8)Applying Runge-Kutta method we numerically calcu-lated the time evolution of the tuning fork response de-scribed by the equation (8) as a function of applied ex-ternal force (in frequency and amplitude) and magneticfield. Calculations took into account transient phenom-ena. Reaching a steady state of the fork motion, thecalculated values were multiplied by the ”reference” sig-nals simulating excitations with aim to obtain the res-onance characteristics i.e. the absorption and the dis-persion component. The magnetic properties of surfacelayer were characterized by the spin-spin relaxation timeconstant T =28 µ sec which served as a fitting parame-ter. Figure 8 shows an example of the tuning fork re-sponse calculated by using equation (8) in form of theresonance characteristics. Calculated resonance charac-teristics were fitted by means of the Lorentz function in order to obtain experimentally measurable parameters:the velocity amplitude, the width and the resonance fre-quency as the function of the excitation and magneticfield. Figure 9 summarizes theoretically calculated de-pendence of the tuning fork velocity drop as a functionof driving force (excitation voltage) and magnetic field.Presented dependencies confirm the presence of the veloc-ity minima at magnetic field corresponding to the Lar-mor resonance condition for He and they are in verygood qualitative agreement with those obtained experi-mentally (see Fig. 5). (cid:1) V e l o c i t y d r o(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)E(cid:9) FIG. 9: (Color online) Theoretical dependence of the tuningfork velocity drop as a function of the magnetic field B andexcitation voltage calculated using equation (8). DISCUSSION
Although, we have presented a simple theoreticalmodel using a phenomenological approach, we obtainedreasonable qualitative agreement with experiment. How-ever, it is worth to say that there is a set of theoreticalpapers dealing with problem of the Andreev-Majoranasurface states in topological superfluid He-B on the levelof the order parameter [5, 27–34]. In light of this, let usdiscuss our experimental results and compare them withtheoretical models.In particular, the spin dynamics and an effect of NMRon the magnetic moment of the surface states had re-cently been theoretically investigated by M. A. Silaev[29]. Using assumptions of a flat surface, i.e. that thevector n representing a rotation axis of the superfluid He-B order parameter is parallel to magnetic field B ,he showed that standard transverse NMR technique doesnot allow to excite the magnetic moment of the surfacestates at Larmor frequency due to two reasons. The first,a mini-gap presented at the surface state spectrum has abroader energy gap E g than corresponding Larmor fre-quency due to Fermi-liquid corrections ( E g ∼ hω L ).The second, the probability of the NMR excitation insurface states spectrum is proportional to the deflectionangle β n of the vector n from the spin quantization axisdefined by magnetic field B . For a flat surface, the an-gle β n is rather small leading to a strong suppression ofthe NMR response from the surface states. However, inorder to excite the Andreev-Majorana surface states us-ing a transverse NMR technique, the vector n has to bedeflected from magnetic field direction by an angle β n ,so that the effective driving field B eff has a componentparallel to the spin quantization axis [29].The Lancaster group [35] performed the NMR mea-surements using superfluid He-B near a surface in ex-perimental configuration, which is similar to that as as-sumed in [29]. The experimental cell was made fromsapphire. However, instead of a flat horizontal surface,their cell had a semi-spherical end cap. Semi-sphericalend cap of the experimental cell formed a texture of n -vectors in broad range of angles β n with respect to thedirection of magnetic field ( z -direction). This is a config-uration for which the theory [29] predicts possibility toexcite and observe response from the Andreev-Majoranasurface states. Pulsed NMR technique allowed them tocreate a long lived state with coherent spin precessionnamed as persistently precessing domain (PPD) [36, 37].Using magnetic field gradient they were able to controlthe position of the PPD with respect to the horizontalwall of the cell [38]. They showed that the closer the Lar-mor resonance condition is to the cell horizontal surface,the shorter the PPD signal life time is. The presence ofthe cell surface reduces the signal life time by four or-ders of magnitude [35]. In the light of theory [29], theinterpretation of this phenomena needs to be elucidated.Our experimental configuration of the NMR detectionusing mechanical resonator is completely different fromthat of a standard transverse NMR technique. The mostimportant difference is that we did not apply any exter-nal rf-field B rf to excite magnetic moments in superfluid He-B. The only magnetic field presented is the staticmagnetic field B . An excitation rf-field B rf is a “vir-tual” field, which is experienced only by the anisotropicmagnetic moment M during fork oscillations.Harmonic oscillations of the tuning fork arms lead tothe oscillation of the whole texture of n vectors causingtheir time dependent deflection from the quantized axisdefined by the constant magnetic field B . The amplitudeof the tuning fork oscillation at maximum excitation was ∼ n vectors in a broad spectrum of angles β n from the fielddirection. This is a configuration, which according tomodel [29], satisfies the condition for the observation oftransverse NMR of the surface states in superfluid He-B. However, according to our opinion, the assumptions of the theoretical model [29] do not fully correspond tothe conditions of the experiment presented here and themodel itself could be extended for them.On the other hand, theoretical model presented in [5]suggests that strong spin anisotropy of superfluid Helayer near the surface is large enough to be observed ex-perimentally. We think that above mentioned experi-mental technique using mechanical resonators (e.g. tun-ing forks) with various surface roughness and resonancefrequencies is a way. However, the open question is aninfluence of the solid He on this phenomenon. Althoughwe assumed a paramagnetic property of the solid Helayer, the magnetic susceptibility of the solid He domi-nates at ultra low temperatures [26]. An influence of solid He could be tested by using He as a coverage on thesurface of the tuning fork, since He atoms remove Heatoms from the surface. However, He simultaneouslycovers the heat exchangers inside nuclear stage and thisreduces the cooling efficiency of the He liquid. As themeasurements are performed in temperature range be-low 200 µ K, the test of influence of the He solid layer onthe observed phenomenon is going to be an experimentalchallenge.
CONCLUSION
In conclusion, we have observed the NMR-like effecton anisotropic magnetic moment of the superfluid sur-face layer, including Andreev-Majorana fermionic exci-tations formed on the resonators’ surface, being detectedas additional magnetic field dependent damping of itsmechanical motion. Further work is required to developthis technique, which in combination with e.g. acousticmethod [7], or non-standard NMR technique based onPPD [35], or thermodynamic method [39] opens a possi-bility to study the spin dynamics of excitations trappedin SABS in topological superfluid He-B and, perhapsto prove their Majorana character experimentally. Fi-nally, a development of a theory considering oscillationsof n vectors representing the order parameter of super-fluid He near a surface at constant magnetic field, i.e. atheory for the condition of presented experiment wouldbe very useful and challenging.
ACKNOWLEDGMENTS
We acknowledge the support of APVV-14-0605, VEGA2/0157/14, Extrem II - ITMS 26220120047 and EuropeanMicrokelvin Platform, H2020 project 824109. We wish tothank to M. Kupka for fruitful discussion, V. Komanick´yfor AFM scans of tuning forks, and to ˇS. Bic´ak and G.Prist´aˇs for technical support. Support provided by theU.S. Steel Koˇsice s.r.o. is also very appreciated. ∗ Electronic Address: [email protected][1] G. E. Volovik, Universe in Helium-3 Droplet, ClarendonPress, Oxford (2003).[2] W. J. Gully, D. D. Osheroff, D. T. Lawson, R. C. Richard-son, and D. M. Lee, Phys. Rev. A , 1633 (1973).[3] V. V. Dmitriev, A. A. Senin, A. A. Soldatov, and A. N.Yudin, Phys. Rev. Lett. , 165304 (2015).[4] N. Zhelev, M. Reichel, T. S. Abhilash, E. N. Smith, K. X.Nguyen, E. J. Mueller, and J. M. Parpia, Nature Comm. , 12975 (2016).[5] Y. Nagato, S. Higashitani, K. Nagai, J. of the Phys. Soc.of Japan , 235301 (2009).[7] S. Murakawa, Y. Tamura, Y. Wada, M. Wasai, M. Saitoh,Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Ya-mamoto, S. Higashitani, and K. Naga, Phys. Rev. Lett. , 155301 (2009).[8] G. E. Volovik, JETP Lett. , 201 (2010).[9] T. Mizhushima, K. Machida, J. Low Temp. Phys.
204 (2011).[10] S. N. Fisher, A. M. Gu´enault, C. J. Kennedy, G. R. Pick-ett, Phys. Rev. Lett. , 2566 (1989).[11] Y. Nagato, M. Yamamoto, K. Nagai, J. Low Temp. Phys. , 1135 (1998).[12] A. B. Vorontsov, J. A. Sauls, Phys. Rev. B , 064508(2003).[13] Y. Aoki, Y. Wada, M. Saitoh, R. Nomura, Y. Okuda,Y. Nagato, M. Yamamoto, S. Higashitani, and K. Nagai,Phys. Rev. Lett. , 075301 (2005).[14] D. I. Bradley, P. Crookston, S. N. Fisher, A. Ganshin, A.M. Gu´enault, R. P. Haley, M. J. Jackson, G. R. Pickett,R. Schanen, V. Tsepelin, J Low Temp Phys , 476(2009).[15] D. I. Bradley, S. N. Fisher, A. M. Gu´enault, R. P. Haley.R. C. Lawson, G. R. Pickett, R. Schanen, M. Skyba, V.Tsepelin, and D. E. Zmeev, Nature Physics, , 1017(2016).[16] P. Zheng, W. G. Jiang, C. S. Barquist, Y. Lee, and H. B.Chan, Phys. Rev. Lett. , 195301 (2016).[17] P. Zheng, W. G. Jiang, C. S. Barquist, Y. Lee, and H. B. Chan, Phys. Rev. Lett. , 065301 (2017).[18] R. Blaauwgeers, M. Blaˇzkov´a, M. ˇCloveˇcko, V. B. Eltsov,R. de Graaf, J. Hosio, M. Krusius, D. Schmoranzer, W.Schoepe, L. Skrbek, P. Skyba, R. E. Solntsev, D. E.Zmeev, J. Low Temp. Phys. , 537 (2007).[19] P. Skyba, J. Low Temp. Phys. , 219 (2010).[20] S. Holt, P. Skyba, Rev. Sci. Instrum. , 064703 (2012).[21] P. Skyba, J. Ny´eki, E. Gaˇzo, V. Makr´oczyov´a, Yu. M.Bunkov, D. A. Sergackov, A. Feher, Cryogenics , 293(1997).[22] D. I. Bradley, M. ˇCloveˇcko, E. Gaˇzo, P. Skyba, J. LowTemp. Phys. , 147 (2008).[23] M. ˇCloveˇcko, E. Gaˇzo, M. Kupka, M. Skyba, P. Skyba,J. Low Temp. Phys.
669 (2011).[24] D. Einzel, H. Højgaard, Jensen, H. Smith, P. W¨olfle, J.Low Temp. Phys. , 695 (1983).[25] D. Einzel, P. W¨olfle, H. Højgaard Jensen, H. Smith, Phys.Rev. Lett. , 1705 (1984).[26] D. I. Bradley, S. N. Fisher, A. M. Gu´enault, R. P. Ha-ley, N. Mulders, G. R. Pickett, D. Potts, P. Skyba, J.Smith, V. Tsepelin, and R. C. V. Whitehead, Phys. Rev.Lett. , 125303 (2010). Jim Robert Smith: PhD thesis,Lancaster 2009.[27] M. A. Silaev, Phys. Rev. B , 144508 (2011).[28] M. A. Silaev, G. E. Volovik, JETP , 1042 (2014).[29] M. A. Silaev, J. Low Temp. Phys. , 393 (2018).[30] G. E. Volovik, JETP Lett. , 440 (2009).[31] J. A. Sauls, Phys. Rev. B , 214509 (2011).[32] Y. Tsutsumi, M. Ichioka, and K. Machida, Phys. Rev. B , 094510 (2011).[33] T. Mizushima, M. Sato and K. Machida, Phys. Rev. Lett. , 165301 (2012).[34] T. Mizushima, Y. Tsutsumi, M. Sato, and K. Machida,J. Phys.: Condens. Matter , 113203 (2015).[35] S. N. Fisher, G. R. Pickett, P. Skyba, and N. Suram-lishvili, Phys. Rev. B , 024506 (2012).[36] Yu. M. Bunkov, S. N. Fisher, A. M. Gu´enault, G. R.Pickett, Phys. Rev. Lett. , 3092 (1992).[37] M. Kupka, P. Skyba, Physics Letters A , 324 (2003).[38] D. I. Bradley, D. O. Clubb, S. N. Fisher, A. M. Gu´enault,C. J. Matthews, G. R. Pickett, P. Skyba, J. Low Temp.Phys.
351 (2004).[39] Yu. M. Bunkov, J. Low Temp. Phys.175