Bose-Einstein condensation of magnons and spin superfluidity in the polar phase of 3 He
S. Autti, V.V. Dmitriev, J.T. Mäkinen, J. Rysti, A.A. Soldatov, G.E. Volovik, A.N. Yudin, V.B. Eltsov
aa r X i v : . [ c ond - m a t . o t h e r] N ov Bose-Einstein condensation of magnons and spin superfluidity in the polarphase of He S. Autti, V. V. Dmitriev, J. T. M¨akinen, J. Rysti, A. A. Soldatov,
2, 3
G. E. Volovik,
1, 4
A. N. Yudin, and V. B. Eltsov Low Temperature Laboratory, Department of Applied Physics,Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland P. L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia (Dated: November 9, 2017)The polar phase of He, which is topological spin-triplet superfluid with the Dirac nodal line in thespectrum of Bogolubov quasiparticles, has been recently stabilized in a nanoconfined geometry. Wepump magnetic excitations (magnons) into the sample of polar phase and observe how they form aBose-Einstein condensate, revealed by coherent precession of the magnetization of the sample. Spinsuperfluidity, which supports this coherence, is associated with the spontaneous breaking of U (1)symmetry by the phase of precession. We observe the corresponding Nambu-Goldstone boson andmeasure its mass emerging when applied rf field violates the U (1) symmetry explicitly. We suggestthat the magnon BEC in the polar phase is a powerful probe for topological objects such as vorticesand solitons and topological nodes in the fermionic spectrum. Introduction.—
The phenomenon of Bose-Einstein con-densation, originally suggested for real particles and ob-served in ultracold gases, has been extended in recent ex-perimental and theoretical works to systems of bosonicquasiparticles, including collective modes. Examples arelongitudinal electric modes [1], phonons [2], excitons[3], exciton-polaritons [4], photons [5], rotons [6], andmagnons [7–17]. In these systems quasiparticles are ex-ternally pumped, but they are sufficiently long-lived, sothat their number N is quasi-conserved. As a result, thechemical potential µ = dE/dN is non-zero during thelifetime of the condensate.Bose-Einstein condensate (BEC) of magnons was firstdiscovered in the B phase of He [7]. In this spin-tripletsuperfluid, magnons are quanta of transverse spin waves,associated with precessing spin of He nuclei. Magnoncondensation results in spontaneous coherence of the pre-cession, which produces a characteristic signal in nuclearmagnetic resonance (NMR) experiments [7]. In the ex-periment magnons, carrying spin − ~ , are pumped usingradio-frequency (rf) pulse, which deflects magnetization M (or spin S ) from the equilibrium direction along themagnetic field H k ˆ z . Alternatively, magnons can becontinuously replenished with small rf field H rf ⊥ H tocompensate magnetic relaxation [8].The coherent precession ( S x + iS y ) ∝ e i ( ωt + φ ) is char-acterized by a common frequency ω and definite phase φ . Formation of the coherent phase φ across the wholesample reveals the spontaneously broken SO (2) spin ro-tation symmetry. In the language of magnon BEC thiscorresponds to the breaking of the U (1) symmetry whichcharacterizes the (approximate) conservation law for thenumber of magnons: N M = R dV ( S − S z ) / ~ , while thechemical potential determines the frequency of precession µ = dE/dN M = ~ ω . Spontaneous breaking of U (1) symmetry related toparticle number conservation is linked to the superfluidphase transition. In the case of magnon BEC this isspin superfluidity. Experiments in He-B demonstratedvarious phenomena which accompany the spin superflu-idity, such as ac and dc Josephson effects, spin supercur-rents, and phase-slip processes [18–20]. Another impor-tant marker of the spontaneous U (1) symmetry breakingis appearance of Nambu-Goldstone (NG) mode (whichis a phonon in a usual superfluid)[21]. For magnon con-densates in He-B such mode was indeed experimentallyfound [22, 23].Besides demonstrating the fascinating phenomenon ofspin superfluidity, magnon BEC in He-B proved to bea sensitive probe for topological structures of the orderparameter, like quantized vortices and their dynamics[24–27], for fermionic quasiparticles [28] and for bosoniccollective modes [29]. This coherent probe can be madelocal by trapping magnons in magnetic and textural traps[10, 12]. For a sufficiently large number of pumpedmagnons, the condensate deforms the trap [13] whichleads to the formation of a self-trapped magnon BEC[30]. The latter is an exact implementation of the Q -ballsstudied in the relativistic quantum field theories, whichshows that magnon BEC can also be used for quantumsimulations.All these features call for a search for magnon conden-sation in other topological superfluids. Coherent preces-sion of magnetization was predicted to exist in super-fluid He-A [31] and its observation was reported in theA-like phase in silica aerogel [32]. Here we demonstratemagnon BEC in the recently discovered polar phase ofsuperfluid He. We observe the coherent precession ofmagnetization using NMR techniques. We also measurethe collective NG mode of the condensate as a function m x yz MH rf H -1 0 1 2 301020304000.51.01.5 ( d e g . ) /(2 ) (kHz) c o s -1 0 1 2 302468100123456 D i s p e r s i on ( a . u . ) /(2 ) (kHz) A b s o r p ti on ( a . u . ) (b)(a) FIG. 1. (color online). (a) The cw NMR signal in thepolar phase of He in nafen showing creation of coherentlyprecessing state on sweeping down magnetic field H with H rf = 0 . µ T at T = 0 . T c , P = 6 . λ = 90 ◦ .On the horizontal axis the frequency shift ω rf − γH is shown.(b) Tipping angle β and phase of precession α of magne-tization M determined from the absorption and dispersionsignals in panel (a). Definitions of the angles are given inthe insert. Magnetization M is in a rotating frame of pre-cession. Absorption and dispersion signals are proportionalto M y = M sin β sin α and M x = M sin β cos α , respectively.Relation between ∆ ω and cos β is linear in accordance withEq. (2). of temperature, rf excitation amplitude, precession fre-quency and magnetic field orientation. Polar phase.—
The polar phase is realized in liquid He confined within nafen [33], a commercially producednanostructured material that consists of nearly parallelAl O strands [34]. The order parameter in the polarphase is A νj = ∆ e iϕ ˆ d ν ˆ m j , (1)where ∆ is the gap parameter, e iϕ is the phase factor, ˆ d and ˆ m are the unit vectors of spin and orbital anisotropy,respectively. In nafen ˆ m is locked parallel to the strands[35]. The polar phase is Dirac superfluid which belongs tothe same class of topological matter as Dirac nodal-linesemimetals [36–38]. As distinct from the fully gapped He-B and from He-A with Weyl nodes, the gap in thepolar phase has a line of zeros in the plane normal to ˆ m . Experiment.—
The nafen sample is a cube with a side of4 mm. It has porosity of 94% and density of 0.243 g/cm .The strands are of diameter 9 nm, separated on averageby 35 nm [34]. Experiments are performed at pressures6.9–7.1 bar using pulsed and continuous-wave (cw) NMRin magnetic field of 11.2 mT, corresponding to the NMRfrequency of 362.8 kHz. The static magnetic field H canbe applied at an arbitrary angle λ with respect to ˆ m . Thesample is cooled down in the ROTA nuclear demagnetiza-tion refrigerator [39] and the temperature is measured bya quartz tuning fork [40]. The fork is calibrated against / ( ) ( H z ) time (s) A b s o r p ti on ( a . u . ) H -1rf ( T) (a) (b) FIG. 2. (color online). (a) The frequency shift in a free in-duction decay signal recorded after turning off the rf excita-tion at the point marked by a dashed line in Fig. 1a. Fre-quency is obtained using sliding fast Fourier transform of araw signal with a time window of 20 ms. Dashed curve showsthe expected initial time dependence of ∆ ω calculated fromcw NMR data in Fig. 1a. (b) Cw NMR absorption frommagnon BEC versus H − at the fixed ∆ ω/ (2 π ) = 435 Hz at T = 0 . T c , P = 7 . λ = 90 ◦ . Solid line is a linearfit through zero. the NMR spectra measured in the linear regime usingknown Leggett frequency in bulk He-B [41, 42] and inthe polar phase [33, 43]. To avoid formation of paramag-netic solid He on the surfaces, the sample is preplatedby about 2.5 atomic layers of He. The magnitude of therf magnetic field H rf ≪ H is calibrated with a π/ He.
Coherent precession.—
Liquid He in our sample be-comes superfluid at 0 . T c , where T c is the superfluidtransition temperature in bulk He. In the temperaturerange of our measurements, down to 0 . T c , only the polarphase is observed. In the polar phase the NMR frequencyis given by [35] ω = ω L + Ω P ω L (cid:20) cos β − sin λ β − (cid:21) . (2)Here β is the deflection angle of the magnetization fromthe magnetic field direction (Fig. 1), Ω P is the Leggettfrequency in the polar phase, ω L = γH is the Larmor fre-quency, and γ = 2 . · s − T − is the absolute value ofthe gyromagnetic ratio of He. Most of our experimentsare performed in transverse magnetic field ( λ = 90 ◦ ). Inthis case in cw NMR, where cos β ≈
1, the frequencyshift ∆ ω = ω − ω L equals zero.Coherent precession of magnetization is stable only if dω/d (cos β ) < dµ/dn M >
0. Here the magnon density n M = ( S − S z ) / ~ = ( χH/γ ~ )(1 − cos β ) and χ is themagnetic susceptibility. In the polar phase the stabilitycondition is satisfied when | tan λ | >
2, while the tippingangle of magnetization β can be arbitrary. The criticalmagnetic field direction λ c = arctan 2 ≈ . ◦ . In thestable region superfluid spin currents act to maintain thecoherent precession by redistributing magnetization (and n M ) across the sample in such a way that the precession FIG. 3. (color online). The mass M of the pseudo-Nambu-Goldstone mode in magnon BEC supported by cw NMR in the polarphase of He as a function of H rf (a), Ω P (b), and sin β (c) at P = 7 . λ = 90 ◦ . Symbols are experimental data, curvesare theoretical predictions of Eq. (4) without fitting. Inset to panel (a) shows an example of excitation spectrum of magnonBEC measured as described in the text. M is given by the frequency of the largest peak, while the error bar is determined asthe peak width. The Leggett frequency Ω P ( T ) is determined from cw NMR spectra at λ = 0. Values of cos α , which dependon H rf and ∆ ω , are calculated from absorption and dispersion signals, while sin β is determined from Eq. (2). frequency ω in Eq. (2) remains uniform even if ω L , λ andΩ P have spatial dependence due to field inhomogeneityand disorder in nafen.The coherent precession is observed in cw NMR ex-periment as follows: We initially apply magnetic field H > ω rf /γ , where ω rf is a fixed frequency of rf excita-tion. Then we gradually decrease H . While resonancecondition is approached, magnetization deflects and β in-creases, which results in a positive frequency shift of pre-cession ∆ ω > ω L becomessmaller than ω rf during the field sweep, this frequencyshift may compensate the difference, and ω in Eq. (2)becomes locked to ω rf despite the fact that ω L is chang-ing. For this locking to occur, the rf excitation shouldbe large enough to compensate the magnetic relaxationwhich is presumably determined by the large surface areaof the nafen sample. An example of the NMR signalsmeasured in this way is shown in Fig. 1(a). As one cansee in Fig. 1(b), M can be deflected by more than 90 ◦ .The dissipation grows with increasing β and eventuallythe precessing state collapses, in this case at β ≈ ◦ .The coherent nature of the created state is revealedduring its decay. After switching off the rf pumping, mag-netic relaxation results in a gradual decrease of N M andof the amplitude of precession q M x + M y ∝ − cos β .Simultaneously the frequency of precession ω changes insuch a way that relation of Eq. (2) remains valid. Thisis only possible if the precession remains coherent dur-ing the decay and dephasing owing to the magnetic fieldinhomogeneity ∆ H/H ≈ · − does not occur. Inthe absence of dephasing the decay rate in Fig. 2(a) isa measure of the energy relaxation, just as the absorp- tion M y in cw NMR spectrum, ˙ E = γH rf HM y . Indeed,pulsed and cw measurements of dissipation agree within30%, providing further evidence for the coherent preces-sion during the decay.The structure of the coherently precessing state andthe resulting energy dissipation ˙ E is essentially givenby the distribution of β over the sample. The distri-bution has only a weak dependence on H rf at fixed ∆ ω .Therefore, the absorption signal ( ∝ M y ) is approximatelyproportional to H − , as seen in Fig. 2(b). In the tem-perature range (0 . ÷ . T c , where we are able tomeasure dissipation in cw NMR at a fixed tipping an-gle β = 90 ◦ , we have found that the dissipation slightlyincreases (by about 15%) on warming. At higher tem-peratures magnon BEC is destroyed before reaching thisvalue of β . Nambu-Goldstone mode.—
The spin rigidity of magnonBEC allows for relatively low-frequency oscillations of themagnetization on the background of the coherent preces-sion. This oscillating mode has a relativistic spectrumΩ = M + c k , (3)where Ω is the frequency, k is the wave vector of the oscil-lations and c is the propagation velocity. For a pure NGmode resulting from spontaneous U (1) symmetry break-ing in magnon BEC, the mass (or gap) M is zero. Ifmagnon BEC is supported by pumping, like in our cwNMR experiments, then explicit breaking of U (1) sym-metry by rf field opens gap in the spectrum, and themode becomes pseudo-Nambu-Goldtsone. In the polarphase this gap is given by [45] M = Ω P H rf H (cid:0) − λ (cid:1) sin β cos α, (4)where the factor cos α accounts for the fact that oscilla-tions of the phase of precession occur around non-zero α owing to the dissipation.Boundary conditions in our sample (vanishing spin cur-rent through the boundary) allow for spatially uniformoscillations with k = 0 and frequency Ω ≡ M . In theexperiment this mode is excited with an alternating fieldgradient along H . Oscillations of α result in periodicvariation of the NMR signal. The absorption/dispersionsignal is detected by a lock-in amplifier at the frequency ω and the output is wired to the input of a second lock-in tuned to the frequency of the gradient modulation.Usign the second lock-in we record secondary absorptionand dispersion signals as a function of the modulation fre-quency, as illustrated in the inset in Fig. 3(a). The mainpeak is fitted by a Lorentzian to obtain the resonancefrequency of the pseudo-NG mode M . The secondaryspectrum also shows other peaks probably correspond-ing to standing waves of a pseudo-NG mode with finite k , but a detailed study of that is beyond the scope of thepresent work.The pseudo-NG mass M is plotted in Fig. 3 as a func-tion of H rf , Ω P (controlled by temperature T ), and ∆ ω ,and in Fig. 4 as a function of λ . Discrepancies at small β in Fig. 3(c) and Fig. 4 probably originate from ∆ ω be-ing comparable with the cw NMR linewidth ( ≈
300 Hz).However, for sin β > . τ − of magnon BEC as a functionof the field orientation λ is shown in the inset of Fig. 4.It is measured by pulsed NMR, and the amplitude of thefree induction decay signal is fitted by exp( − t/τ ). Com-pared to cw NMR measurements in Fig. 1, typical β inthis measurement is smaller, β . ◦ , but the precessionis still coherent. As expected, the magnon BEC showsmaximum stability in the transverse field H ⊥ ˆ m . Withdecreasing λ the relaxation rapidly increases and close tothe critical angle λ c it is difficult to resolve the coherentprecession. Conclusions.—
We have created a coherently precess-ing spin state in the polar phase of superfluid He con-fined in nafen. The coherent state is observed in cw andpulsed NMR when large enough number of magnons ispumped by rf field. This state has all the signatures ofmagnon BEC, supported by superfluid spin currents. Inparticular, its decay in the absence of pumping proceedsonly via magnon loss. No dephasing of precession oc-curs and coherence is preserved by spin supercurrents.The broken U (1) symmetry is manifested by the pseudo-Nambu-Goldstone collective mode of coherent precession[46, 47]. We have measured this mode using resonant
65 70 75 80 85 90024681012141618
60 65 70 75 80 85 90102030405060708090 sin (0.4,0.5) sin (0.3,0.4) sin (0.2,0.3) M / ( c o s ) ( H z ) (deg.) T = 0.54T c T = 0.48Tc R e l a x a ti on ( / s ) FIG. 4. (color online). The mass M of the pseudo-NG modeas a function of the magnetic field orientation λ . Symbolsrepresent ranges of sin β over which the mass has been aver-aged. The curve is the theoretical dependence for sin β = 0 . β theoretical line goes lower.The measurements have been done with H rf increasing from0.16 µ T to 0.48 µ T as λ decreases from 90 ◦ to 65 ◦ , but arescaled in the plot with Eq. (4) to H rf = 0 . µ T, which coin-cides with the data in Fig. 3(c). Temperature is within therange between 0.44 T c and 0.49 T c . (Inset) The relaxation rateof magnon BEC as a function of λ at different temperatures. excitation and found that its frequency is in close agree-ment with the theory.Magnon BEC proved to be an excellent tool to studytopological superfluid He-B. The polar phase opens newpossibilities to use magnon BEC as an instrument toprobe various topological objects, like half-quantum vor-tices [43], to manipulate effective metric for NG bosonsincluding modelling black-hole horizon using dependenceof c in Eq. (3) on λ [45], and to investigate the physics of“relativistic” fermions living in the vicinity of the Diracline, where a new type of the quantum electrodynamicsemerges [48].This work has been supported by the Academy ofFinland (Projects No. 284594 and No. 298451) and bythe European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme (Grant Agreement No. 694248). We used facil-ities of the Low Temperature Laboratory infrastructureof Aalto University. Some preliminary experiments havebeen performed at the Kapitza Institute and were sup-ported by Basic Research Program of the Presidium ofRussian Academy of Sciences and by Russian Foundationfor Basic Research Grant No. 16-02-00349. [1] H. 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