Superfluid β phase in liquid 3 He
aa r X i v : . [ c ond - m a t . o t h e r] D ec Superfluid β phase in liquid He V. V. Dmitriev, ∗ M. S. Kutuzov, A. A. Soldatov, and A. N. Yudin P.L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia Metallurg Engineering Ltd., 11415 Tallinn, Estonia (Dated: December 14, 2020)We report the first observation of a new superfluid phase of He – the β phase. This phase appearsin He in nematic aerogel in presence of high magnetic field right below the superfluid transitiontemperature. We use a vibrating aerogel resonator to detect the transition to the β phase andmeasure the region of existence of this phase. Introduction.—
The free energy and the superfluidtransition temperature in bulk superfluid He (with p -wave, spin-triplet Cooper pairing) are degenerate withrespect to spin and orbital momentum projections. Thisallows a variety of superfluid phases with the same tran-sition temperature, but in zero magnetic field only twophases (A and B) with the lowest energy are realized [1].Anisotropy of the space may lift the degeneracy and otherphases can be stabilized. In particular, the anisotropicscattering of He quasiparticles may lift the degeneracywith respect to orbital angular momentum projectionsand make favorable new phases – polar, polar-distortedA and polar-distorted B phases [2–8]. These phases wererecently observed and investigated in He confined in ne-matic aerogel [9–18]. Nematic aerogels consist of nearlyparallel strands that resulting in essentially anisotropicscattering of He quasiparticles inside the aerogel [19, 20].If the anisotropy is strong enough, then the superfluidtransition of He in nematic aerogel occurs into the po-lar phase, and, on further cooling, transitions into polar-distorted A and polar-distorted B phases may occur.Both polar and A phases belong to the class of Equal SpinPairing (ESP) phases and contain Cooper pairs with only ± ↑↑ and ↓↓ pairs), but, in contrast to the A phase, the polar phaseis not chiral and has a Dirac nodal line in the energyspectrum of Bogoliubov quasiparticles in the plane per-pendicular to the direction of the aerogel strands [2].The degeneracy with respect to spin projections maybe lifted by magnetic field. In bulk He in strong mag-netic fields the transition temperature for different spincomponents is splitted leading to a formation of new (A and A ) phases instead of the pure A phase. Then in-stead of the temperature of superfluid transition at zerofield ( T c ) there are two transitions: to the A phase at T = T A c > T c and to the A phase at T = T A c < T c .The A phase contains only ↑↑ pairs and exists in a nar-row range of temperatures ( ∼ . T c in field of 10 kOe)which increases proportionally to the field [21–25]. The A phase contains also ↓↓ pairs, which fraction rapidlygrows with cooling, and this phase is continuously trans-formed to the A phase, where fractions of both spin com-ponents are equal to each other. Similar splitting shouldalso occur in the polar phase [26, 27]. On cooling of He in nematic aerogel in a strong magnetic field, the su-perfluid transition should occur to the so-called β phase[1] (or P phase in notation of Refs. [26, 27]) instead ofthe pure polar phase. On further cooling, the second-order transition into the distorted β (or P ) phase isexpected which is continuously transformed to the purepolar phase. The orbital part of the order parameterof β and distorted β phases is the same as in the polarphase, but the β phase contains only ↑↑ Cooper pairs,while in the distorted β phase, the ↓↓ component is alsopresent. In this sense, the P –P splitting is analogousto the splitting of the A phase into A and A phases inbulk He. Worthy to mark that, although the superfluidA-like phase of He in silica aerogel corresponds to theA phase of bulk He, the A -A splitting in pure He insilica aerogel was not observed [28]. Theory explains thisfact by suppression of the splitting due to the presenceof solid He atomic layers on the aerogel strands [29].In this Letter, we present results of high magneticfield experiments in superfluid He in nematic aerogel,wherein the solid He layers on the aerogel strands havebeen replaced by He. We use a vibrating wire (VW) res-onator with the aerogel attached to it, as in previous VWexperiments with He in silica aerogel [30, 31]. We havemeasured temperature dependencies of resonance prop-erties of the resonator – the full width at half-maximum(FWHM) and the resonance frequency ( f a ) – and de-tected superfluid transitions, which we attribute to tran-sitions between normal and β , β and distorted β phases.We also have found that the region of existence of the β phase is proportional to the magnetic field value. How-ever, the measured P –P splitting does not agree withtheoretical calculations in Ref. [26] made in frames of theGinzburg-Landau theory. Theory.—
The order parameters of the β and dis-torted β phases are given by [26] A P µj = ∆ √ d µ + ie µ ) m j , (1) A P µj = ∆ √ d µ + ie µ ) m j + ∆ √ e iϕ ( d µ − ie µ ) m j (2)correspondingly, where ∆ and ∆ are gap parameters, e iϕ is a phase factor, d and e are mutually orthogonalunit vectors in spin space (which are perpendicular to themagnetic field H ), and m is a unit vector in orbital spacealigned along the direction of nematic aerogel strands.The β phase contains only ↑↑ Cooper pairs, while thedistorted β phase is a condensate of ↑↑ (the first term inEq. (2)) and ↓↓ (the second term in Eq. (2)) pairs. For∆ = ∆ Eq. (2) corresponds to the order parameter ofthe pure polar phase.In the presence of magnetic field, on cooling from thenormal phase of He, a superfluid transition should occurinto the β phase at the temperature T P c = T ca + ηH, (3)where T ca is a superfluid transition temperature of Hein nematic aerogel in zero field and η ∼ − kOe − [26].On further cooling, a second-order transition to the dis-torted β phase is expected at the temperature T P c = T ca − ηH β − β , (4)where β = β + β , and so on, β i , i ∈ { , . . . , } arecoefficients of the fourth-order invariants of the order pa-rameter in the Ginzburg-Landau free energy functional[1], or beta parameters.From Eqs. (3) and (4) we obtain that the width ofthe domain of existence of the β phase T P c − T P c = ηH β − β is proportional to H , and the P –P splitting ischaracterized by the following equation: T P c − T ca T ca − T P c = − β β . (5)Assuming the bulk He beta parameters [32], the fractionin Eq. (5) equals 0.93 at s.v.p. and 1.36 at 15.4 bar.
Sample and methods.—
We used the sample of mul-lite nematic aerogel (Metallurg Engineering Ltd.) withdensity ≈
150 mg/cm and porosity ≈ . ≤
14 nm (from the scanningtransmission electron microscope images) and a charac-teristic separation of 60 nm. The effective mean freepaths of He quasiparticles in directions parallel andtransverse to the aerogel strands in the limit of zero tem-perature are 900 nm and 235 nm correspondingly [17].The sample had a form of cuboid with a size alongstrands ≈ . ≈ × µ m NbTi wire, bent into theshape of an arch with a total height of ≈
10 mm and adistance between the legs of 4 mm. Strands of the aerogelwere oriented along the oscillatory motion. The aerogelwire is mounted in cylindrical experimental cell (internaldiameter is 6 mm) made from Stycast-1266 epoxy resinsurrounded by a superconducting solenoid so that thesample is located at the maximum of the magnetic field
ForkSampleSolenoidSintered(cid:3)heat exchanger mm mm mm FIG. 1. The sketch of the experimental cell. (with homogeneity of ∼ .
1% at distances ± He atomic layers on the aerogel strands and tostabilize the polar phase [15], we added 1.55 mmole of Heinto the empty cell at T ≤
100 mK and then filled it with He. This amount of He is enough to completely removesolid He and, according to our estimations, correspondsto 2.5–3.2 atomic layers of He [33].The experiments were carried out at a pressure of15.4 bar and in magnetic fields 0.5–8.9 kOe. The neces-sary temperatures were obtained by a nuclear demagneti-zation cryostat. A measurement procedure of the aerogelresonator is similar to that of a conventional wire res-onator [34]. An alternating current, with amplitude I varying in our experiments from 0.5 to 8.9 mA (depend-ing on H and being set to keep the amplitude of oscilla-tions constant at all fields), is passed through the NbTiwire. The Lorentz force sets the wire into oscillations.For this resonator, the resonance frequency in vacuum is621 Hz. Motions of the wire in the magnetic field gener-ates a Faraday voltage proportional to the wire velocity.This voltage is amplified by a room-temperature step-up transformer 1:30 and measured with a lock-in ampli-fier. In-phase and quadrature signals are joint fitted toLorentz curves in order to extract the FWHM and f a .In liquid He the maximum velocity of the wire in theused range of temperatures did not exceed 0.2 mm/s. Ina given field additional experiments with 2 times smallerexcitation current were also done and showed the sameresults.Similar mullite samples (cut from the same originalpiece of the nematic aerogel) were used in nuclear mag-netic resonance (NMR) experiments in He [16] and inVW experiments in low magnetic fields [35]. Correspond-ingly, we can expect that the present sample should havenearly the same T ca and the temperature width of thesuperfluid transition about 0.001 T ca . We note that inexperiments with VW [35] an additional (the second)resonance mode was observed, existing only below T ca .This second mode is an analog of the second-sound-likemode (called also as slow sound mode) observed in silicaaerogel in superfluid helium [36, 37] and corresponds tomotions (in opposite directions) of the normal componentinside the aerogel together with the aerogel strands andof the superfluid component. On cooling from T = T ca ,the resonant frequency of this additional mode increasesvery quickly from 0 up to ∼ . T ca becomesclose to the resonance frequency of the main mechanicalVW resonance resulting in a strong interaction of thesemodes (see Ref. [35] for details). In present experimentswe focused on measurements only of the main resonance,which intensity was significantly greater. Temperature measurements.—
Depending on H and temperature, a quartz tuning fork, located near theVW, was immersed in either normal, or in A , or inA phases. Unfortunately, we were not able to makegood enough temperature calibration of the fork reso-nance properties in A and A phases due to their depen-dence on H . Therefore we used the following procedurefor determining the temperature.All the experiments were done on a slow warming of thenuclear demagnetization stage starting from T ≈ . T c ,because at 15.4 bar and in low magnetic field the expectedsuperfluid transition temperature of He in our sampleis ≈ . T c [16, 35]. High final demagnetization fieldswere used, and the stage was warmed by a large fixed heatflux applied directly to the stage. In this case the heatcapacity of the stage determines the warming rate andinverse temperature of the stage should decrease linearlywith time.Overheating of the sample with respect to the stage( δT ) depends on a residual heat leak ( ˙ q ) to the cell, heatcapacity, and effective heat conductivity of He. We haveperformed computer simulations of warming of the stagetogether with the cell in the temperature range of (0.95–1.05) T c taking into account the magnetic field distribu-tion, ˙ q (in the range of 0–5 nW), and realistic parameters(effective heat conductivity and heat capacity) of normaland superfluid (A , A , and B) phases of He which, de-pending on temperature and H , occupy different partsof the cell. We have found that if the sample tempera-ture is above T A c or below T A c , then in this model thesample is warming with practically the same rate as thestage. The point is that at T > T A c all He in the cell isnormal while at
T < T A c the spatial distribution of thesuperfluid phases in the used small range of temperaturespractically does not depend on T : in the high field region He is in the A phase and near the heat exchanger – in F W H M ( H z ) time (hours) R e s on a n ce fr e qu e n c y ( H z ) T A c T A c T P c T P c FIG. 2. The FWHM and resonance frequency of the mainresonance of the VW versus time measured in magnetic fieldof 7.8 kOe at I = 0 .
57 mA. Heat flux, applied to the stage,is 32 nW and the final demagnetization field at its maximumis 6.6 kOe. Different types of straight arrows (see the legendbox) indicate the features we associate with T P c and T P c in He in the sample, with T A c and T A c in bulk He. the B phase. This provides a very weak dependence of δT on T above T A c or below T A c , although δT at T > T A c is greater. Significant variations of He parameters and ˙ q (and its spatial distribution) used in the simulations donot change the picture qualitatively, that is, the rates ofthe sample warming at T < T A c and at T > T A c remainthe same as the warming rate of the stage.Transitions in bulk He at T = T A c and T = T A c areclearly detected by the VW and may be used as refer-ence points [23, 24]. Correspondingly, for each warm-upexperiment we determined the “reference” warming rateof the sample using the time dependence of the FWHMmeasured on warming above the first reference point T A c ,where the FWHM is inversely proportional to T [38].Then we determined the time-dependence of the sampletemperature below T A c using this “reference” warmingrate. In a fixed H experiments were repeated 2–4 timesusing different warming rates of the stage. No differenceshave been found in the obtained dependencies of the VWresonance characteristics on the temperature calculatedas explained above. Results.—
In Fig. 2 we show an example of a warm-up experiment, where we measure the FWHM and thefrequency of the main resonance of the VW. As it isdescribed above, we get a temperature calibration ofthe x -axis (time) in each experiment using the data at T > T A c . The obtained temperature dependencies atvarious H are shown in Fig. 3. Note that in Fig. 3 theexperimental data at T & T A c are not shown because at T A c < T < T A c the sample warming rate changes. Inboth figures by straight arrows we mark different super- (b) R e s on a n ce fr e qu e n c y ( H z ) T/T c T A c T P c T P c H = 8.9 kOeH = 6.7 kOeH = 4.4 kOe0.96 0.97 0.98 0.99100120140 F W H M ( H z ) T/T c T A c T P c T P c H = 8.9 kOeH = 6.7 kOeH = 4.4 kOe (a)
FIG. 3. Temperature dependencies of FWHM (panel (a))and frequency (panel (b)) of the main resonance of the VWresonator measured in magnetic fields of 8.9 kOe (squares),6.7 kOe (triangles), and 4.4 kOe (circles) at correspondingexcitation currents of 0.5 mA, 0.67 mA, and 1 mA. Differenttypes of straight arrows (see the legend box) indicate the fea-tures we associate with T P c , T P c , and T A c . T c is a superfluidtransition temperature of bulk He in zero magnetic field. fluid transitions. On warming from the lowest tempera-tures, we observe the first feature, “step” on the FWHMplot (Fig. 3(a)) or “kink” on the resonance frequencyplot (Fig. 3(b)), which we refer to the transition fromthe distorted β phase to the pure β phase at T = T P c .At this temperature the frequency of the second modeis ∼ T P c / T c ( T P c / T c ) H (kOe) (T P c T ca )/(T ca T P c ) = 0.63T ca /T c = 0.983 H (kOe) ( T P c T P c ) / T c FIG. 4. P –P splitting of the superfluid transition of Hein nematic aerogel in magnetic field. Open and filled circlesindicate transitions between distorted β and β , β and normalphases respectively. Lines are linear approximations (withthe same zero field value) of experimental data. Inset: thedependence of the temperature interval of existence of the β phase versus H . The line is the best fit through the zero. higher temperatures becomes stronger, and in the mainresonance we observe a peak-like change of the linewidthas well as the rapid change of the resonance frequency.Similar behavior of the VW with nematic aerogel wasobserved near T ca in experiments in low magnetic field[35]. Approximately at the local minimum of the FWHMplot the second mode vanishes meaning that He in aero-gel is no longer superfluid and T = T P c . During furtherwarming, on the FWHM and the frequency plots onemore kink is seen marking the A –A phase transitionin surrounding bulk He ( T = T A c ). Both dependen-cies reach their maximum (the FWHM) and minimum(the frequency) at the superfluid transition temperatureof bulk He ( T = T A c ) (see Fig. 2).In Fig. 4 we summarize results of our experiments andshow the measured at 15.4 bar dependencies of T P c , T P c and ( T P c − T P c ) on the applied magnetic field.In order to test our way of determining the temper-ature, we have done a separate low field experiment( H ≈
500 Oe), where we have determined T P c usingas the additional reference point the equilibrium B–Atransition in bulk He (accompanied by a jump on theFWHM and f a ) which in this field occurs near T ca , i.e.at T ≈ . T c (see the leftmost data point in Fig. 4).Mutual fit of the data in Fig. 4 by two lines gives usa zero field superfluid transition temperature in aerogel T ca = 0 . T c , which is close to the value ( ≈ . T c )measured in NMR experiments with the similar sample(see Ref. [16]). As it was expected, the width of the do-main of existence of the β phase is nearly proportional to H (inset of Fig. 4). However, the obtained ratio of slopesof the fit lines ( T P c − T ca ) / ( T ca − T P c ) = 0 .
63 is surpris-ingly smaller than expected value (1.36) if we considerthe beta parameters of bulk He [32].
Conclusions.—
Using the VW techniques, we haveobserved the β phase and measured a P –P splittingof the superfluid transition temperature of He in ne-matic aerogel in strong magnetic field. We have foundthat the temperature range of existence of the β phaseis nearly proportional to H . We also have observed that( T P c − T ca ) < ( T ca − T P c ) that contradicts a common pic-ture confirmed in the bulk A phase of He [23–25]. Thus,further experimental studies at low and high pressuresare needed to understand reasons for the discrepancy.This work was supported by the Russian Science Foun-dation (project no. 18-12-00384). We are grateful toI.A. Fomin and E.V. Surovtsev for useful discussions. ∗ [email protected][1] D. Vollhardt and P. W¨ o lfle, The Superfluid Phases ofHelium 3 (Tailor & Francis, London, 1990).[2] K. Aoyama and R. Ikeda, Phys. Rev. B , 060504(2006).[3] J.A. Sauls, Phys. Rev. B , 214503 (2013).[4] I.A. Fomin, J. Exp. Theor. Phys. , 765 (2014).[5] R. Ikeda, Phys. Rev. B , 174515 (2015).[6] I.A. Fomin, J. Exp. Theor. Phys. , 933 (2018).[7] G.E. Volovik, JETP Lett. , 324 (2018).[8] I.A. Fomin, J. Exp. Theor. Phys. , 29 (2020).[9] V.V. Dmitriev, A.A. Senin, A.A. Soldatov, andA.N. Yudin, Phys. Rev. Lett. , 165304 (2015).[10] R.Sh. Askhadullin, V.V. Dmitriev, D.A. Krasnikhin,P.N. Martynov, A.A. Osipov, A.A. Senin, andA.N. Yudin, JETP Lett. , 326 (2012).[11] V.V. Dmitriev, A.A. Senin, A.A. Soldatov, E.V. Surovt-sev, A.N. Yudin, J. Exp. Theor. Phys. , 1088 (2014).[12] V.V. Dmitriev, A.A. Soldatov, and A.N. Yudin,JETP Lett. , 643 (2016).[13] N. Zhelev, M. Reichl, T. Abhilash, E.N. Smith,K.X. Nguyen, E.J. Mueller, and J.M. Parpia, Nat. Com-mun. , 12975 (2016).[14] S. Autti, V.V. Dmitriev, J.T. M¨akinen, A.A. Solda-tov, G.E. Volovik, A.N. Yudin, V.V. Zavjalov, andV.B. Eltsov, Phys. Rev. Lett. , 255301 (2016).[15] V.V. Dmitriev, A.A. Soldatov, and A.N. Yudin,Phys. Rev. Lett. , 075301 (2018).[16] V.V. Dmitriev, M.S. Kutuzov, A.A. Soldatov, andA.N. Yudin, JETP Lett. , 734 (2019). [17] V.V. Dmitriev, A.A. Soldatov, and A.N. Yudin,J. Exp. Theor. Phys. , 2 (2020).[18] V.B. Eltsov, T. Kamppinen, J. Rysti, and G.E. Volovik,arXiv:1908.01645.[19] V.E. Asadchikov, R.Sh. Askhadullin, V.V. Volkov,V.V. Dmitriev, N.K. Kitaeva, P.N. Martynov, A.A. Os-ipov, A.A. Senin, A.A. Soldatov, D.I. Chekrygina,A.N. Yudin, JETP Lett. , 556 (2015).[20] V.V. Dmitriev, L.A. Melnikovsky, A.A. Senin,A.A. Soldatov, A.N. Yudin, JETP Lett. , 808(2015).[21] V.J. Gully, D.D. Osheroff, D.T. Lawson, R.C. Richard-son, and D.M. Lee, Phys. Rev. A , 1633 (1973).[22] D.D. Osheroff and P.W. Anderson, Phys. Rev. Lett. ,686 (1974).[23] U.E. Israelson, B.C. Crooker, H.M. Bozler, andC.M. Gould, Phys. Rev. Lett. , 1943 (1984).[24] D.C. Sagan, P.G.N. deVegvar, E. Polturak, L. Friedman,S.S. Yan, E.L. Ziercher, and D.M. Lee, Phys. Rev. Lett. , 1939 (1984).[25] H. Kojima and H. Ishimoto, J. Phys. Soc. Jpn. ,111001 (2008).[26] E.V. Surovtsev, J. Exp. Theor. Phys. , 477 (2019).[27] E.V. Surovtsev, J. Exp. Theor. Phys. , 1055 (2019).[28] G. Gervais, K. Yawata, N. Mulders, and W.P. Halperin,Phys. Rev. B , 054528, 2002.[29] J.A. Sauls and P. Sharma, Phys. Rev. B , 224502(2003).[30] P. Brussaard, S.N. Fisher, A.M. Gu´enault, A.J.Hale, andG.R. Pickett, J. Low Temp. Phys. , 555 (2000).[31] P. Brussaard, S.N. Fisher, A.M. Gu´enault, A.J. Hale,N. Mulders, and G.R. Pickett, Phys. Rev. Lett. , 4580(2001).[32] H. Choi, J.P. Davis, J. Pollanen, T.M. Haard, andW.P. Halperin, Phys. Rev. B , 174503 (2007).[33] V.V. Dmitriev, M.S. Kutuzov, A.Y. Mikheev, V.N. Mo-rozov, A.A. Soldatov, and A.N. Yudin, Phys. Rev. B ,144507 (2020).[34] D.C. Carless, H.E. Hall, and J.R. Hook,J. Low Temp. Phys. , 583 (1983).[35] V.V. Dmitriev, M.S. Kutuzov, A.A. Soldatov, andA.N. Yudin, JETP Lett. , issue 12 (2020)[arXiv:2011.13747].[36] M.J. McKenna, T. Slawecki, and J.D. Maynard,Phys. Rev. Lett. , 1878 (1991).[37] A. Golov, D.A. Geller, and J.M. Parpia, Phys. Rev. Lett. , 3492 (1999).[38] R. Blaauwgeers, M. Blazkova, M. ˘Clove˘cko, V.B. Eltsov,R. de Graaf, J. Hosio, M. Krusius, D. Schmoranzer,W. Schoepe, L. Skrbek, P. Skyba, R.E. Solntsev, andD.E. Zmeev, J. Low Temp. Phys.146