Exceeding the Landau superflow speed limit with topological Bogoliubov Fermi surfaces
S. Autti, J.T. Mäkinen, J. Rysti, G.E. Volovik, V.V. Zavjalov, V.B. Eltsov
EExceeding the Landau Speed Limit with Topological Bogoliubov Fermi Surfaces
S. Autti,
1, 2
J.T. M¨akinen,
1, 3, 4
J. Rysti, G.E. Volovik,
1, 5
V.V. Zavjalov,
1, 2 and V.B. Eltsov ∗ Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland Department of Physics, Lancaster University, Lancaster LA1 4YB, UK Department of Physics, Yale University, New Haven, Connecticut, 06520, USA Yale Quantum Institute, Yale University, New Haven, Connecticut, 06520, USA L.D. Landau Institute for Theoretical Physics, Moscow, Russia
A common property of topological systems is the appearance of topologically protected zero-energyexcitations. In a superconductor or superfluid such states set the critical velocity of dissipationlessflow v cL , proposed by Landau, to zero. We check experimentally whether stable superflow is never-theless possible in the polar phase of p-wave superfluid He, which features a Dirac node line in theenergy spectrum of Bogoliubov quasiparticles. The fluid is driven by rotation of the whole cryostat,and superflow breakdown is seen as the appearance of single- or half-quantum vortices. Vorticesare detected using the relaxation rate of a Bose-Einstein condensate of magnons, created withinthe fluid. The superflow in the polar phase is found to be stable up to a finite critical velocity v c ≈ . v cL but below v c is provided by the accumulation of the flow-induced quasi-particles into pockets in the momentum space, bounded by Bogoliubov Fermi surfaces. In the polarphase this surface has non-trivial topology which includes two pseudo-Weyl points. Vortices formingabove the critical velocity are strongly pinned in the confining matrix, used to stabilize the polarphase, and hence stable macroscopic superflow can be maintained even when the external drive isbrought to zero. I. INTRODUCTION
The stability of superflow in superfluids and supercon-ductors is supported by both topology and the Landaucriterion. Via quantization of circulation, the topologi-cal stability protects gradual decay of flow around vor-tices and in a ring geometry. The Landau criterion pro-tects the superflow against decay via creation of quasi-particles for velocities below the Landau critical velocity v cL = min[ E ( p ) /p ], where E ( p ) is the energy spectrumof quasiparticles with momentum p . In Fermi superflu-ids and superconductors v cL ≈ ∆ /p F , where ∆ is the gapin the energy spectrum of Bogoliubov quasiparticles and p F is the Fermi momentum. In topological systems, ap-pearance of sub-gap (in particular, zero-energy) states orpresence of nodes in the energy gap is ubiquitous. Howsuch zero-energy states affect the stability of superflow intopological superconductors and superfluids is an openquestion.Remarkably, in the topological superfluid phases of He, superflow may persist when one [1, 2], or even both[3–5], of those constraints are violated. In particular,topological protection is absent in the chiral superfluid He-A, where the circulation is topologically unstable to-wards a phase slip with the formation of skyrmions [4, 5].However, contrary to the statement in Ref. [5], the super-flow persists up to velocity v ∼ . He-A due to presence of two point nodes [ E ( p ) = 0] inthe energy spectrum. On one hand, the absence of topo-logical stability does not exclude local stability of super- ∗ vladimir.eltsov@aalto.fi flow, supported by anisotropy of the superfluid density,effects of boundaries, applied magnetic field, or spin-orbitinteraction. On the other hand, superflows exceeding theLandau critical velocity do not necessarily lead to the de-struction of superfluidity in Fermi superfluids [6]. In thesuper-Landau superflow some Bogoliubov quasiparticlestates acquire negative energy. Fermionic quasiparticlesstart to occupy those energy levels forming a Fermi sur-face. Such a Fermi surface is called the Bogoliubov Fermisurface (BFS). In superfluid He and in cuprate supercon-ductors [7–10] the BFS appears in the presence of super-flow, while in systems with multiband energy spectrumor with broken time-reversal symmetry the BFS may ex-ist even in the absence of superflow [11–18]. Note that ina nodal topological superfluid or in a cuprate supercon-ductor the superflow explicitly breaks time-reversal andinversion symmetries, and thus origin of the BFS can beconsidered on a common ground in different systems.Appearance of the BFS gives rise to a non-zero den-sity of states at zero energy and thus to a non-zero nor-mal component density ρ n even at T = 0. When allthe negative states are occupied, the equilibrium valueof ρ n ( T = 0) is reached, and the non-dissipative super-flow is restored, though with smaller superfluid density, ρ s ( T = 0) = ρ − ρ n ( T = 0). The superflow above theLandau critical velocity remains stable until some othercritical velocity v c is reached. This can be either thevelocity at which ρ n ( T = 0) = ρ and thus the super-fluid density ρ s = ρ − ρ n vanishes, or the critical velocityat which quantized vortices or other topological defects,such as skyrmions, are created.The topology and other properties of the p-wave super-fluid He can be tuned on a wide range via controllingtemperature, pressure, or magnetic field [19], or by in- a r X i v : . [ c ond - m a t . o t h e r] J un v s Bogoliubov Fermi surface pseudo-Weylpointpseudo-Weylpoint (a) (b) d D ˆ m p x p y p z gapDirac node line N (pseudo)= +1 p x p y − p F p z N (pseudo)= -1 FIG. 1. The polar phase of superfluid He engineered withnanostructured confinement. (a) The confining matrix is aset of parallel solid strands, realized using commercial nafenmaterial with d ≈ D ≈
35 nm [25]. In the stationarypolar phase, the energy spectrum of Bogoliubov quasiparti-cles includes a Dirac node line in the plane perpendicular tostrands. (b) In the presence of superflow v s the node linetransforms to the Bogoliubov Fermi surface, consisting of twoFermi pockets, which touch each other. Here the superflow isapplied along the x axis, and touching points at p = ± p F ˆ y are pseudo-Weyl points. Their topology is illustrated as thehedgehog in momentum space, with the topological invari-ant in Eq. (4). Arrows show direction of the ˆ n vector andthe parameters in Eq. (2) are chosen as m ∗ c/p F = 1 /
12 and v s /c = 1 / troducing engineered nanoscale confinement [20–23]. Re-cently a new phase of He, the time reversal symmetricpolar phase has been engineered using such confinement[24–26]. The polar phase, Fig. 1, features a Dirac nodalline, robust to disorder and impurities owing to the ex-tension of the Anderson theorem [27–29]. Due to thepresence of the nodal line with E ( p ) = 0, the Landaucriterion in the polar phase is violated for any non-zerovelocity.The purpose of the current Report is twofold: First,we experimentally demonstrate that the superflow in thepresence of the nodal line remains stable until the fluidvelocity at the sample boundaries reaches 0 .
24 cm/s, wellabove the zero Landau critical velocity. At higher veloc-ity the flow, driven by rotation of the sample container,becomes unstable towards formation of quantized vor-tices. The appearance of vortices, strongly pinned to thestrands of the confining matrix, is detected as the in-creased relaxation rate of a Bose-Einstein condensate ofmagnon quasiparticles [30, 31]. Vortices remain in thesample for days after the rotation is stopped maintaininglong-living superflow exceeding the Landau critical veloc-ity even in a stationary sample. These observed featuresof vortex dynamics in the polar phase are supported bynumerical simulations. Second, we discuss the topologyof the resulting Bogoliubov Fermi surface and providesuggestions for characterization of the effects of the BFSon superfluid properties in future experiments. ˆm ππ ˆd α H φ ˆ d soliton Single -quantum vortex(SQV)
Half -quantum vortex(HQV)
FIG. 2. Types of quantized vortices in the polar phase. Theorder parameter phase φ (background color) winds by 2 π around a single-quantum vortex, and by π around a half-quantum vortex. To keep the order parameter single-valued,vector ˆ d (red arrows) also rotates around the HQV core, sothat ˆ d → − ˆ d when φ → φ + π . In non-axial magnetic field(green arrows) this leads to the formation of ˆ d solitons con-necting HQVs pairwise (blue dash line). Vortex cores, vectorˆ m , nafen strands, and the axis of rotation are perpendicularto the plane of the picture. II. POLAR PHASE
The order parameter in the polar phase is A µj = ∆ P ˆd µ ˆm j e iφ . (1)Here, ∆ P is the maximum gap in the quasiparticle energyspectrum, φ is the superfluid phase, and ˆd is a unit vec-tor of spontaneous anisotropy in the spin space. Orbitalanisotropy vector ˆm is locked along the nafen strands,while the node line in the energy spectrum develops inthe plane perpendicular to the strands, Fig. 1a. In oursample the strands are oriented along the rotation axis,labeled ˆ z .This order parameter allows for both usual single-quantum phase vortices (SQV), around which φ → φ +2 π , and half-quantum vortices (HQV), where φ → φ + π and α → α + π [32], Fig. 2. Here α is the azimuthal an-gle of ˆd in the plane perpendicular to the magnetic field.The ˆ d vector is kept in this plane by the Zeeman energyin the magnetic field of applied in our experiments. Ad-ditionally pure spin vortices with winding α → α + 2 π and φ = const around the core can exist, but they arenot relevant for the critical velocity in applied mass flow,so we do not discuss them here.The quasiparticle energy spectrum in the polar phase,which is Doppler shifted in the presence of a superflow v s , takes the form E ( p ) = (cid:15) ( p ) + p · v s , (cid:15) ( p ) = c p z + v ( p − p F ) , (2)where c = ∆ P /p F . For any v s not collinear with the ˆ z axis, this spectrum contains states with E ( p ) < v cL = 0. III. MEASUREMENTS
We study the stability of superflow starting from theinitial state prepared by slowly cooling the stationary (rad/s) M - ( M - ( ) ( s - ) (a) HQV SQV 0 0.02 0.04 0.06 0.08 Normalized satellite intensity I sat M - ( I s a t M - ( ) ( s - ) (b) FIG. 3. Increase of the relaxation rate of the magnon BEC τ − M due to vortices in the polar phase. (a) The relaxationrate grows as a function of rotation velocity Ω applied at thetransition to the superfluid state. For HQVs the shown slopeis extracted from the data in Ref. [32] with the contributionof vortices, created by the Kibble-Zurek mechanism (KZM),removed. For SQVs KZM is suppressed by the symmetry-violating bias [40]. The measured points are shown by sym-bols and the line is a fit to Ω / dependence. (b) For HQVs τ − M (circles) is proportional to the total volume of the ˆ d -solitons between HQV cores, measured by the area of thesatellite peak I sat in the normalized NMR spectrum, like inFig. 4. The line is a linear fit. sample through the superfluid transition in a transversemagnetic field to suppress the formation of both HQVsand SQVs [40]. All measurements are performed on the4 × × cubic sample container at 7 bar pres-sure and T = 0 . T c in the magnetic field H = 12 mT.The container is filled with nanomaterial called nafen,which consists of parallel columnar Al O strands with0.243 g/cm volume density. Then, at constant T , wegradually increase the rotation velocity in small steps,reducing the applied field to zero before changing thevelocity.When the change is finished, we restore the trans-verse magnetic field and measure the relaxation rate oflong-living magnons, pumped to the sample by a radio-frequency excitation pulse. Under conditions of thiswork, magnons form a Bose-Einstein condensate (BEC)within the sample [30, 31, 41]. The magnon BEC is man-ifested by coherent precession of magnetization with thesame frequency and coherent phase, despite the inhomo-geneity of the magnetic field or variation in the spin-orbitinteraction strength. The precession slowly decays due tomagnon loss. In He-B such condensates are thoroughlyexplored and were used as sensitive probes of temper-ature [33, 34], collective modes [35], spin supercurrents[36], analogue event horizon [37], and of vortices [38, 39].Their usefulness as sensors in the polar phase has notbeen known before this work.We have found that the decay rate τ − M of the magnonBEC is sensitive to the presence of vortices also in thepolar phase, Fig. 3. To calibrate this effect, we createan equilibrium array of vortices by rotating the sampleat the angular velocity Ω while slowly cooling it down to the superfluid state. HQVs are produced with cooling inthe zero or axial field [32].For HQVs, Fig. 3b, τ − M is proportional to the intensity I sat of the characteristic satellite peak in the continuous-wave (cw) nuclear magnetic resonance (NMR) spectrummeasured in the transverse field, Fig. 4a. The I sat ∝ Ω / is essentially a fractional volume occupied by theˆ d solitons connecting HQVs pairwise [32]. We concludethat the magnon BEC relaxation is concentrated in thesesolitons, where the spin configuration deviates from theequilibrium.If the magnetic field is applied transverse to the rota-tion axis and the nafen strands, HQVs become energet-ically unfavorable [42], and an array of SQVs is createdon cooling through T c . SQVs cannot be easily identifiedbased on cw NMR spectra, since SQVs are not associatedwith a soliton structure and their cores are too small toprovide noticable contribution to the signal. However,a definite increase of the magnon BEC relaxation as afunction of Ω is observed, Fig. 3a.The origin of the magnon BEC relaxation in the Bphase is conversion of the magnons from the conden-sate to longitudinal spin waves (light Higgs quasiparticles[35]) in the distorted orbital texture surrounding non-axisymmetric vortex cores [43]. In that case τ − M ∝ Ωin the equilibrium vortex state. In the polar phase thestructure of vortex cores is not known and expected to besubstantially affected by the nafen strands with diameter ∼ . τ − M ∝ Ω / remain unclear, but we nevertheless can use τ − M as a marker for appearance of vortices. Moreover,usage of the magnon BEC probe proved to be essentialfor fulfilling the main goal of this work, establishing sta-bility of the flow in the polar phase, as it turns out thatthe stability is lost with formation of SQVs (see the nextsection), which are invisible to classic linear NMR. IV. CRITICAL VELOCITY
The measured magnon BEC relaxation rate τ − M asa function of Ω, when a change of rotation velocity isstarted from the vortex-free state at Ω = 0, is presentedin Fig. 5. It shows that there is a clear velocity, abovewhich the relaxation rapidly grows. We interpret thispoint as a critical velocity for vortex formation. Thesevortices remain in the sample even when Ω is returned tozero, as expected if the vortices are pinned on the nafenstrands. After the Ω cycle is finished, we identify thetype of the formed vortices by measuring the cw NMRspectrum of the sample, Fig. 4a. Since we find no satel-lite peak characteristic to HQVs, we infer that vorticesformed during the Ω cycle are SQVs. In our experimen-tal conditions the critical velocity for SQV formation is −8 −6 −4 −2 0 200.020.040.060.080.10.120.140.160.180.2 f − f (kHz) N o r m a li z ed a m p li t ude ( k H z − ) satellite main line with SQVswith HQVs time (h) −2−1012 Ω (r ad / s ) time (h) Ω (r ad / s ) (a)(c)(b) FIG. 4. Continuous-wave NMR spectra of the polar phasemeasured in the magnetic field transverse to nafen strandswith HQVs and SQVs present in the sample. (a) The ab-sorption normalized to the total spectrum area is plotted ver-sus the frequency shift from the Larmor value f = | γ | H/ π ,where γ is the gyromagnetic ratio of He. The HQVs producea satellite in the NMR spectrum, while for SQVs no clear dis-tinguishing features are seen. (b) SQVs are produced with theslow sweep of the angular velocity Ω( t ). (c) Rapid changes inrotation velocity produce HQVs in addition to SQVs. Lastof the ten periods of such drive applied in the course of themeasurement in zero magnetic field is shown. thus lower than that for forming HQVs. This is the casealthough the energy of an HQV pair is smaller than thatof a single SQV in zero magnetic field (which is appliedwhen Ω changes). A similar situation is observed in He-A, where the critical velocity for the double-quantum vor-tex skyrmions [44] is lower than that of SQVs, while theenergy preference is the opposite [45].In our cubic container the flow is non-uniform and atΩ = 1 rad/s, which is seen as a characteristic angular ve-locity for vortex formation in Fig. 5, the maximum flowvelocity of 0.2 cm/s is reached in the middle of each sidewall of the square container cross-section. This value issomewhat lower than v c ≈ He-B, where it was also found to depend stronglyon the surface conditions [46]. For confined samples con-trol of the surface conditions, especially at the bound-aries of the confining matrix, remains a challenge for thefuture.Remarkably, we are also able to create HQVs in the su-perfluid state by changing the rotation velocity rapidlyenough, see Fig. 4c. To produce HQVs we vary the rota-tion velocity between Ω = +2 .
25 rad/s and − .
25 rad/sfor several hours, in which case the SQV creation andannihilation is not able to compensate for changes in therotation velocity fast enough and local flow velocity canexceed the critical velocity for HQV formation. While theamount of SQVs created in this process increases magnon
SQVs nucleated at v c ≈ fl ow vortices do not decaydue to pinning (rad/s)1.50.5 100510152025303540 τ M ( s − ) − FIG. 5. Change of the magnon BEC relaxation τ − M when therotation velocity Ω is gradually increased from 0 to 2 rad/sand then decreased back to 0 at constant temperature, start-ing from the state with no vortices. Two traces for two in-dependently prepared initial states are shown, the respectiveΩ( t ) dependences and final spectra are plotted in Fig. 4(b,a).Single-quantum vortices form first at Ω ≈ v c ≈ . v cL = 0 in the polar phase. BEC relaxation beyond what can be measured, HQVs arealso created, as seen from the appearance of the satellitein the cw NMR spectrum (magenta trace in Fig. 4a).Therefore it is possible to set bounds for the HQV crit-ical velocity for the conditions of the measurements: itexceeds 0.2 cm/s but is below 1 cm/s.Before the first vortices are formed, the superfluid is ina stable flow with non-zero velocity with respect to thewalls of the container and to nafen strands, despite thefact that the Landau critical velocity is zero in the polarphase.
V. NUMERICAL SIMULATIONS
We can gain qualitative understanding of the processthat creates vortices during Ω sweep in the superfluidstate with a simple numerical model, Fig. 6. In the modelwe consider a square sample container with 4 × cross section containing a grid of 201 ×
201 pinning pointand rotated about its center along the axis perpendicularto the cross section. The flow velocity is calculated as thesum of the potential flow in a box, and the contributionfrom vortices, which in the two-dimensional simulationare points. Vortices carry one quantum of circulationeither along the rotation velocity or in the opposite di-rection (antivortices).We start from the configuration with no vortices andzero rotation velocity Ω. Then Ω is increased in small -0.2 0 0.2 x, cm -0.200.2 y , c m -0.2 0 0.2 x, cm (rad/s) V o r t e x nu m be r positivenegativetotal (a)(b) (c) FIG. 6. Simulation of the vortex formation with pinning: (a)Number of vortices (red dash dot line), antivortices (blue dashline) and the sum of the two populations (solid black line),as a function rotation velocity Ω which is changed from 0 to2.5 rad/s and back. (b) The configuration of vortices (redcircles) and antivortices (blue crosses) at Ω = 2 . steps of 5 · − rad/s. After each step we calculate the su-perfluid velocity in the rotating frame at each grid point.If the flow velocity magnitude exceeds v c , simul = 0 . v c , simul . These vortex avalanchesemerge from the middle of each of the four containerwalls, where v c , simul is first reached. This feature resem-bles the substantial jump in τ − M at the critical velocityin the experiments. When the rotation velocity is laterdecreased, many vortices with positive circulation areannihilated by antivortices. However, the total numberof vortices and antivortices remains relatively constant, which agrees with the observation that the magnon BECrelaxation rate never decreases with changes in the rota-tion velocity. VI. TOPOLOGY OF THE BOGOLIUBOVFERMI SURFACE
Detection of stable superflow in the polar phase is anindirect indication of the formation of a BFS, given bysolution of E ( p ) = 0 in Eq. (2). Fig. 1b demonstrates theBFS for the superflow in the ( x, y ) plane transverse to theˆ m vector. Such BFS possesses quite remarkable featurescompared to the BFS expected to form in super-Landauflow in He-A or cuprate superconductors. In the case of He-A with the point nodes and in the available range ofstable superflow velocities [3], the BFS is formed as tinyellipsoidal pockets around the nodes. In case of cuprates,cylinders are formed around separate line nodes [7, 50].Both cases are topologically trivial. In the polar phase,the BFS is formed by two (electron and hole) pockets,which extend across the whole momentum space even atthe smallest velocities and touch each other at two pointswith non-trivial topology. Such Fermi surface resemblesthat in graphite, where the chain of touching electronand hole pockets is present [51–55].The non-trivial topology of the BFS in the polarphase is associated with the conical touching points at p = ± p F ˆ v s × ˆ z . It is similar but not identical to that ofthe Weyl point in Weyl semimetals and in He-A. Thisfollows from the Bogoliubov-de Gennes Hamiltonian: H = τ n ( p ) + n ( p ) + τ n ( p ) , (3)where τ and τ are Pauli matrices in the particle-holespace. As distinct from the Weyl Hamiltonian, the ma-trix τ is missing and thus we call those points pseudo-Weyl points [55]. The components of the vector n ( p ) are n ( p ) = cp z , n ( p ) = p · v s and n ( p ) = v F ( p − p F ). Theinvariant, which is similar to that for the Weyl points, is: N (pseudo) = 18 π e ijk (cid:90) S dS k ˆ n · (cid:18) ∂ ˆ n ∂p i × ∂ ˆ n ∂p j (cid:19) , (4)where ˆ n = n | n | − is a unit vector and S is the sphericalsurface around the touching point. Topological chargesof the two pseudo-Weyl points are N (pseudo) = ± He [56, 57]. Wethus suggest that topologically non-trivial BFS could berealized also in those systems, provided that pseudo-Weylpoints turn out to be robust against impurities.Let us now discuss predictions for observables resultingfrom the appearance of the non-thermal normal compo-nent in the polar phase. The BFS leads to a finite densityof states (DoS): N (0) = (cid:90) d p (2 π ) δ ( E ( p )) = N F v s c , (5)where N F = p F m ∗ /π is the DoS in the normal He and m ∗ = p F /v F is the effective mass. This results in a fi-nite density of the normal component at T = 0 and anadditional heat capacity, which both are linear in v s : ρ n ( T = 0) ρ = v s c m ∗ m , C ( T ) C F ( T ) = v s c . (6)Here C F ( T ) is heat capacity of the normal liquid. For v s ∼ . . N F , which, in principle, is detectable.Additionally, the presence of superflow suppresses thegap amplitude. According to Muzikar and Rainer [58],the suppression of the gap at T = 0 and v s (cid:28) c is(∆ P ( v s ) − ∆ P (0)) / ∆ P (0) = − v / c , and at experimen-tally relevant temperatures∆( v s , T )∆(0 ,
0) = (cid:20) − α T T − α v c TT c (cid:21) , v s /c (cid:28) T /T c (cid:28) , (7)where the parameters α , and α are of order of unity[59]. As a result, the spin-orbit interaction F D = g D (ˆ d · ˆ m ) is also suppressed with δg D /g D = − v / c at T = 0. A well-established method to measure thestrength of the spin-orbit interaction in superfluid He isthrough the shifts of the characteristic lines in the NMRspectra. The smallness of the expected effect (relativefrequency shift ∼ − for the temperature and velocityreached in the present experiment) will, however, makethis measurement challenging. VII. CONCLUSIONS
We have experimentally observed stable superflow inthe polar phase of He at velocities exceeding the zeroLandau critical velocity in the nodal line superfluid. Thestability of superflow, provided by formation of the Bo-goliubov Fermi surface and of the non-thermal normalcomponent at super-Landau velocities, is limited by cre-ation of the single-quantum vortices (at velocities of about 0.2 cm/s) and of the half-quantum vortices (at ve-locities below 1 cm/s). The next development will be toobserve the contribution of the flow-induced quasiparticlestates to thermodynamic quantities, e.g. those predictedby Eqs. (6) and (7). We thus confirm that appearanceof protected zero-energy states in topological superfluidsand superconductors does not prevent existence of sta-ble superflow in such systems. When the the originalzero-energy states belong to a closed line node, we pre-dict that the resulting BFS possesses non-trivial topol-ogy with the pseudo-Weyl points. It will be appealing tofind a topologically non-trivial BFS existing even withoutthe applied flow in systems with broken symmetries. Itwill also be interesting to elucidate further consequencesof the symmetries, broken by the superflow in topologi-cal superfluids and superconductors, beyond formation ofthe BFS. One example here is provided by the predictionof the spin-stripe phases [60, 61].In the case where BFS is formed by the flow at super-Landau velocities, like in the polar phase, the initial pro-cess of filling the negative energy states is also a fascinat-ing problem for future research, as it proceeds via radia-tion of quasiparticles which has similarities to the Hawk-ing radiation from the black-hole horizon [62–64]. Finallywe note that recent successes in stabilization of uniformultracold quantum gases [65] open possibilities to studythe evolution of the super-Landau superflow [66] in theBEC-BCS crossover, where the spectrum of Bogoliubovexcitations in the stationary system changes drastically.
ACKNOWLEDGMENTS
We thank V.V. Dmitriev for instructive discussionsand providing the nafen sample. This work has been sup-ported by the European Research Council (ERC) underthe European Union’s Horizon 2020 research and inno-vation programme (Grant Agreement No. 694248). Theexperimental work was carried out in the Low Tempera-ture Laboratory, which is part of the OtaNano researchinfrastructure of Aalto University. S. Autti acknowledgesfinancial support from the Jenny and Antti Wihuri foun-dation.S.A. and J.M. contributed equally to this work. [1] D.I. Bradley, S.N. Fisher, A.M. Guenault, R.P. Haley,C.R. Lawson, G.R. Pickett, R. Schanen, M. Skyba, V.Tsepelin, D.E. Zmeev, Breaking the superfluid speedlimit in a fermionic condensate, Nature Phys. , 1017(2016).[2] J. A. Kuorelahti, S. M. Laine and E. V. Thuneberg, Mod-els for supercritical motion in a superfluid Fermi liquid,Phys. Rev. B , 144512 (2018).[3] V. M. H. Ruutu, J. Kopu, M. Krusius, . Parts, B. Plaais,E. V. Thuneberg, and W. Xu, Critical Velocity of Vor-tex Nucleation in Rotating Superfluid He-A, Phys. Rev. Lett. , 5058 (1997).[4] P.W. Anderson and G. Toulouse, Phase slippage withoutvortex cores: vortex textures in superfluid He, Phys.Rev. Lett. , 508 (1977).[5] Onur Erten, Po-Yao Chang, Piers Coleman and AlexeiM. Tsvelik, Skyrme insulators: insulators at the brink ofsuperconductivity, Phys. Rev. Lett. , 057603 (2017).[6] G.E. Volovik, The Universe in a Helium Droplet , Claren-don Press, Oxford (2003).[7] G.E. Volovik, Superconductivity with lines of GAPnodes: density of states in the vortex, JETP Lett. ,
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