Fundamental dissipation due to bound fermions in the zero-temperature limit
S. Autti, R. P. Haley, A. Jennings, G. R. Pickett, R. Schanen, A. A. Soldatov, V. Tsepelin, J. Vonka, T. Wilcox, D. E. Zmeev
FFundamental dissipation due to bound fermions in the zero-temperature limit
S. Autti, ∗ R. P. Haley, A. Jennings, G. R. Pickett, R. Schanen, V. Tsepelin, J. Vonka, † T. Wilcox, and D. E. Zmeev
Department of Physics, Lancaster University, Lancaster LA1 4YB, UK.
A. A. Soldatov
P.L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia
The ground state of a fermionic condensate is well protected against perturbations in the presenceof an isotropic gap. Regions of gap suppression, surfaces and vortex cores which host Andreev-bound states, seemingly lift that strict protection. Here we show that the role of bound states ismore subtle: when a macroscopic object moves in superfluid He at velocities exceeding the Landaucritical velocity, little to no bulk pair breaking takes place, while the damping observed originatesfrom the bound states covering the moving object. We identify two separate timescales that governthe bound state dynamics, one of them much longer than theoretically anticipated, and show thatthe bound states do not interact with bulk excitations.
INTRODUCTION
Interfaces in fermionic condensates and the bound states hosted by them have recently attracted research focusin numerous condensed matter systems. Contemporary superfluid research is moving towards nanoscale probes ,nanoconfinement , and microscopic dissipation mechanisms related to quantum turbulence . Unconventional su-perconductivity in various materials and the dynamics of certain atomic BECs is characterised by the physics ofbound excitations. Nanofabrication of superconducting hybrid structures has enabled tailoring bound states withdesired properties. For instance, engineered superconducting systems containing Andreev-bound states (ABS) areused in studies of Majorana fermions aimed at producing components of, say, a quantum computer . In thisarticle we show that bound states covering the surface of a fermionic condensate not only determine the propertiesof its surfaces, but ABS dynamics are also central in understanding one of the most elementary of bulk phenomena:the drag an object experiences when moving at constant velocity through the fermionic superfluid.A recent experiment showed, contrary to the classic textbook picture, that a macroscopic object moving quasi-uniformly in superfluid He-B can exceed the predicted Landau velocity in the zero-temperature limit withoutdestroying the condensate . Phenomenologically the dissipation experienced by objects moving rapidly can beexplained in terms of the Lambert picture . This scenario is based on assuming that there is a well-defined 2Dgapless spectrum of bound states available within roughly a coherence length of all surfaces, and that the fermionsoccupying these states can escape to the bulk promoted by macroscopic flow fields, thereby providing dissipation.Simultaneously the surface layer shields the bulk condensate from interacting with the moving object, preventing pairbreaking even if the object’s velocity exceeds the Landau velocity. Assuming a free 2D gas of surface excitations isclearly a simplification, but the model captures the main features of the experiment well enough for the presentationof the results.The above approach still lacks rigorous theoretical justification. In particular, its critics have pointed out thatthermalisation of the surface-bound states should be so fast, of the order of nanoseconds, that the equilibrium cannotbe disturbed by the relatively slow motion (typically milliseconds) of a macroscopic moving object . Here weprovide experimental evidence to end the controversy. We show that this criticism correctly identifies only one of thetwo timescales that govern the dynamics of surface-bound states, and that this process is very fast as theoreticallyanticipated. Crucially, we also observe a second, much slower process, which allows for the surface states’ contributionto dissipation. Our experiment shows, remarkably, that the observed dissipation is independent of the density ofthermal bulk excitations. This observation justifies the assumption that surface-bound states are involved in theprocess in the first place.An ideal experiment aimed at probing ABS-originating dissipation in superfluid He-B would be carried out at atemperature so low that thermal bulk excitations vanish completely. Otherwise trivial collisions with them providea thermal dissipation background. In our experiments the sample temperature was between 140 µ K and 240 µ K.Therefore the density of thermal bulk quasiparticles is not negligible, but it is low enough that they propagateballistically, and their contribution to observed dissipation can be confidently subtracted.In these conditions at saturated vapour pressure the Landau critical velocity is v L = ∆ /p F = 27 mm/s (∆ is thesuperfluid gap and p F the Fermi momentum). We can reach and exceed this velocity in both AC and DC measurementsby driving a large current through a goalpost-shaped superconducting wire (legs 25 mm, crossbar 9 mm, thickness135 µ m). An external magnetic field is oriented parallel to the legs of the wire, and the resulting Lorentz force movesthe wire. The wire is surrounded by a volume of superfluid He-B, and the drag force experienced by the wire is a r X i v : . [ c ond - m a t . o t h e r] F e b inferred from the number of emitted quasiparticles, seen as a heat pulse measured by the thermometers. RESULTS
Let us study the consequences of a gapless spectrum of bound states covering the surface of the moving wire crossbarat a temperature so low that there are no thermal bulk excitations. In the presence of macroscopic flow, both bulk andgapless surface dispersion curves for quasiparticles (and holes) move according to ± v fl p F (in one dimension). Here v fl isthe local flow velocity of the superfluid. In the frame of the crossbar, the bulk far away from the wire flows at v fl = − v ,where v is the velocity of the crossbar in the laboratory frame. Near the cylindrical crossbar the flow is enhancedand reaches a maximum of v fl = − v , assuming perfect potential flow. These two contributions narrow down theenergy mismatch ∆ between available bulk states and populated ABS states by 3 vp F , and the flow therefore enablesABS escape from the surface of the wire to the bulk when the wire moves faster than v c = 1 / · ∆ /p F = 9 mm/s,the well known critical velocity in AC measurements . Strikingly, if v is held constant, the ABS spectrum finds anequilibrium and the drag force disappears .Such difference between oscillatory and uniform motion must arise from the time constants that govern the pro-cess described above. Basically, the repeating reversal of direction in oscillatory motion replenishes the reserve ofquasiparticles that can escape to bulk when the velocity exceeds the critical velocity, while in uniform motion suchreplenishment is not available. If we compare equally-long and otherwise identical trajectories where velocity is eitherreversed or not, the reversal should increase the observed dissipation. In the absence of the bound state contributionsuch directionality dependence is not expected. In order to experimentally substantiate this picture, and to remove the effect of the finite density of thermal bulkexcitations that cannot be avoided in experiments, we move the wire crossbar in two phases (Fig. 1). Each measurementstarts from standstill long enough to remove any history dependence. The wire is then quickly accelerated to v , thevelocity is kept constant for 0.5 mm distance, before reducing the velocity back to zero . The acceleration time wasvaried from 3 ms to 5 ms, and the obtained results were independent of it. After time ∆ t , the ramp is repeated fromthis new starting location with either v (“up ramp”) or − v (“down ramp”). This allows subtracting the dissipationmeasured for up ramps Q up from that measured for down ramps Q down , ∆ Q = Q down − Q up . As described in detailin Methods, the dissipation due to ambient bulk quasiparticles is cancelled assuming the up and down trajectoriesare equally long.The above measurement shows that bound-state dissipation is characterised by two regimes: First, the differencein the measured dissipation, ∆ Q , is zero when v ≤ v c = 9 mm/s. Second, for velocities higher than9 mm/s at ∆ t = 0, up ramps experience less dissipation than down ramps. Summing over the solid angle of allowedescape directions to bulk and over the available bound quasiparticle states on the wire surface (see Methods) yields thepower law ∆ Q ∝ (( v − v c ) /v c ) . , which is in good agreement with the experiment. We emphasise that v c = 9 mm/sis the well known AC critical velocity, confirming that the results of the DC measurement directly apply to dissipationexperienced by moving objects in superfluid He-B in general. This confirms that direct bulk pair breaking is replacedby a much weaker drag force originating from surface-bound quasiparticles, as speculated earlier .We can now vary ∆ t to measure the dynamics of bound states on the wire surface (Fig. 2). The difference indissipation, ∆ Q , disappears exponentially as exp( − ∆ t/τ ), where τ ≈ µ K – 230 µ K. Within this range of temperatures the thermal bulk quasiparticle density varies by almost two ordersof magnitude, while measured τ is constant. Together with the above observations this justifies the assumption thatbound quasiparticles are responsible for the dissipation. It also rules out any direct interaction between surface-boundand bulk quasiparticles as the source of thermalisation.The dynamics of bound quasiparticles on the wire surface can be described by two time constants in our toymodel. The first one, τ , gives the probability of the inter-branch process where a quasiparticle scatters from the wireexchanging momentum going from, say, p to − p . This process enables drag, as the exit channel to bulk is open onlyin the direction of wire motion while only quasiparticles with the momentum in the opposite direction gain energydue to the flow. That is, drag is produced by those quasiparticles that escape to bulk right after scattering with thewire, removing momentum from the wire. The second time constant, τ , describes the rate at which quasiparticlesthat did not escape to bulk relax within each dispersion branch to the thermal equilibrium distribution distorted byprocess one. If τ = ∞ , then the imbalance produced by moving the wire and stopping in the middle, measured bycomparing up and down ramps, disappears diffusively as determined by τ . Assuming τ = 0, the imbalance stilldisappears at the rate given by τ , but this time the process is deterministic. Therefore, regardless of τ , we havemeasured τ ≈ τ = 6 ms.It is also possible to set an experimental upper bound for τ . We have measured the AC critical velocity of a set ofprobes with resonance frequencies f ranging from 350 Hz to 158 kHz (Fig. 3). The critical velocity is seen as kinkin the velocity of the probe measured as a function of force. It corresponds to a sudden increase in the force requiredto increase the velocity by a given amount. As speculated by Lambert , when f > /τ , one should see a clearreduction in the observed AC critical velocity as quasiparticles would be able to “climb up” the dispersion curvesdiffusively by scattering back and forth between p and − p . Say, by a two-step process one would get escape to bulkstarting at 4.5 mm/s, and so on.We observe no critical velocity reduction up to f = 158 kHz. In fact the observed critical velocity slowly andsmoothly increases as a function of resonance frequency. This happens despite the varying geometries of the fourprobes used: the two low frequency probes are vibrating superconducting wires, while the two kHz probes are the firstharmonic and an overtone of a custom-made quartz tuning fork with smooth surfaces. While studying the reason forthe observed slow power-law increase of v c ( v c ∝ f . ) systematically remains a task for the future, this result impliesthat τ < ∼ /f ≈ µ s. We believe that τ describes the fast thermalisation process anticipated in Ref. 16.Surface specularity may play an important role in the bound state escape process. In the B phase the gap suppressiondoes not significantly depend on the details of surface scattering, and therefore the bound states’ spectrum itself isapproximately independent of specularity . On the other hand, one would expect that τ depends on specularityas it describes the process of quasiparticle scattering with the wire. It is known that the surface scattering can betuned from diffuse to specular continuously by preplating the sample with slightly over two monolayers of He . Itis possible that He coverage also changes the scattering qualitatively by removing magnetic spin exchange with thesurface layer . However, reaching fully specular scattering requires a 2D superfluid layer of He, which flows to thecoldest part of the sample container (the heat sink) and cannot therefore be stabilised on the wire in our experiment .We have resorted to measuring the effect of adding two monolayers of solid He on the wire surface (and all othersurfaces in the sample container). While this causes the heat exchangers to become less efficient , we observe verylittle change in the bound state dynamics or dissipation. In particular, τ remains ≈ DISCUSSION
In conclusion, our experiments confirm that quasiparticles (and holes) bound to the region of gap suppression nearsolid surfaces in the B phase of superfluid He are responsible for the zero-temperature dissipation experienced bymacroscopic objects moving in the superfluid. The dissipation begins when the flow velocity is sufficient for releasingthe bound quasiparticles to the bulk of the superfluid. We have demonstrated experimentally that the bound statesre-equilibrate through a two-stage process involving a fast thermalisation step with τ < µ s, but importantly also atheoretically unforeseen slow scattering step characterised by τ = 6 ms. To be clear, τ is the effective time constantdescribing the process where the quasiparticle momentum is changed from p F to − p F . In a proper two-dimensionaltreatment of the process that τ emerges from, this may involve a large number of scattering events which collectivelyallow the quasiparticle to eventually find the very opposite momentum. We speculate that this process is potentiallythe underlying reason as to why τ is not of the order of nanoseconds. Furthermore, we vary the bulk quasiparticleconcentration to show that the bulk states do not contribute to bound state dynamics on any observable timescale.Therefore our results can be readily generalised to the description of transient phenomena in 2-dimensional Diracsystems such as graphene , if the escape process to bulk is neglected. It is worth emphasising that the bulk escapeprocess is temperature independent at low temperatures and, hence, provides dissipation even in the zero-temperaturelimit.We acknowledge the preliminary nature of the bound-state model presented above. In particular, it remains an openquestion how exactly bound state dynamics should be described in a full 2-dimensional model of the wire surface witha flow velocity distribution, diffusion or transport between various parts of the surface etc. Competing theoreticalsuggestions are sparse, but there have been some discussions related to a layer of vortices covering the wire acting as abuffer and therefore masking the Landau velocity . On the other hand, theoretical work on superflows exceeding theLandau velocity have recently been published studying other systems, such as polaron-polaritons and graphene .For these reasons it remains of interest to provide additional experimental insight. For example, one could changethe Landau velocity by studying flow in a confined geometry, say, in the recently-discovered polar phase of superfluid He . The gap spectrum of the polar phase contains a nodal line, meaning that the Landau speed limit is zero inthat plane. On the other hand, it seems essential to enhance the experiment presented in this article by building adetector or spectrometer fast enough to distinguish the dissipation associated with the separate phases of the wiremotion. This requires designing and devising nano-sized instruments . AUTHOR CONTRIBUTIONS
All authors contributed to gathering the results and writing the manuscript.
ACKNOWLEDGEMENTS
We thank Erkki Thuneberg for stimulating discussions. This work was funded by UK EPSRC (grant No.EP/P024203/1)and EU H2020 European Microkelvin Platform (Grant Agreement 824109). We acknowledge M.G. Ward and A.Stokes for their excellent technical support. S.A. acknowledges financial support from the Jenny and Antti WihuriFoundation.
COMPETING INTERESTS
The authors declare no competing interests
METHODS
An ideal experiment aimed at probing bound-state-originating dissipation in superfluid He-B would be carriedout at a temperature so low that thermal bulk excitations are rare enough to be neglected completely. Otherwise,trivial collisions with them provide a thermal dissipation background. In our experiments we used a nested nucleardemagnetisation cryostat , capable of holding the sample temperature between 100 µ K and 200 µ K for several days.Temperature was measured with a quartz tuning fork , and a vibrating wire thermometer . Pressure in allmeasurements was the saturated vapour pressure, corresponding to superfluid transition temperature T c =929 µ K.Therefore the density of thermal bulk quasiparticles is not negligible, but it is low enough so that they propagate bal-listically, and their contribution to the observed dissipation can be subtracted. We can vary the thermal quasiparticledensity by two orders of magnitude within the ballistic regime. Subtracting the thermal quasiparticle contributionis robust and reliable because the gap spectrum of the B phase is isotropic in zero magnetic field, and only slightlydistorted in small external magnetic field.The main probe used in the experiment is the goalpost-shaped wire. The bolometric volume that surrounds the wireis calibrated by resonant AC measurements by fitting the measured calibration data to known BCS heat capacityusing the effective volume of the sample as a fitting parameter. The fitted volume is 16 cm , which falls betweenthe free volume of the sample container, 15 cm , and the total volume of the sample container including the volumewithin the heat exchangers, 32 cm .We simultaneously monitor the position of the wire by picking up a high-frequency signal mixed in with the drivingcurrent using nearby coils. This method is not sensitive enough for measuring the drag force , but together withthe known wall-to-wall distance ( ≈
10 mm) it calibrates the range of motion. Finally, we record the induced voltageacross the wire with a 4-point measurement. The induced voltage reveals whether the main AC resonance or somehigher mode of oscillation is excited by the DC drive, allowing us to ensure that dissipation due to these modes wasminimised during all measurements.We access zero-temperature dissipation by moving the wire crossbar in two phases (Fig. 1). Each measurementstarts from standstill long enough to remove any history dependence. The wire is then quickly accelerated to v ,the velocity is kept constant for 0.5 mm distance, before decelerating back to zero. After time ∆ t , the ramp isrepeated from this new starting location with either v (“up ramp”) or − v (“down ramp”). This allows subtractingthe dissipation measured for up ramps Q up from that measured for down ramps Q down , ∆ Q = Q down − Q up .Ideally, both the trajectories would meet identical dissipation from scattering of thermal quasiparticles in the bulk.Subtracting the dissipation measured for up ramps Q up from that measured for down ramps Q down would leave onlythe ABS contribution ∆ Q = Q down − Q up , which for long enough ∆ t is expected to be zero. In practice, the motionof the wire is hysteretic due to the motion of flux lines in the superconductor, driven by the current used for movingthe wire in the magnetic field . Therefore the crossbar travels a shorter distance with down ramps than with upramps, and as a result scatters a different amount of thermal quasiparticles, corresponding to ∆ Q <
0. The hystereticcontribution follows ∆ Q = a ( v/v c ) / (Fig. 4a), where a ∝ (∆ I ) (Fig. 4c). That is, the force moving the wire iskept constant in all the measurements (for a given velocity v ), carried out at different magnetic fields H . Hence, thechange in the current passed through the wire ∆ I ∝ /H .Once the hysteretic contribution is subtracted, ∆ Q = 0 when v ≤ t = 0, up ramps experience less dissipationthan down ramps. Assuming perfect potential flow and constant density of states around Fermi energy, one canintegrate the total solid angle where the escape condition to bulk is fulfilled: The number of available quasiparticlestates at a fixed location on the wire surface increases as proportional to v − v c . The solid angle of directions whereescape to bulk is allowed from that given location also scales like v − v c . Finally, the surface area on the wirecrossbar which contributes to the dissipation expands like √ v − v c . Assuming the damping scales proportionally tothe available solid angle for escape, this yields the power law ∆ Q = b (( v − v c ) /v c ) . . This is in good agreement withthe measurements at ∆ t = 0 with b ≈ . DATA AVAILABILITY
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Critical flow velocity in superfluid He-B , Ph.D. thesis, Lancaster University, UK (2016). D.I. Bradley, M. lovecko, M.J. Fear, S.N. Fisher, A.M. Gunault, R.P. Haley, C.R. Lawson, G.R. Pickett, R. Schanen,V. Tsepelin, and P. Williams, “A new device for studying low or zero frequency mechanical motion at very low temperatures,”Journal of Low Temperature Physics , 114–131 (2011). Http://dx.doi.org/10.17635/lancaster/researchdata/xxx. a a b cd x , mm Time, ms ∆ t τ τ b cd v fl =0v fl =2vv FIG. 1.
Schematic illustration of the measurement at zero temperature: (Centre) The crossbar of the wire is movedin two phases. Comparing two different, equally long trajectories reveals the bound state dissipation. Bound state dynamics isprobed by varying the waiting time ∆ t between the two phases. All ramps begin by accelerating the wire from zero to a constantvelocity (here v = 40 mm/s), which in the coordinate system of the wire creates a flow around the wire. The flow pattern isshown with contour lines in the inset. The flow shifts both the bound state spectrum and the available bulk states as describedin the main text. After the acceleration, the population of bound quasiparticles (red circles) and quasiholes (blue circles) onthe wire surface reaches a steady state (panel a ). The dispersion curves are drawn for the top (or bottom) generatrices of thewire, where the flow velocity is maximal. During ∆ t , momentum exchange with the wire surface allows exchange of boundquasiparticle populations between the branches ( τ ). Within a branch the population relaxes with τ (down ramp, panel b ).If the first phase of motion is followed by the second one in the same direction and ∆ t < ∼ τ (up ramp, panel c ), the remainingimbalance in populations results in less dissipation from quasiparticles escaping to bulk than a fully symmetric one would. Ifthe direction of motion is reversed (panel d ) and ∆ t < ∼ τ , then the dissipation will be enhanced. Temperature ( K) ( m s ) t (ms) Q ( p J ) Thermal quasiparticle density, ab
160 170 180 190 200 210 220 µarb. units
FIG. 2.
Bound state dynamics: ( a ) Measured dissipation as a function of ∆ t for up ramps (blue upward triangles) and downramps (red downward triangles) reveals the characteristic time of bound-state dynamics on the wire surface. The solid linesare exponential fits to the data. At large waiting times ∆ Q ≈ − . b ) The fitted time constants with He preplating (magenta squares) and without it (black circles) are independentof temperature. The relative change in thermal quasiparticle density is shown on the top axis. All data in this figure wasmeasured at H = 130 mT and v = 45 mm/s. P ea k v e l o c i t y , m s - Force, N -9 -10 -11 -12 -13 -8 -14 -3 -2 -4 f , Hz C r i t i c a l v e l o c i t y , mm s - FIG. 3.
AC critical velocity measurements as a function of frequency:
In AC measurements, critical velocity v c isseen as a sharp increase in the force needed to increase peak oscillation velocity above the critical velocity. The exact criticalvelocity depends on details of the flow field around the probe, and therefore on the shape of the object. The measured probesare, in order from left to right: a 1 µ m thick vibrating wire operating at 843 Hz (red points, measured at 110 µ K), a 4 . µ mthick vibrating wire at 355 Hz (green points, 110 µ K), a custom-made quartz tuning fork at 25.7 kHz (blue points, 120 µ K),and an overtone of the same fork at 158kHz (magenta points,110 µ K). The inset shows critical velocities extracted from themain figure. Black dash line is a guide to the eye that corresponds to v c ∝ f . .
40 60 80 100 120140-30-10-3-1 a ( p J )
40 60 80 100 120 140
Field (mT) b ( p J ) c d Velocity (mm/s) -20-10010203040 ∆ Q ( p J ) ∆ Q ( p J ) b Velocity (mm/s) a Field (mT)
FIG. 4.
Bound state dissipation and critical velocity: ( a ) The measured hysteretic difference in dissipation with ∆ t =50 ms (cyan circles) follows the empirical power law ∆ Q = a ( v/v c ) / (solid line), as clearly seen in log scale. ( b ) When ∆ t = 0(cyan circles), the hysteretic contribution can be removed by fitting the data at v ≤ Q = b (( v − v c ) /v c ) . (solid blue line) above v c = 9 mm/s,and ∆ Q = 0 for v < v c , as explained in the text. The sum of the two fits is shown by the cyan line. The fitted values of a are shown in panel ( c ): Fits to data where ∆ t = 0 are shown with black circles, ∆ t = 50 ms corresponds to blue squares,and ∆ t = 100 ms is shown as red triangles. The dashed line is a guide to the eye that corresponds to a = − . /H nJ mT .Panel ( d ) shows fitted values of b (black circles), which within the scatter of the data is independent of H as expected. Thetemperature varied from 150 µ K to 190 µµ