Lifshitz transition in the double-core vortex in 3He-B
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Lifshitz transition in the double-core vortex in He-B
M. A. Silaev, E. V. Thuneberg, and M. Fogelstr¨om Department of Theoretical Physics, KTH-Royal Institute of Technology, SE-10691 Stockholm, Sweden Department of Physics, University of Oulu, FI-90014, Finland Department of Microtechnology and Nanoscience, Chalmers, SE-41296 G¨oteborg, Sweden (Dated: August 27, 2018)We study the spectrum of fermion states localized within the vortex core of a weak-coupling p-wave superfluid. The low energy spectrum consists of two anomalous branches that generate largedensity of states at the locations of the half cores of the vortex. Fermi liquid interactions significantlystretch the vortex structure, which leads to Lifshitz transition in the effective Fermi surface of thevortex core fermions. We apply the results to rotational dynamics of vortices in superfluid He-Band find explanation for the observed slow mode.
PACS numbers: 67.30.he, 67.30.hj, 74.25.nj
The double-core vortex is an amazing structure be-cause it is the unique answer to a simple question: whatis the vortex structure of a weak-coupling p-wave-pairingsuperfluid. The ground state in this case is the Balian-Werthamer (BW) state [1]. The lowest energy vortex hasthe double-core structure, where the core is split into two“half cores” as depicted in Fig. 1(a,b) [2–8]. This is notonly of theoretical interest since superfluid He is closeto being weak coupling, and its B phase was identifiedas the BW state. Two vortex types have been found ex-perimentally in He-B [9–12]. The vortex being stable inthe major, low-pressure part of the phase diagram wasidentified as the double-core vortex. The vortex stable athigher pressures was identified as the axially symmetricA-phase-core vortex. Available experimental evidence isconsistent with the theoretical identification of the vor-tex structures. In particular, the broken axial symmetryof the double-core vortex was used to explain the pe-culiar dynamical properties that have been observed forthe low pressure vortex using homogeneous precessingdomain (HPD) mode of NMR [13, 14]. A similar double-core vortex structure has been suggested to appear inspin-triplet heavy fermion superconductor UPt [15].One of the most interesting properties of quantizedvortices in superconductors and Fermi superfluids is thepresence of fermionic quasiparticles localized within vor-tex core at energies smaller than the bulk energy gap[16, 17]. Generally fermionic bound states determineboth thermodynamic and dynamic properties of vorticesat low temperatures [18–22]. In the rotational dynam-ics of the double-core vortex they are predicted to giverise to resonance absorption at the frequency comparablewith the spacing of the localized energy eigenstates [23].Recently much attention has been focused on the topo-logically protected zero energy vortex-core and surfacestates in superfluid He [24–27]. Particularly motivat-ing is a predicted existence of self-conjugated Majoranastates localized on half-quantum vortices in p-wave su-perfluids [28].In this letter we calculate the low-energy fermionic ex-citation spectrum of the double-core vortex. We find thatthe low-energy excitations mostly are localized in the two
FIG. 1: (Color online) (a,b) The double-core vortex structuremade visible by the pair density | Ψ | = P µ,i | A µ,i | plotted inthe x − y plane perpendicular to the vortex axis at temperature T /T c = 0 .
9, and (b)
T /T c = 0 .
1. (c,d) The normalized localdensity of states profiles demonstrating the quasiparticle wavefunction at the Fermi level ε = 0, ˆ p z = 0 at T /T c = 0 . T /T c = 0 .
1. All plots correspond to pressure P = 24 bar. half cores. This is visualized in Fig. 1(c,d) which show thefermionic local density of states (LDOS) profiles aroundthe vortex core. We can interpret the two half coresas potential wells for quasiparticles. The motion of theexcitations between the wells depends on the potentialbarrier between them. We find that this barrier changesessentially as the distance of the wells changes as a func-tion of pressure and temperature (see Fig. 1 to comparevortex structures at T /T c = 0 . T /T c = 0 . p -wave statesis described by the matrixˇ∆( r , ˆ p ) = X α,i A αi ( r ) i ˇ σ α ˇ σ y ˆ p i , (1)where ˇ σ x,y,z are Pauli matrices, p is the momentum closeto the Fermi surface p ≈ p F = ~ k F , and ˆ p = p /p . Thegap function ˇ∆ (1) is determined by the 3 × A αi . Here α = x, y, z and i = x, y, z are spin and orbital indices,respectively.In the weak coupling theory of p -wave superfluid, thestable state has the Balian-Werthamer (BW) form [1].In BW state, the order parameter far from the vortexaxis is A αi = ∆ exp( iϕ ) R αi . Here ϕ is the azimuth withrespect to the vortex axis, R αi is a constant rotationmatrix and ∆ the order parameter amplitude. Near thevortex axis a more sophisticated structure appears [2–8]. It is energetically favorable to change the sign of theorder parameter across the vortex axis by spin rotationof the BW-state matrix A αi by π [4]. This effectivelyresults in splitting of a singly-quantized vortex to a pairof half-quantum vortices that are bound together by aplanar-phase domain wall. For illustration see Figs. 1(a)and 1(b), which show the pair density | Ψ | = P α,i | A αi | in the x − y plane. The pair density has two distinctminima, whence the name double-core vortex.To determine the vortex structure we calculate self-consistently the order parameter and the Fermi-liquid selfenergy [31]. The numerics is performed as described inRef. 6, i.e. using the explosion trick to solve the Eilen-berger transport equation. We extend the previous work[6, 7] to higher accuracy, lower temperatures and differentvalues of the Fermi-liquid parameter F s correspondingto different pressures. The parameter F s determines thefeedback of superfluid mass current on the order parame-ter and can significantly change both the vortex structureand spectrum of bound fermions.The distance a between the half cores is shown Fig. 2.Its scale is R = (1 + F s / ξ , where ξ = ~ v F / πT c isthe coherence length and v F the Fermi velocity. As F s1 in liquid He ranges from 5.4 to 14.6 depending on pres-sure P [32], the two length scales can differ essentially.Thus at large values of F s1 , corresponding to high pres-sures, the vortex size at a low temperature is much largerthan the coherence length. For example, a = 46 ξ in thecase of Fig. 1(b). Fig. 2 also shows strong temperatureand pressure dependence. The distance of the half coresgrows almost 3-fold when the temperature decreases from0 . T c to 0 . T c at 24 bar. Similarly as a function of pres-sure the distance a measured in units of ξ , grows almost F s1 =12.0 ( P = 24 bar) F s1 = 9.0 ( P = 12 bar) F s1 = 6.0 ( P = 2 bar) F s1 = 3.0 F s1 = 0.0 a / R T/T c FIG. 2: (Color online) Distance a between the half cores inthe double-core vortex at different values of the Fermi liquidparameter F s . The locations of the half cores are determinedfrom zeros of supercurrent density. p of a low energy excitation is closeto the Fermi surface, p ≈ p F . The classical trajecto-ries are straight lines parallel to p . In studying a vortexwe fix the z axis as the vortex axis, and we parameter-ize the momentum direction ˆ p = (ˆ p ⊥ cos θ p , ˆ p ⊥ sin θ p , ˆ p z ).The direction on the trajectory is fixed by giving ˆ p z and θ p . The location of the trajectory is given by the im-pact parameter b , the coordinate measuring the distancefrom the vortex axis. The parameterization is visual-ized in Fig. 3(a). The impact parameter is related tothe projection of the angular momentum µ on the vor-tex axis through the usual classical mechanics formula µ = p ⊥ b . The quasiclassical energy spectrum is givenby ε = ε i (ˆ p z , θ p , b ), where the parameters ˆ p z , θ p , b specifythe classical trajectory and integer i counts the eigen-values of Andreev equations on a given trajectory [33].Figure 3(b) shows a bunch of trajectories at the Fermilevel and ˆ p z = 0. The concentration of the trajectoriesat the two half cores results in the large LDOS at thehalf cores. The concave triangular shape of the causticof the trajectories at the half cores is clearly visible inthe LDOS shown in Fig. 1(d). Also the classically non-allowed region around the vortex axis in Fig. 3(b) can berecognized in Fig. 1(c) as a valley in the LDOS profile inthe region between the half cores.Due to the lifted spin degeneracy, singly-quantized vor-tices in He-B have two anomalous branches of quasipar-ticle spectrum [34]. At low energy compared to the bulkenergy gap, | ε | ≪ ∆ , they can be represented as ε i (ˆ p z , θ p , b ) = − ω i p F ( b − b i ) , (2)where i = 1 ,
2. Here b i (ˆ p z , θ p ) is the impact parame- sby pr x q p quasiclassicaltrajectory FIG. 3: (Color online) (a) Schematic plot of a quasiclassicaltrajectory in the x − y plane in the direction of (cos θ p , sin θ p )passing the vortex axis at distance b (impact parameter). Apoint r = ( x, y ) on the trajectory is determined by the coor-dinate s . (b) A bunch of quasiparticle trajectories (straightlines) at the Fermi level ε = 0 and ˆ p z = 0 superimposedon the pair-density contour plot at T /T c = 0 . F s = 0.(c) Anomalous branches of the quasiparticle spectrum ε = ε , ( b, θ p ) at ˆ p z = 0, F s = 12 and T = 0 . T c . (d) Two sheetsof effective Fermi surface b = b , ( θ p ) at ˆ p z = 0, F s = 12, T = 0 . T c (solid lines) and T = 0 . T c (dashed lines). ter that corresponds to vanishing excitation energy, and ω i (ˆ p z , θ p ) indicates the slope of the energy at b = b i . Thenew feature in a non-axisymmetric vortex is that theseparameters depend on the trajectory direction θ p in the x − y plane.Fig. 3(c) shows the calculated quasiparticle energies asa function of impact parameter b and different directionsof the trajectory θ p . The curves cross the Fermi level at afinite b in accordance with Eq. (2). These locations in theenergy spectrum b , ( θ p ) are shown by the black dots inFig. 3(c). The states at the Fermi level in the spectrum(2) form a 2D effective Fermi surface b = b , (ˆ p z , θ p )in the 3D space formed by the quasiclassical quantumnumbers (ˆ p z , θ p , b ) in the vortex core. Because of twonondegenerate branches (2), there are two sheets in theFermi surface. One more representation of this is givenin Fig. 3(d). It shows b , as a function of θ p . The curvesdepend also on ˆ p z but that dependence is less importantin the following because ˆ p z is conserved. For comparison,the trajectories passing precisely through the half cores at y = ± a/ b ( θ p ) = ∓ a cos θ p . The topology of the effective Fermi surface is deter-mined by the behavior of zero energy lines b , ( θ p ) at θ p = π ( n + 1 /
2) with n = 1 ,
2. At these angles the quasi-particle trajectories pass through both half cores. In gen-eral there is overlap of the quasiparticle wave functionslocalized at different half cores. This makes that there isno sign change of b , ( θ b ). That is, there is anticrossing ofthe two branches and a finite splitting 2 δb = | b − b | > θ p = π/
2, as shown by the solid lines in Fig. 3(d).Physically this means that an excitation created at onehalf core will jump periodically between the half cores.The growing core separation (compared to ξ ) at lowtemperatures and large pressures reduces the overlap ofthe quasiparticle wave functions located at different halfcores. As a result the splitting 2 δb becomes extremelysmall as shown by dashed lines in Fig. 3(d) for F s = 12and T = 0 . T c . In this case Landau-Zener (LZ) tunnel-ing between the quasiclassical branches (2) becomes im-portant. The probability W of these transitions can befound from the conventional approach [35, 36] by taking θ p and the angular momentum µ = p ⊥ b as the conju-gate variables. Near the anticrossing point at θ p = π/ b , ( θ p ) ≈ ± p δb + ( aθ/ , where θ = θ p − π/
2. The transition probability is given by W = exp[ − k ⊥ Im R iθ ∗ ( b − b ) dθ ] where iθ ∗ = 2 iδb/a is the intersection point of quasiclassical branches inthe complex plane. A simple calculation yields W =exp[ − π ˆ p ⊥ ( δb/ ∆ b ) ], where ∆ b = p a/k F has a physicalmeaning of the quantum mechanical uncertainty of theimpact parameter.Once the transition probability becomes large, W ≈ b ( θ p ) = b ( θ p ) for − π/ < θ p < π/ b ( θ p ) = b ( θ p )for π/ < θ p < π/
2. The calculated LZ probability W ( T, P, ˆ p z ) is shown in supplemental material [37] todemonstrate that the condition W ≈ b , ( θ p ) to intersectingisoenergetic lines ˜ b , ( θ p ) is an analog of the Lifshitz tran-sition [39] changing the topology of the Fermi surface.The transition leads to a formation of two spatially sep-arated low-energy fermionic states localized at the half-cores. Whether there are Majorana states precisely at theFermi level [28] or not [27] is beyond our quasiclassicalapproach. The transition affects the rotational dynamicsof the double-core vortex. The bound fermions in the corerespond to oscillation of the core orientation. A frictiontorque acting on a rotating vortex core can be expressedby a friction coefficient f = f p F ( k F ξ ) , where f ∼ p F ( k F ξ ) is determinedby the density of quasiparticles in the vortex core. Theexpression for the friction torque [23] yields resonancepeaks in f located at angular frequencies ω ≈ nE m / ~ where n is integer. Here E m is the spacing of quantized P = 2 bar P = 12 bar P = 24 bar T = 0 . T c
27 kHz 71 kHz 98 kHz T = 0 . T c
22 kHz 65 kHz 106 kHzTABLE I: Values of the minigap E m /h at different pressuresand temperatures.FIG. 4: (Color online) Demonstration of the effect of the Lif-shitz transition on the rotational friction coefficient f plottedas a function of frequency ω for (a) W = 0, (b) W = 1. Thevortex structure and the excitation spectrum are calculatedat T = 0 . T c and (a) P = 2 bar, (b) P = 24 bar. The mini-gap values are given in Table I. According to mutual frictionmeasurements [12] ~ /E m τ = 0 . P = 24 bar but a widerrange is given to illustrate the influence of relaxation on theshape of absorption peak. energy levels obtained from the quasiclassical spectrum(2) using the Bohr-Sommerfeld quantization rule for theangular momentum [23], E m = ~ h ω − (ˆ p z = 0) i − , where h ... i denotes the average over θ p . The scale of the minigapis determined by ~ /τ n = (2 πT c ) /v F p F , which is on theorder of the quasiparticle relaxation rate in the normalstate. The calculated values of the minigap are listed inTable I.The amplitudes of the resonances are determined bythe Fourier amplitudes of the zero-energy curves shownin Fig. 3, A n ∼ | R dθ p e inθ p b ( θ p ) | . From the plots inFig. 3(d) one can see that at pressures below the Lifshitztransition the largest components are those with doublefrequency ~ ω = 2 E m . At pressures above the transitionthe amplitudes are determined by the harmonics of theintersecting curves ˜ b i ( θ p ), which have strongest matrixelement at ~ ω = E m . The difference in the friction co-efficient f in the two cases is demonstrated in Fig. 4.In the following we show that at least the low frequencylimit of the curves in Fig. 4 is experimentally accessible.Kondo et al [13, 14] have studied a sample of rotating He-B using the homogeneously precessing domain. Inthis mode the magnetization M is tipped by a large an-gle ( > ◦ ) from the field direction B . It was found thatthe contribution of vortices to the relaxation changed ona few minute time scale [13]. This was interpreted that the double-core vortex gets twisted as its end points (at z = ± L/
2) are pinned but in the bulk the rotating mag-netization exerts a torque on the core. A quantitativemodel was constructed for the vortex core rotation angle φ ( t, z ) as a function of time t and z . The parameters ofthe model were determined by fitting to the experiment[14]. These include the friction parameter f , the dipoletorque T D , which drives the vortex in the presence of ro-tating magnetization, and the rigidity K , which gives theenergy caused by twisting the core, F twist = K ( ∂ z φ ) .By precise calculation of the vortex structure we cannow calculate the vortex parameters. We find a value of f that is three orders of magnitude larger than fitted byKondo et al [14]. Thus a serious revision of the modelhas to be made. The large value of f means that only anegligible fraction of energy dissipation comes from vor-tex core rotation. Thus essentially all dissipation has toarise from normal-superfluid disequilibrium [40], spin dif-fusion, and radiation of spin waves. Without going intodetails, these can be incorporated by allowing an elasticvortex structure, where the rotation angle α ( t, z ) at a dis-tance from the vortex axis (where the dipole torque acts)can be different from the vortex core angle φ ( t, z ). Theseare bound by elasticity energy T A ( α − φ ) , and both an-gles have their own friction coefficients: f ˙ φ = − δF/δφ , g ˙ α = − δF/δα . This model results in the diffusion equa-tion [37] ˙ φ = Kf ∂ z φ + P g ωf , (3)where P g is the power absorption per vortex length. Animportant virtue of this model is that based on our cal-culations of f and K , Eq. (3) predicts the time scale L f /π K of several minutes. Thus Eq. (3) gives a sim-ple explanation for the observed slow mode [13, 29, 30],which remained unexplained in previous models [14, 30].In summary, we have investigated the spectrum ofbound fermion states localized within the vortex coreof weak-coupling p-wave superfluid. We predicted a Lif-shitz transition, which separates low-energy quasiparticlestates at the half cores and affects the rotational dynam-ics. Applying our results to Ref. 14 explains the observedlong time scale and thus gives one more piece of evidenceof the double-core nature of the low pressure vortex in He-B.
Acknowledgments
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Lifshitz transition in the double-core vortex in He-B
I. LANDAU-ZENER TUNNELING PROBABILITY BETWEEN THE QUASICLASSICAL SPECTRUMBRANCHES
The topology of quasiparticle spectrum in double-core vortices is determined by the behavior of quasiclassicalspectrum branches ε = ε , (ˆ p z , θ p , b ) near the intersection points at θ p = π/ πn , see Fig. 3(d) in the main text.In the Fig. 5 we show in detail these curves in the vicinity of an anticrossing point θ p = π/ δb as function of temperature T , pressure P ,momentum projection to the vortex axis ˆ p z .Quantitatively the splitting δb is determined by the overlap between quasiparticle wave functions localized atdifferent half-cores as shown in the Fig. 1 in the main text. At low temperatures the characteristic localization scaleis determined by the coherence length ξ . On the other hand the distance between vortex cores a is determinedby the scale R = (1 + F s1 ) ξ so that the ratio R /ξ increases with growing pressure. Hence the overlap at largepressures is weaker and the splitting δb decreases as can be seen comparing the curves for F s1 = 9 ( P = 11 . F s1 = 12 ( P = 23 .
75 bar) in Fig. 5(a). At larger temperatures the quasiparticle localization is determined bythe temperature-dependent coherence length ξ ≈ ~ v F / ∆ where ∆ = ∆( T ) is the gap amplitude. On the other handthe distance between vortex cores for temperatures up to T = 0 . T c has a much weaker temperature dependence for F s1 = 6 , ,
12 as shown in Fig. 2 in the main text. Thus the overlap between half-core states and the splitting δb strongly decrease with decreasing temperature which can be seen from the comparison of the curves b = b , ( θ p ) for T = 0 . T c and 0 . T c in Fig. 5(b) .The most striking is the dependence of splitting δb on the quasiparticle momentum projection on the vortex axisˆ p z . The absolute value of momentum is fixed and determined by the Fermi momentum. Hence different values of ˆ p z correspond to the different angles of quasiclassical trajectories with respect to the vortex axis. For the finite ˆ p z theeffective distance between half-cores along the trajectories is elongated by the factor of 1 / p − ˆ p z which decreasesthe overlap of localized states and suppresses the splitting δb . As can be seen in Figs. 5(c) and 5(d), the splitting candecrease to the very small values at large ˆ p z ∼ δb determines the tunneling between the quasiclassical branches. The Landau-Zener tunneling prob-ability W = W ( T, P, ˆ p z ) calculated according to the general expression from the main text is shown in Figs. 6(a) and6(b) as function of temperature for several values of P and ˆ p z . These curves demonstrate a general tendency of thetunneling probability to increase with decreasing temperatures and increasing pressure. The probability is stronglyenhanced at large ˆ p z . Thus the Lifshitz transition determined by W ∼ P and T fordifferent ˆ p z . From Fig. 6 one can see that at high pressures and low temperatures the condition W ∼ p z . II. MODEL OF ROTATIONAL DYNAMICS
The rotational dynamics of the double-core vortex was studied by Kondo et al [14]. Here we present a modifiedmodel that allows azimuthal shear of the vortex structure. Such a case appears because the driving dipole torqueacts on the asymptotic order parameter, typically a distance ξ D ≈ µ m from the vortex axis, whereas the opposingrotational friction occurs in the vortex cores, on the scale of coherence length ξ ≈
10 nm. Our notation follows closelyRef. 14 and the model of Ref. 14 is obtained in the limiting case T A → ∞ and g = 0.We study the model where the free energy of a single vortex parallel to z is F = Z dz (cid:20) − T H ( ˆ S α R αi ˆ b i ) + T D (ˆ a · ˆ n ) − T A (ˆ a · ˆ b ) + 12 K ( ∂ z φ ) (cid:21) (4)The magnetic field term (coefficient T H ) depends on the hard core anisotropy axis ˆ b = ˆ x cos φ + ˆ y sin φ . The dipole term(coefficient T D ) depends on the soft core anisotropy axis ˆ a = ˆ x cos α + ˆ y sin α . The azimuthal shear term (coefficient T A ) is supposed to be strong enough to keep the two anisotropy vectors nearly parallel. Other quantities are the spin-orbit rotation matrix R αi parametrized by axis ˆ n = ˆ y cos ωt + ( ˆ z sin η − ˆ x cos η ) sin ωt and angle θ = arccos( − / S α = R αi ˆ H i , and the static field H = H ( ˆ x sin η + ˆ z cos η ). This modelneglects the anisotropy of the dipole energy in the double-core vortex in the plane perpendicular to ˆ a ≈ ˆ b . In principle,we should allow ˆ a to have component in the z direction also, but this will not affect the main results and thus is FIG. 5: (Color online) Detailed structure of zero-energy lines b = b , ( θ p ) near the anticrossing point θ p = π/ T = 0 . T c and different values of pressure F s1 = 9 (red), F s1 = 12 (blue); (b) F s1 = 12 and T = 0 . T c (blue), T = 0 . T c (red); (c) F s1 = 12, (d) F s1 = 9 and different momentum projections ˆ p z = 0 (red), 0 .
23 (blue), 0 .
45 (green). dropped here for simplicity. We get F = Z dz (cid:20) − T H ( ˆ H · ˆ b ) + T D (ˆ a · ˆ n ) − T A (ˆ a · ˆ b ) + 12 K ( ∂ z φ ) (cid:21) = Z dz (cid:20) − T H sin η cos φ + T D (cos ωt sin α − cos η sin ωt cos α ) + T A ( α − φ ) + 12 K ( ∂ z φ ) (cid:21) (5)Note that for ˆ H = ˆ x , cos η = 0 and both field and dipole terms are minimized by φ = α = nπ , that is ˆ a = ˆ b = ± ˆ x and ˆ n rotating around it.We suppose the frictional equations of motion g ˙ α = − δFδα , f ˙ φ = − δFδφ . (6)We obtain g ˙ α = − δFδα = − T D sin ωt sin η sin 2 α − T D cos 2 ωt (1 − cos η ) sin 2 α + T D cos η sin 2( ωt − α ) − T A ( α − φ ) . (7)Using the same approximation as Kondo et al this simplifies to g ˙ α = − T D sin η sin 2 α + T D cos η sin 2( ωt − α ) − T A ( α − φ ) . (8) FIG. 6: (Color online) Temperature dependencies of the Landau-Zener tunneling probability between quasiclassical spectralbranches in the double-core vortex for (a) F s1 = 9 ( P = 11 . F s1 = 12 ( P = 23 .
75 bar) and different values ofmomentum projection ˆ p z . This expression is exact in second (but not fourth) power in η . The other equation is f ˙ φ = − δFδφ = − T H sin η sin 2 φ + 2 T A ( α − φ ) + K∂ z φ. (9)Suppose now that f is very large, which means φ ( t ) is so slow that it can be taken as a constant during one cycle.We study g ˙ α = − M sin 2 α + G sin 2( ωt − α ) − T A ( α − φ ) (10)where in the approximation above M = T D sin η and G = T D cos η . Suppose α ( t ) = α + α ( t ) with small α g ˙ α = − M sin 2 α − T A ( α − φ ) + G sin 2( ωt − α ) − T A + M cos 2 α ) α − G cos 2( ωt − α ) α . (11)Assuming α = A cos 2( ωt − χ ) − ωgA sin 2( ωt − χ )= − M sin 2 α − T A ( α − φ ) + G sin 2( ωt − α ) − A ( T A + M cos 2 α ) cos 2( ωt − χ ) − GA cos 2(2 ωt − α − χ ) − GA cos 2( χ − α ) . (12)At frequencies 0 and 2 ω we get 0 = − M sin 2 α − T A ( α − φ ) − GA cos 2( χ − α ) − ωgA sin 2( ωt − χ ) = G sin 2( ωt − α ) − A ( T A + M cos 2 α ) cos 2( ωt − χ ) (13)The latter equation can be written2 A ( T A + M cos 2 α ) cos 2( ωt − χ ) − ωgA sin 2( ωt − χ ) = G sin 2( ωt − α ) (14)2 A ( T A + M cos 2 α ) cos 2 ωt − ωgA sin 2 ωt = G sin 2( ωt + χ − α )= G [sin 2 ωt cos 2( χ − α ) + cos 2 ωt sin 2( χ − α )] (15)2 A ( T A + M cos 2 α ) = G sin 2( χ − α ) − ωgA = G cos 2( χ − α ) (16) A = G / ω g + ( T A + M cos 2 α ) ≈ G / ω g + T A = T D cos ηω g + T A . (17)The absorbed power is determined by A , P g = −h ˙ α δFδα i = g h ˙ α i = 2 gω A . (18)Note that here P denotes the absorption per vortex length whereas in Ref. 14 P denotes the total absorption.Applying the latter of (16) to the first equation (13) gives M sin 2 α + 2 T A ( α − φ ) − ωgA = 0 . (19)From (9) we have f ˙ φ = − T H sin η sin 2 φ + 2 T A ( α − φ ) + K∂ z φ. (20)Because φ is a slow variable, it is only the low frequency limit of f that appears. What remains to be solved is φ and α from equations (19) and (20). Two different cases need to be studied.1) Rocking oscillations. We assume a solution where the left hand side of (20) vanishes, − δFδφ = − T H sin η sin 2 φ + 2 T A ( α − φ ) + K∂ z φ = 0 . (21)Supposing φ − α ≪ φ = α − T H sin η sin 2 α − K∂ z α T H sin η cos 2 α + T A ) ≈ α − T H sin η sin 2 α − K∂ z α T A . (22)The lag of φ behind α is an increasing function of φ ≈ α for 0 < φ ≈ α < π/
4. Substituting (22) into (19) gives( T H + 12 T D ) sin η sin 2 α − K∂ z α − ωgA = 0 (23)and thus sin 2 α = 2 ωgA + K∂ z α ( T H + T D ) sin η . (24)This solution is possible only if the right hand side of (24) is less than unity. At equality, α = π/ nπ . The stabilitycondition at ∂ z α = 0 can be writtentan η > tan η = ωgT D ( T H + T D )( ω g + T A ) . (25)Interestingly, this can also be written sin η > sin η = P g ω ( T H + T D ) . (26)These results are essentially (tan η ≈ sin η ) the same as in Ref. 14 if one makes the replacement f → g + T A /gω .2) In the case solution (24) is not possible, we get slow rolling motion of φ ( t ). For simplicity we consider η = 0only. From (19) we get 2 T A ( α − φ ) = 2 ωgA (27)and substitution to (20) gives ˙ φ = Kf ∂ z φ + P g ωf . (3)Equation (3) is a diffusion equation ˙ φ = D∂ z φ + C (28)The solution for 0 < z < L and t > φ ( z = 0) = 0 = φ ( z = L ) can be found as Fourierseries φ = C D ( Lz − z ) + X n B n e − Dk n t sin k n z (29)with k n = nπ/L . The slowest component has the rateΓ = π D/L = π K/f L . (30)We compare our results to Ref. 14. For that we give numerical values of our weak-coupling results correspondingto 29.3 bars pressure, T /T c = 0 . L = 7 mm. At this pressure F s = 13 . f ≈ × − J s/m, see Section III. In section IV we calculate K ≈ × − Jm. This gives the time constant Γ − ≈ f = 4 . × − J s/m and K = 1 . × − Jm that were obtained by fittingthe experiment to the model in Ref. 14.
III. ROTATIONAL FRICTION
The calculation of the rotational friction coefficient f in the main text applies the kinetic equation of Ref. 23 toour numerical solution of the excitation spectrum in the vortex core. The results are presented in Fig. 4. Evaluatingthis at the conditions of Ref. 14 (see above) gives f = 1 . × − J s/m in the low frequency limit. The result at theLarmor precession frequency at H = 14 . ~ ω/E m = 3 .
9) would be f ( ω L ) = 0 . × − J s/m.An alternative estimate of f is obtained as follows. In the case of strong Landau-Zener tunneling, the half coresbehave as separated half-quantum vortices except that they are bound together at the distance a . The standardmutual friction force [12, 21] applied to both half cores gives f = 14 a κρ s d k , (31)where κ = π ~ /m is the circulation quantum, ρ s the superfluid density and d k the mutual friction parameter. Taking a from Fig. 2 and d k from Ref. 12 gives f = 1 . × − J s/m at the conditions stated above.
IV. TWIST RIGIDITY
The twisting of a double-core vortex is measured by a twist wave vector k = ∂ z φ . For a small twist the additionalfree energy is supposed to be quadratic in k , F twist = Kk . (32)Here we aim to calculate the twist-rigidity coefficient K .The reduced order parameter is expected to satisfy the boundary condition ˜ A ( r, ϕ, z ) → I e iϕ for r → ∞ , where I isa unit matrix. An axially symmetric vortex satisfies the following symmetry [4]˜ A ( r, φ ) = e iθ R ( ˆ z , θ )˜ A ( r, φ − θ ) R ( ˆ z , − θ ) , (33)where R is a rotation matrix parametrized by an axis and angle of rotation. A twisted vortex apparently has theproperty ˜ A ( r, φ, z ) = e ikz R ( ˆ z , kz )˜ A ( r, φ − kz, R ( ˆ z , − kz ) . (34)Taking the z derivative and evaluating it at z = 0 gives ∂ ˜ A ∂z = k B , (35)where B = i ˜ A + R ′ ˜ A − ˜ AR ′ + y ∂ ˜ A ∂x − x ∂ ˜ A ∂y , R ′ = − . (36)In the Ginzburg-Landau region the twist-rigidity coefficient K is given by K = 2 λ G2 Z d r X µ =1 X j =1 | B µj | + 2 | B µz | . (37)In order to get a good approximation of the twist rigidity at general temperature, we have written the prefactor using[38] λ G2 = ~ ρ (1 − Y )40 mm ∗ . (38)This form of λ G2 is valid since we assume F a1 = F a3 = 0. Physically, the twisting (34) causes axial spin flow inthe asymptotic region and axial mass flow in the vortex center, but the F a1 dependence of the former and the F s1 dependence of the latter are not properly included in (37).The asymptotic form of the order parameter (neglecting dipole-dipole coupling) far from the vortex axis is of theform ˜ A = R ( θ ) + O ( r − ), where θ = C cos φr (cid:18) sin φ c ˆ r + cos φ ˆ φ (cid:19) + C sin φr (cid:18) − cos φ c ˆ r + sin φ ˆ φ (cid:19) . (39)In general c = λ G1 / λ G2 but in the present approximation F a1 = F a3 = 0 we have c = 1 [38]. Based on this, we cancalculate the asymptotic contribution to the twist rigidity. Since (39) is expressed in cylindrical coordinates, the twistenergy can be calculated more directly than above, by replacing φ by φ − kz . The additional energy can be evaluatedfrom the gradient energy [38] F Gz = 2 λ G2 (1 + c ) Z d r ∂θ k ∂z ∂θ k ∂z = 2 λ G2 (1 + c ) k Z d r ∂θ k ∂φ ∂θ k ∂φ = 2 λ G2 (1 + c ) k Z dφ Z dr r (cid:20) (cos φ − sin φ ) (1 + c ) + 4 cos φ sin φ (cid:21) ( C − C ) = 2 πλ G2 k c + c c ( C − C ) Z dr r . (40)Comparison to (32) allows to identify K tail = 4 πλ G2 c + c c ( C − C ) Z dr r . (41)The dipole-dipole energy suppresses the contribution at distances beyond the dipole length ξ d .We estimate numerical values under conditions explained above [below Eq. (30)]. Substituting the numerical orderparameter into (37) gives K = 3 . × − Jm in the region r < ξ . Adding the asymptotic part from (41) with C = 3 . R and C = 0 . R we get K = 4 . × − Jm.Another estimation of the twist rigidity is to calculate the twisting energy of a pair of half-quantum vortices.Estimating energy by the length increase of the twisted pair gives K mass flow = ρ s κ a π ln ar c (42)where a is the distance between the half cores and r c their radius. Estimating a = 5 . R gives K mass flow = 1 . × − Jm. Estimating this without Fermi-liquid interaction, i.e. as we did in (37), we would get K mass flow0 = 0 . × − Jm. Supposing this difference is just missing in the formula based on (37), the corrected value for total twist coefficientis K = 5 . × −27