Half-quantum vortices and walls bounded by strings in the polar-distorted phases of topological superfluid 3 He
J.T. Mäkinen, V.V. Dmitriev, J. Nissinen, J. Rysti, G.E. Volovik, A.N. Yudin, K. Zhang, V.B. Eltsov
HHalf-quantum vortices and walls bounded by strings in the polar-distorted phases of topologicalsuperfluid He J.T. M¨akinen ∗ , V.V. Dmitriev , J. Nissinen , J. Rysti , G.E. Volovik , , A.N. Yudin , K. Zhang , , and V.B. Eltsov Low Temperature Laboratory, Department of Applied Physics,Aalto University, FI-00076 AALTO, Finland; *E-mail: jere.makinen@aalto.fi P. L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia. University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68 FIN-00014, Helsinki, Finland (Dated: November 12, 2018)
Symmetries of the physical world have guided formu-lation of fundamental laws, including relativistic quantumfield theory and understanding of possible states of mat-ter. Topological defects (TDs) often control the universalbehavior of macroscopic quantum systems, while topologyand broken symmetries determine allowed TDs. Takingadvantage of the symmetry-breaking patterns in the phasediagram of nanoconfined superfluid He, we show thathalf-quantum vortices (HQVs) – linear topological defectscarrying half quantum of circulation – survive transitionsfrom the polar phase to other superfluid phases with polardistortion. In the polar-distorted A phase, HQV cores in2D systems should harbor non-Abelian Majorana modes.In the polar-distorted B phase, HQVs form composite de-fects – walls bounded by strings hypothesized decades agoin cosmology. Our experiments establish the superfluidphases of He in nanostructured confinement as a promis-ing topological media for further investigations rangingfrom topological quantum computing to cosmology andgrand unification scenarios.
TDs generally form in any symmetry-breaking phase tran-sitions. The exact nature of the resulting TDs depends on thesymmetries before and after the transition. Our universe hasundergone several such phase transitions after the Big Bang.As a consequence, a variety of TDs might have formed dur-ing the early evolution of the Universe, where phase tran-sitions lead to unavoidable defect formation via the Kibble-Zurek mechanism.
Experimentally accessible energy scales (cid:46) TeV are currently limited to times t (cid:38) − s after theBig Bang by the Large Hadron Collider. Theoretical under-standing may be extended up to the Grand Unification energyscales (cid:46) GeV of the electroweak and strong forces ( t (cid:38) − . . . − s). The nature of the interactions before thisepoch remains unknown, but yet unobserved cosmic TDs,the nature of which depends on the Grand Unified Theory(GUT) in question, may help us limit the possibilities. Pre-dictions exist for point defects, such as the t’Hooft-Polyakovmagnetic monopole, linear defects or strings, surface de-fects or domain walls, and three-dimensional textures. Even though cosmic TDs have not been detected, many oftheir condensed-matter analogs have been reproduced in thelaboratory, where they have an enormous impact on the be-havior of the materials they reside in. Examples include vor-tices in superconductors, vortices and monopoles in ultra-cold gases, and skyrmions in chiral magnets. Superfluidphases of He offer an experimentally accessible system to
Polar-distortedA phase PolarphaseNormal P r e ss u r e , ba r T/T c (a)(c) d d RotationaxisMagneticfield NMR pick-up coils
Polar-distortedB phase PdA phase on cooling,PdB phase on warming (b)
FIG. 1.
The experimental setup and superfluid phase diagram innanoconfinement. (a) The He sample is confined within a cylindri-cal container filled with commercially available nanomaterial callednafen-90 (where the number refers to its density in mg/cm ) withuniaxial anisotropy, which consists of nearly parallel Al O strandswith d ≈ nm diameter, separated by d ≈ nm on average.The strands are oriented predominantly along the axis denoted as ˆ z . The sample can be rotated with angular velocities up to 3 rad/saround the same axis ˆ z . The sample is surrounded by rectangularnuclear magnetic resonance (NMR) pick-up coils. The static mag-netic field transverse to the NMR coils can be oriented at an arbitraryangle µ with respect to the ˆ z axis. (b) The magnetic field, orientedalong the y -direction ( µ = π/ ) in this figure, locks the ˆ e -vectorin the polar-distorted B phase order parameter, Eq. (4). Vectors ˆ d and ˆ e are free to rotate in the xz -plane by angle θ . (c) Sketch ofthe superfluid phase diagram in our sample in units of T c of the bulkfluid. The purple arrows illustrate the thermal cycling used in themeasurements and the purple marker shows a typical measurementpoint within the region where either polar-distorted phase can exist,depending on the direction of the temperature sweep. The thermalcycling is performed at constant 7 bar pressure. study a variety of TDs and the consequences of symmetry-breaking patterns owing to its rich order-parameter structureresulting from the p -wave pairing. Analogs of exotic TDs,such as the Witten string – the broken-symmetry-core vor-tex in superfluid He-B, the skyrmion texture in super-fluid He-A, and the Alice string – half-quantum vortex a r X i v : . [ c ond - m a t . o t h e r] N ov (HQV) in the polar phase of superfluid He, have been ob-served.Here we focus on composite defects – combina-tions of TDs and/or non-topological defects of differentdimensionality. Such defects appear in some GUTs andeven in the Standard Model, where the Nambu monopolemay terminate an electroweak string.
There are two mech-anisms for the formation of composite defects: the hier-archy of energy/interaction length scales, and the hi-erarchy (sequential order) of the symmetry-breaking phasetransitions.
Composite defects originating from the hier-archy of length scales of condensation, magnetic, and spin-orbit energies are well-known in superfluid He. For example,the spin-mass vortex in He-B has a hard core of the co-herence length size, defined by the condensation energy, and asoliton tail with thickness of the much larger spin-orbit length.A half-quantum vortex (HQV) originally predicted to exist inthe chiral superfluid He-A has a similar structure with thesoliton tail, which makes these objects energetically unfavor-able.Composite defects related to the hierarchy of symmetry-breaking phase transitions were discussed in the context ofthe GUT scenarios by Kibble, Lazarides, and Shafi. Herethe GUT symmetry, such as
Spin (10) , is broken into the Pati-Salam group SU (4) × SU (2) × SU (2) , which in turn is bro-ken to the Standard Model symmetry group SU (3) × SU (2) × U (1) . At the first transition the linear defects – cosmic strings,become topologically stable, while after the second transitionthey are no longer supported by topology and form the bound-aries of the nontopological domain walls, henceforth referredto as Kibble-Lazarides-Shafi (KLS) walls. To the best of ourknowledge, observations of KLS walls bounded by stringshave not been reported previously.In this work we explore experimentally the composite de-fects formed by both the hierarchy of energy scales and thehierarchy of symmetry-breaking phase transitions allowed bythe phase diagram of superfluid He confined in nematicallyordered aerogel-like material called nafen. In our sample a se-quence of the polar, chiral polar-distorted A (PdA) and fullygapped polar-distorted B (PdB) phases occurs on cooling fromthe normal state, see Fig. 1 (c). Previously we establisheda procedure to form topologically protected HQVs in the po-lar phase. At the transition from the polar phase to the PdAphase we expect the HQVs to acquire spin-soliton tails withthe width of the spin-orbit length which is much larger thanthe coherence-length size of vortex cores. On a subsequenttransition to the PdB phase, the symmetry breaks in such away that HQVs lose topological protection and may exist onlyas boundaries of the non-topological KLS walls. Simultane-ously, the spin solitons between HQVs are preserved in thePdB phase and such an object becomes a doubly-compositedefect. Naively, however, one would expect that a muchstronger tension of the KLS wall compared to that of the spinsoliton, would lead to collapse of an HQV pair, possibly to asingly quantized vortex with an asymmetric core.
Here we report evidence that HQVs do exists in the super-fluid PdA and PdB phases of He. We create an array ofHQVs by rotating the container with the angular velocity Ω in zero magnetic field during the transition from the normalfluid to the polar phase and proceed by cooling the samplethrough consecutive transitions to the PdA and PdB phases.The HQVs are identified based on their NMR signature as afunction of temperature and Ω . A characteristic satellite peakpresent in the NMR spectrum confirms that the HQVs survivein the PdA phase, where they provide experimental access tovortex-core-bound Majorana states. Moreover, the HQVsare found to survive the transition to the PdB phase. The ob-served features of the NMR spectrum in the PdB phase sug-gest that a KLS wall emerges between a pair of HQVs alreadyconnected by the spin soliton. Evidently the tension of theKLS wall is not sufficient to overcome the pinning of HQVs innafen. Vortex pinning allows us to study the properties of theout-of-equilibrium vortex state created during the superfluidphase transitions while suppressing the vortex dynamics. Si-multaneously pinning does not affect the symmetry-breakingpattern leading to formation of the KLS walls. Our resultsshow that pinned TDs, once created, may be transferred tonew phases of matter with engineered topology.
RESULTS
The superfluid phase diagram under confinement bynafen – a nanostructured material consisting of nearly paral-lel strands made of Al O , c.f. Fig. 1 (b) – differs from that ofthe bulk He; the critical temperature is suppressed and, moreimportantly, new superfluid phases - the polar, polar-distortedA (PdA), and polar-distorted B (PdB) phases - are observed.We refer to the Supplementary Note 1 for a detailed discus-sion on these phases and their symmetries and focus on ourobservations regarding the HQVs in the PdA and PdB phases.
Half-quantum vortices in the PdA phase
The order parameter of the PdA phase can be written as A αj = (cid:114) b PdA e iφ ˆ d α ( ˆ m j + ib ˆ n j ) , (1)where the orbital anisotropy vectors ˆ m and ˆ n form an orthog-onal triad with the Cooper pair orbital angular momentum axis ˆ l = ˆ m × ˆ n , and ˆ d is the spin anisotropy vector. Vector ˆ m isfixed parallel to the nafen strands. The amount of polar distor-tion is characterized by a dimensionless parameter < b < and ∆ PdA ( T, b ) is the the maximum gap in the PdA phase.The order parameter of the polar phase is obtained for b = 0 ,while b = 1 produces the order parameter of the conventionalA phase.In our experiments, we use continuous-wave NMR tech-niques to probe the sample, see Methods for further details. Inthe superfluid state the spin-orbit coupling provides a torqueacting on the precessing magnetization, which leads to a shiftof the resonance from the Larmor value ω L = | γ | H, where γ = − . × s − T − is the gyromagnetic ratio of He.The transverse resonance frequency of the bulk fluid with
FIG. 2.
Survival of HQVs during phase transitions.
The plotshows the measured NMR spectra in transverse ( µ = π/ ) mag-netic field in the presence of HQVs. HQVs were created by rotationwith 2.5 rad/s during the transition from normal phase to the polarphase. The NMR spectrum includes the response of the bulk liquidand the ˆ d -solitons, which appear as a characteristic satellite peak atlower frequency. The satellite intensity in the PdA phase remainsunchanged after thermal cycling presented in Fig. 1 (c). The NMRspectrum in the PdB phase at the same temperature, measured be-tween the two measurements in the PdA phase, is shown for refer-ence. magnetic field in the direction parallel to the strand orienta-tion, i.e. µ = 0 in Fig. 1 (a), is ∆ ω PdA = ω PdA − ω L ≈ Ω ω L , (2)where Ω PdA is the frequency of the longitudinal resonance inthe PdA phase at µ = π/ . The NMR line retains its shapeduring the second order phase transition from the polar phasebut renormalizes the longitudinal resonance frequency due toappearance of the order parameter component with b .Quantized vortices are linear topological defects in theorder-parameter field carrying non-zero circulation. In thePdA phase quantized vortices involve phase winding by φ → φ + 2 πν and possibly some winding of the ˆ d vector. The typ-ical singly quantized vortices, also known as phase vortices,have ν = 1 and no winding of the ˆ d -vector, while the HQVshave ν = and winding of the ˆ d -vector by π on a loop aroundthe HQV core so that sign changes of ˆ d and of the phase factor e iφ compensate each other. The reorientation of the ˆ d -vectorleads to the formation of ˆ d -solitons – spin-solitons connect-ing pairs of HQVs. The soft cores of the ˆ d -solitons providetrapping potential for standing spin waves. Since the ˆ m -vector is fixed by nafen parallel to theanisotropy axis, the ˆ l -vector lies on the plane perpendicular toit, prohibiting the formation of continuous vorticity like thedouble-quantum vortex in He-A. Some planar structures inthe ˆ l -vector field, such as domain walls or disclinations, re-main possible but the effect of the ˆ l -texture on the trapping potential for spin waves is negligible due to the large polardistortion (i.e. for b (cid:28) ). Recent theoretical work pro-vides arguments why formation of HQVs in the polar phaseis preferred compared to the A phase. Indeed, in confinedgeometry where the PdA phase is observed immediately be-low T c , no HQVs are found. In our case the PdA phase isobtained via the second-order phase transition from the po-lar phase with preformed HQVs. We already know that themaximum tension from the spin-soliton in the polar phase (for µ = π/ ) is insufficient to overcome HQV pinning. Thus,survival of HQVs in the PdA phase is expected. Moreover,we note that even for | b | = 1 and in the absence of pinning, apair of HQVs, once created, should remain stable with finiteequilibrium distance corresponding to cancellation of vortexrepulsion and tension from the soliton tail. In the presence of HQVs the excitation of standing spinwaves localized on the soliton leads to a characteristic NMRsatellite peak in transverse ( µ = π/ ) magnetic field, c.f.Fig. 2, with frequency shift ∆ ω PdAsat = ω PdAsat − ω L ≈ λ PdA Ω ω L , (3)where λ PdA is a dimensionless parameter dependent on thespatial profile (texture) of the order parameter across the soli-ton. For an infinite 1D ˆ d -soliton, one has λ PdA = − , cor-responding to the zero-mode of the soliton. The mea-surements in the supercooled PdA phase, Fig. 3 (a), at tem-peratures close to the transition to the PdB phase give value λ PdA ≈ − . , which is in good agreement with theoreticalpredictions and earlier measurements in the polar phase witha different sample. This confirms that the structure of the ˆ d -solitons connecting the HQVs is similar in polar and PdAphases and the effect of the orbital part to the trapping poten-tial can safely be neglected. Detailed analysis of the satellitefrequency shift as a function of magnetic field direction in thePdA phase remains a task for the future. Half-quantum vortices in the PdB phase
Since the HQVs are found both in the polar and PdAphases, a natural question is to ask what is their fate in the PdBphase? The number of HQVs in the polar and PdA phases canbe estimated from the intensity (integrated area) of the NMRsatellite, a direct measure of the total volume occupied by the ˆ d -solitons. When cooling down to the PdB phase from thePdA phase, one naively expects the HQVs and the relatedNMR satellite to disappear since isolated HQVs cease to beprotected by topology in the PdB phase. However, the mea-sured satellite intensity in the PdA phase before and after vis-iting the PdB phase remained unchanged, c.f. Fig. 2, whichis a strong evidence in favor of the survival of HQVs in thephase transition to the PdB phase. Theoretically it is possiblethat HQVs survive in the PdB phase as pairs connected by do-main walls, i.e. as walls bounded by strings. For very shortseparation between HQVs in a pair and ignoring the order-parameter distortion by confinement, such construction may
FIG. 3.
NMR spectra and spin-solitons in the polar-distorted phases. (a) Frequency shift of a characteristic satellite peak in the NMRspectrum expressed via parameter λ as a function of temperature in the PdA and PdB phases. In the PdA phase the measured values resideslightly above the theoretical prediction for a ˆ d -soliton with π winding, shown as the red dashed line. The difference is believed to be causedby disorder introduced by nafen, as in the polar phase. The corresponding values in the PdB phase for the lowest-energy ˆ d -soliton (marked“soliton”) and its antisoliton (marked “big soliton”), as well as the combined π -soliton (see text) are shown as dashed blue lines. The π -solitonvalues turn out to be in the same ratio with respect to the experimental points as in the PdA phase. The error bars denote the uncertainty in theposition of the satellite peak by 1.0 kHz and 0.5 kHz in the PdB and PdA phases, respectively. The uncertainty is taken as the full width at halfmaximum (FWHM) of the satellite peak in the PdB phase and as half of the FWHM due to improved signal-to-noise ratio in the PdA phase.(b) The plot shows the measured NMR spectrum in the PdB phase at 0.38 T c for different HQV densities, controlled by the angular velocity Ω at the time of crossing the T c . The presence of KLS walls produces characteristic features seen both as widening of the main line (with smallpositive frequency shift) and as a satellite peak with a characteristic negative frequency shift. The inset shows magnified view of the satellitepeak. (c) The satellite intensity in the PdA phase at . T c (blue circles) and in the PdB phase multiplied by a factor of 9 (red triangles) at . T c show the expected √ Ω -scaling. The solid black line is a linear fit to the measurements including data from both phases. The non-zero Ω = 0 intersection corresponds to vortices created by the Kibble-Zurek mechanism. (d) The FWHM of the main line, determined fromthe spectrum in panel (b), gives FWHM ≈ kHz for 2.5 rad/s. FWHM for other angular velocities is recalculated from the amplitude of themain NMR line, shown in panel (b), assuming constant area. resemble the broken-symmetry-core single-quantum vortex ofthe B phase. In our case, however, the HQV separation in apair exceeds the core size by three orders of magnitude. Letus now consider this composite defect in more detail.The order parameter of the PdB phase can be written as A αj = (cid:114) q PdB e iφ (ˆ d α ˆ z j + q ˆ e α ˆ x j + q ˆ e α ˆ y j ) , (4)where | q | , | q | ∈ (0 , , | q | = | q | ≡ q describes the rela-tive gap size in the plane perpendicular to the nafen strands, ˆ e and ˆ e are unit vectors in spin-space forming an orthogo-nal triad with ˆ d , and ∆ PdB ( T, q ) is the maximum gap in thePdB phase. For q = 0 one obtains the order parameter of thepolar phase, while q = 1 recovers the order parameter of theisotropic B phase. We extract the value for the distortion fac-tor, q ∼ . at the lowest temperatures from the NMR spec-tra using the method described in Ref. 47, see SupplementaryNote 7 for the measurements of q in the full temperature range.In transverse magnetic field H exceeding the dipolar field,the vector ˆ e becomes locked along the field, while vectors ˆ d and ˆ e are free to rotate around the axis ˆ y , directed along H ,with the angle θ between ˆ d and ˆ z , c.f. Fig. 1 (b). The orderparameter of the PdB phase in the vicinity of an HQV pairhas the following properties. The phase φ around the HQVcore changes by π and the angle θ (and thus vectors ˆ d and ˆ e ) winds by π . Consequently, there is a phase jump φ → φ + π and related sign flips of vectors ˆ d and ˆ e along some directionin the plane perpendicular to the HQV core. In the presenceof order-parameter components with q > , Eq. (4) remainssingle-valued if, and only if, q also changes sign. We con-clude that the resulting domain wall separates the degeneratestates with q = ± q and together with the bounding HQVs hasa structure identical to the domain wall bounded by strings –the KLS wall – proposed by Kibble, Lazarides, and Shafi inRefs. 23,29.The KLS wall and the topological soliton have distinctdefining length scales – the KLS wall has a hard core ofthe order of ξ W ≡ q − ξ , where ξ is the coherence length,and the soliton has a soft core of the size of the dipole length ξ D (cid:29) ξ W . The combination of these two objects may emergein two different configurations illustrated in Fig. 4. The min-imization of the free energy (Supplementary Notes 3 and 4)shows that in the PdB phase the lowest-energy spin-solitoncorresponds to winding of the ˆ d -vector by π − θ , where sin θ = q (2 − q ) − , on a cycle around an HQV core. Ad-ditionally, the presence of KLS walls results in winding of the ˆ d -vector by θ . These solitons can either extend between dif-ferent pairs of HQVs, Fig. 4 (a), while walls with total change ∆ θ = π are also possible if both solitons are located betweenthe same pair of HQVs, Fig. 4 (b). (a) (b) Soliton - reorientation of Virtual jump - continuous order parameterKLS wall - sign change
FIG. 4.
Kibble-Lazarides-Shafi (KLS) wall configurations in the PdB phase.
Each HQV core terminates one soliton - reorientation of thespin part of the order parameter denoted by the angle θ - and one KLS wall. The orientation of the ˆ d -vector is shown as arrows where theircolor indicates the angle θ , based on numerical calculations (Supplementary Note 2). (a) The KLS wall is bound between a different pair ofHQV cores as the soliton. Ignoring the virtual jumps, the angle θ winds by π − θ across the soliton and by θ across the KLS wall. Theorder parameter is continuous across the virtual jumps, where φ → φ + π , θ → θ + π , and q → − q . (b) The soliton and the KLS wall arebound between the same pair of HQV cores. The total winding of the ˆ d -vector is π across the structure. In principle, the KLS wall may lieinside or outside the soliton. Here the KLS wall and the soliton are spatially separated for clarity. The appearance of KLS walls and the associated ˆ d -solitonshas the following consequences for NMR. The frequency shiftof the bulk PdB phase in axial field for q < / is ∆ ω PdB , (cid:107) = ω PdB , (cid:107) − ω L ≈ (cid:18) q (cid:19) Ω ω L , (5)where Ω PdB is the Leggett frequency of the PdB phase, de-fined in the Supplementary Note 6. In transverse magneticfield the bulk line has a positive frequency shift ∆ ω PdB , ⊥ = ω PdB , ⊥ − ω L ≈ (cid:0) q − q (cid:1) Ω ω L , (6)and winding of the ˆ d -vector in a soliton leads to a character-istic frequency shift ∆ ω PdBsat = ω PdBsat − ω L ≈ λ PdB Ω ω L , (7)where the dimensionless parameter λ PdB is characteristic tothe defect. Numerical calculations in a 1D soliton model(Supplementary Note 4) for all possible solitons shown inFig. 3 (a) give the low-temperature values λ soliton ∼ − . for π − θ -soliton (“soliton”) and λ big ∼ − . for its anti-soliton, which has π +2 θ winding (“big soliton”). The ( θ )-soliton (“ KLS soliton”) related to the KLS walls outside spin-solitons gives rise to a frequency shift experimentally indistin-guishable from the frequency shift of the bulk line. The lastpossibility, the “ π -soliton” consisting of a KLS soliton and asoliton, c.f. Fig. 4 (b), gives λ π ∼ − . at low temperatures.The measured value, λ PdB ∼ − . at the lowest tempera-tures, as seen in Fig. 3 (a). The measured values for λ PdB , together with the fact that the total winding of the ˆ d -vector isalso equal to π in the PdA and polar phases above the tran-sition temperature, suggest that the observed soliton structurein the PdB phase corresponds to the π -soliton in the presenceof a KLS wall.In addition, the KLS wall possesses a tension ∼ ξq ∆ N , where N is the density of states. Thusthe presence of KLS walls applies a force pulling the twoHQVs at its ends towards each other. The fact that the numberof HQVs remains unchanged in the phase transition signifiesthat the KLS wall tension does not exceed the maximum pin-ning force in the studied nafen sample. This observation isin agreement with our estimation of relevant forces (see Sup-plementary Note 5). Strong pinning of single-quantum vor-tices in B-like phase in silica aerogel has also been observedpreviously. An alternative way to remove a KLS wall is tocreate a hole within it, bounded by a HQV. Creation of sucha hole, however, requires overcoming a large energy barrierrelated to creation of a HQV with hard core of the size of ξ .Moreover, growth of the HQV ring is prohibited by the strongpinning by the nafen strands. We also note that for larger val-ues of q there may exist a point at which the KLS wall be-comes unstable towards creation of HQV pairs and as a resultthe HQV pairs bounded by KLS walls would eventually shrinkto singly-quantized vortices. For the discussion of the effectof nafen strands on the KLS walls see Supplementary Note 5. Effect of rotation
The density of HQVs created in the polar phase is con-trolled by the angular velocity Ω of the sample at the time ofthe phase transition from the normal phase, n HQV = 4Ω κ − ,where κ is the quantum of circulation. The integral of theNMR satellite depends on the total volume occupied by thesolitons, whose width is approximately the spin-orbit lengthand the height is fixed by the sample size mm. The averagesoliton length is equal to the intervortex distance ∝ Ω − / .Since the number of solitons is half of the number of HQVs,the satellite intensity scales as ∝ Ω · Ω − / = √ Ω whichhas been previously confirmed by measurements in the polarphase. Here we observe similar scaling in the PdA and PdBphases, c.f. Fig 3 (c).Although the satellite intensity scales with the vortex den-sity in the same way in both phases, there is one striking differ-ence – the satellite intensity normalized to the total absorptionintegral in the PdB phase is smaller by a factor of ∼ relativeto the PdA phase. Simultaneously, the original satellite inten-sity in the PdA phase is restored after a thermal cycle shownin Fig. 1 (b). Our numerical calculations of the soliton struc-ture do not indicate that the PdB phase soliton width nor theoscillator strength would decrease substantially to explain theobserved reduction in satellite size and the reason for the ob-served spectral intensity remains unclear – see SupplementaryNote 8 for the calculations.Another effect of rotation in the PdB phase transverse( µ = π/ NMR spectrum is observed at the main peak,c.f. Fig. 3 (b). The full-width-at-half-maximum (FWHM), ex-tracted from the amplitude of the main peak assuming w · h =const , where w is its width and h is height, scales as ∝ √ Ω ;Fig. 3 (d). Increase in the FWHM may indicate that the pres-ence of KLS walls enhances scattering of spin waves and thusresults in increased dissipation. Further analysis of this effectis beyond the scope of this Article. DISCUSSION
To summarize, we have found that HQVs, created in thepolar phase of He in a nanostructured material called nafen,survive phase transitions to the PdA and PdB phases. Pre-viously HQVs have been reported in the polar phase, atthe grain boundaries of d -wave cuprate superconductors, inchiral superconductor rings, and in Bose condensates. Of these systems, only the polar phase contains vortex-core-bound fermion states as others are either Bose systems orlack the physical vortex core altogether. The domain wallswith the sign change of a single gap component in He-Bwere suggested to interpret the experimental observations inbulk samples ( q = 1 ) and in the slab geometry. Suchwalls, however, differ from those reported here as they arenot bounded by strings but rather terminate at container walls.In the slab geometry such walls are additionally topologicallyprotected by a Z symmetry due to pinning of the ˆ l vector bythe slab. The survival of HQVs in the PdA and PdB phases has sev-eral important implications. First, HQVs in 2D p x + ip y topological superconductors (such as the A or PdA phases)are particularly interesting since their cores have been sug-gested to harbor non-Abelian Majorana modes, which can beutilized for topological quantum computation. This fact hasattracted considerable interest in practical realization of suchstates in various candidate systems.
While the PdA phasehas the correct p x + ip y type order parameter, scaling thesample down to effective 2D remains a challenge for future.However, the presence of the nafen strands, smaller in diam-eter than the coherence length, increases the separation of thezero-energy Majorana mode from other vortex-core-localizedfermion states to a significant fraction of the superfluid energygap, making it easier to reach relevant temperatures ( k B T (cid:46) energy separation of core-bound states ) in experiments. Second, we have shown how in the PdB phase the HQVs,although topologically unstable as isolated defects, surviveas composite defects known as “walls bounded by strings”(here KLS walls bounded by a pair of HQVs) – first discusseddecades ago by Kibble, Lazarides and Shafi in the context ofcosmology. Although the present existence of KLS walls inthe context of the Standard Model is shown to be unaccept-able, as they either dominate the current energy density (first-order phase transition) or disappeared during the early evolu-tion of the Universe (second-order phase transition), they oc-cur in some GUTs and beyond-the-Standard-Model scenarios,especially in ones involving axion dark matter.
Any signof similar defects in cosmological context would thus immedi-ately limit the number of viable GUTs. Under our experimen-tal conditions the transition from the PdA phase to the PdBphase is weakly first-order ( q (cid:28) at transition), but in prin-ciple the order parameter allows a second-order phase transi-tion to the PdB phase directly from the polar phase. Such aphase transition may be realized in future e.g. by tuning con-finement parameters. Studying the parameters affecting theamount of supercooling of the metastable PdA state (“falsevacuum”) before it collapses to the lowest-energy PdB state(“true vacuum”) may also give insight on the nature of phasetransitions in the evolution of the early Universe.In conclusion, we have shown that the creation and sta-bilization of HQVs in different superfluid phases with con-trolled and tunable order parameter structure is possible inthe presence of strong pinning by the confinement. The sur-vival of HQVs opens up a wide range of experimental andtheoretical avenues ranging from non-Abelian statistics andtopological quantum computing to studies of cosmology andGUT extensions of the Standard Model. Additionally, our re-sults pave way for the study of a variety of further problems,such as different fermionic and bosonic excitations living inthe HQV cores and within the KLS walls, and the interplayof topology and disorder provided by the confining matrix. A fascinating prospect is to stabilize new topological objectspossibly in novel superfluid phases by tuning the confinementgeometry, temperature, pressure, magnetic field, or scat-tering conditions. METHODSSample geometry and thermometry
The He sample is confined within a 4-mm-long cylindri-cal container with ∅ kHz resonance frequency, commonly used forthermometry in He.
The fork is calibrated close to T c against NMR signal from bulk He-B surrounding the nafen-filled volume. At lower temperatures we use a self-calibrationscheme by determining the onset of the ballistic regime fromthe fork’s behavior. Sample preparation
To avoid paramagnetic solid He on the surfaces, the sam-ple is preplated with approximately 2.5 atomic layers of He. The HQVs are created by rotating the sample in zeromagnetic field with angular velocity Ω while cooling the sam-ple from the normal phase to the polar phase. Then the rota-tion is stopped since, based on our observations, the HQVs re-main pinned (and no new HQVs are created) over all relevanttime scales, at least for two weeks after stopping the rotation.The typical cooldown rate close to the critical temperature wasof the order of . T c per hour to reduce the amount of vor-tices created by the Kibble-Zurek mechanism. Once the statehad been prepared the temperature was kept below the polarphase critical temperature until the end of the measurement. NMR spectroscopy
Static magnetic field of 12–27 mT corresponding to NMRfrequencies of 409–841 kHz, is created using two coils ori-ented along and perpendicular to the axis of rotation. Themagnetic field can be oriented at an arbitrary angle in the planedetermined by the two main coils. Special gradient coils areused to minimize the field gradients along the directions ofthe main magnets. The magnetic field inhomogeneity along the rotation axis is ∆ H ax /H ax ∼ − and in the transversedirection an order of magnitude larger, ∆ H tra /H tra ∼ − .The NMR pick-up coil, oriented perpendicular to both mainmagnets, is a part of a tuned tank circuit with quality factor Q ∼ . Frequency tuning is provided by a switchable ca-pacitance circuit, thermalized to the mixing chamber of thedilution refrigerator. We use a cold preamplifier, thermalizedto a bath of liquid helium, to improve the signal-to-noise ratioin the measurements. Rotation
The sample can be rotated about the vertical axis with angu-lar velocities up to rad/s, and cooled down to ∼ µ K us-ing ROTA nuclear demagnetization refrigerator. The refriger-ator is well balanced and suspended against vibrational noise.The earth’s magnetic field is compensated using two saddle-shaped coils installed around the refrigerator to avoid parasiticheating of the nuclear stage. In rotation, the total heat leak tothe sample remains below 20 pW. ACKNOWLEDGEMENTS
We thank V.V. Zavyalov and V.P. Mineev for useful discus-sions and related work on spin-solitons and HQVs. This workhas been supported by the European Research Council (ERC)under the European Union’s Horizon 2020 research and in-novation programme (Grant Agreement No. 694248) and bythe Academy of Finland (grants no. 298451 and 318546).The work was carried out in the Low Temperature Labora-tory, which is part of the OtaNano research infrastructure ofAalto University.
AUTHOR CONTRIBUTIONS
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Halperin,“Strong coupling corrections to the Ginzburg-Landau theory ofsuperfluid He ,” Phys. Rev. B , 174503 (2007). M. M. Salomaa and G. E. Volovik, “Half-quantum vortices in su-perfluid He - A ,” Phys. Rev. Lett. , 1184 (1985). SUPPLEMENTARY NOTE 1: SYMMETRIES OF LIQUID HE IN CONSTRAINED GEOMETRY
Here we discuss the symmetries possessed by the nor-mal fluid and the superfluid phases under confinement. Forschematic illustration of the superfluid gaps in differentphases, see Supplementary Figure 5.
Normal phase
Above the superfluid transition bulk He possesses the sym-metry group G = SO (3) L × SO (3) S × U (1) φ × T × P (8)which includes continuous symmetries: three-dimensional ro-tations of coordinates SO (3) L , rotations of the spin space SO (3) S , and the global phase transformation group U (1) φ , aswell as discrete symmetries; T is the time-reversal symmetryand P is the space parity symmetry. The transitions from nor-mal fluid to superfluid phases as well as transitions betweendifferent superfluid phases are accompanied by the sponta-neous breaking of continuous and/or discrete symmetries in G (in addition to the broken U (1) φ symmetry of the super-fluid). In bulk He three superfluid phases can be realized;the fully-gapped superfluid B phase characterized by brokenrelative spin-orbit symmetry, the chiral p x + ip y state knownas the superfluid A phase, and finally, the spin-polarized A phase in high magnetic fields.In nanostructured confinement, i.e. in thin slabs or invarious aerogels, the phase diagram, as well as the sym-metry group of the normal phase, can be altered in a controlledfashion. In the presence of commercially available nemati-cally ordered material called nafen, the three-dimensionalcontinuous rotational symmetry SO (3) L in Eq. (8) is explic-itly broken in the real space by the confinement. As a result,the total symmetry group of the normal phase is reduced to G (cid:48) = D ∞ L × SO (3) S × U (1) φ × T × P , (9)where D ∞ L contains rotations about axis ˆ z and π rotationsabout perpendicular axes. The resulting phase diagram dif-fers from that of the bulk He; the critical temperature issuppressed and, more importantly, new superfluid phases, c.f.Supplementary Figure 5 - the polar, polar-distorted A (PdA),and polar-distorted B (PdB) phases - are observed.
Polar phase
In our samples, the phase transition with the highest criticaltemperature always occurs between the normal phase and thepolar phase. The order parameter of the polar phase can bewritten as A αj = 1 √ P e iφ ˆ d α ˆ m j , (10) where ∆ P ( T ) is the maximum superfluid gap in the polarphase, φ is the superfluid phase, ˆ d is the unit vector of spinanisotropy, and ˆ m is the unit vector of orbital anisotropy par-allel to the anisotropy axis of the confinement. That is, inthe transition to the polar phase the orbital part is fixed by thenafen strands and rotational symmetry is preserved only in theplane perpendicular to ˆ m . As for any superconducting or su-perfluid state, the phase acquires an expectation value and thephase gauge symmetry U (1) φ is broken in the transition. Thegroup describing the remaining symmetries of the polar phasein zero magnetic field is H P = ˜ D ∞ L × ˜ D ∞ S × T × ˜ P . (11)Here the discrete symmetry ˜ P is the inversion P combinedwith the phase rotation e πi . The symmetries ˜ D ∞ L and ˜ D ∞ S are the symmetries D ∞ L and D ∞ S in L and S spaces, wherethe π rotations about transverse axes are combined with aphase rotation e πi . The homotopy group π ( G (cid:48) /H P ) = Z × Z provides the topological stability of the phase vorticesand the half-quantum vortex. The topological stability of spinvortices is determined by spin-orbit interaction and orientationof the magnetic field. Polar-distorted A phase
At certain nafen densities and pressures the polar phasetransforms on cooling to the polar-distorted A (PdA) phasevia a second-order phase transition. The order parameter ofthe PdA phase is A αj = (cid:114) b PdA e iφ ˆ d α ( ˆ m j + ib ˆ n j ) , (12)where the vector ˆ n is an orbital anisotropy vector both per-pendicular to vector ˆ m and the Cooper pair orbital angularmomentum axis ˆ l = ˆ m × ˆ n , and < b < is a dimen-sionless parameter characterizing the gap suppression by theconfinement. The anisotropy vector ˆ l defines the axis of theWeyl nodes in the PdA phase quasiparticle energy spectrum.The remaining symmetry group in the PdA phase in zero mag-netic field is H PdA = ˜ D × ˜ D ∞ S × ˜ P . (13)The time-reversal symmetry is explicitly broken in the PdAphase, while ˜ P combined with π orbital rotation about ˆ m re-mains a symmetry. Together with π rotation about the axis ˆ m × ˆ n combined with the phase rotation e πi the orbital sym-metry forms the ˜ D -group. The group H PdA is the subgroupof H P , which reflects the fact that the PdA phase can be ob-tained by the second-order phase transition from the polarphase. The homotopy group π ( G (cid:48) /H PdA ) = Z × Z × Z provides the topological stability of the phase vortices, thehalf-quantum vortex and also the orbital disclination in thevector ˆ l .1 Polar phase m ^ PdB phase z ^ PdA phase
Weylpoints l ^ m ^ m ^ n ^ FIG. 5. The figure shows schematic illustration (not to scale) of superfluid gaps in all superfluid phases encountered under confinementby nafen. The polar phase and PdB phase gaps are symmetric under rotation by the vertical axis, and the PdA phase gap is shown in twoprojections as it lacks the rotational symmetry.
Polar-distorted B phase
The lowest temperature phase transition to the polar-distorted B phase (PdB) may in principle occur via a first-order transition from the PdA phase, or via a second-orderphase transition from the polar phase. For the experimentalconditions studied here, the transition occurs via a first-orderphase transition. The order parameter of the PdB phase canbe written as A αj = (cid:114) q PdB e iφ ( ˆ d α ˆ z j + q ˆ e α ˆ x j + q ˆ e α ˆ y j ) , (14)where | q | , | q | ∈ (0 , , | q | = | q | ≡ q describes the rela-tive gap size in the plane perpendicular to the strands. Vectors ˆ e and ˆ e are unit vectors in the spin-space. The maximumgap ∆ PdB ( T, q ) is achieved along the direction parallel to thestrand orientation. For q = 0 , we obtain the order parameterof the polar phase and for q = 1 , we obtain the order param-eter of the isotropic B phase. In zero magnetic field the totalsymmetry group describing the PdB phase can be written as H PdB = D ∞ J × T × ˜ P , (15)where the notation J refers to the symmetry of the combinedrotation of L and S simultaneously. The group H PdB is again a subgroup of H P , which reflects the fact that the PdB phasecan in principle be obtained by the second-order phase transi-tion from the polar phase.The homotopy group π ( G (cid:48) /H PdB ) = Z × Z provides thetopological stability of phase vortices and combined orbitaland spin disclinations, but not of the half-quantum vortices. Itis the lack of the last factor Z in the homotopy group whichgives rise to the topologically unstable domain wall terminat-ing on HQVs in the PdB phase: the KLS wall bounded byHQV strings.2 FIG. 6.
Supplementary Note 2 – Extended data figure for 2D calculations:
2D numerical calculations of the distribution of angle θ insideand in the vicinity of KLS walls. In both (a) and (b) panels the upper half corresponds to q > and the lower half to q < . The KLS wallsare located on the y = 0 axis between X/ξ ∈ ( − , and virtual jumps in the order parameter on the same axis between X/ξ ∈ [ − , − and X/ξ ∈ [5 , . Plot (a) corresponds to the situation where the KLS walls and ( π − θ ) -solitons are located between different HQV pairs.Plot (b) corresponds to the situation where the KLS Walls and ( π − θ ) -solitons extend between the same HQV pair. Parameter value q = 0 . was used in the calculations. SUPPLEMENTARY NOTE 3: FREE ENERGY OF THEPOLAR-DISTORTED B PHASE
The Landau-Ginzburg free-energy of the PdB phase isgiven as (summation over repeated indices is assumed) F = (cid:90) d x ( f grad + f bulk + f nafen + f H + f so ) (16) f grad = K ( ∇ i A µj )( ∇ i A ∗ µj )+ K + K ( ∇ i A µi )( ∇ j A ∗ µj ) (17) f so = g so (cid:0) | Tr ( A ) | + Tr ( AA ∗ ) (cid:1) (18) f nafen = η ij A µi A ∗ µj , η ij = ηδ ij + ∆ η ˆ z i ˆ z j , (19) f H = − HχH , χ αβ = χ N δ αβ − ˜ αA αi A ∗ βi , (20)where f grad is the gradient energy, f bulk is the standard bulkcondensation energy of He, with order parameter matrix A µi corresponding to d µi in the notation of Ref. 39. Withthis convention, the spin-orbit coupling corresponds to g so = λ D N F of Ref. 39. The effect of the nafen confinement f nafen , with the uniaxial anisotropy ∆ η along (cid:107) ˆ z , is to renor-malize the quadratic coefficients ∝ ∆ since T c, nafen < T c in bulk. The magnetic susceptibility tensor χ and the coeffi-cient ˜ α for the PdB phase are also found in Ref. 47.The order parameter of polar distorted He-B in nafen isparametrized by Eq. (14). Here we concentrate on the limit oflarge polar anisotropy of the superfluid with the magnetic field H (cid:54) = 0 transverse to the uniaxial anisotropy, i.e. the condition | q , | (cid:28) holds. The corresponding ansatz for the spin part of the order parameter is ˆ d = R ( ˆ y , θ ) ˆ x = cos θ ˆ x − sin θ ˆ z ˆ e = − R ( ˆ y , θ ) ˆ z = − cos θ ˆ z − sin θ ˆ x (21) ˆ e = ˆ y . We emphasize that with this parametrization ˆ e × ˆ e = ˆ d and the polar phase is obtained by setting q = 0 , whereasthe bulk B-phase corresponds to q = ± q = 1 . The de-generacy parameters for the bulk B phase are given by therotation axis ˆ n = ˆ y and angle cos θ ˆ n = sin θ , which isthe spin-orbit resolved Leggett angle. To first order in q , ,the gradient energy is simply f grad = K θij ( ∇ i θ )( ∇ j θ ) with K θij = K ( δ ij − ˆ z i ˆ z j ) + ( K + K + K ) ˆ z i ˆ z j . With theansatz (14), the spin-orbit interaction takes the form f so = 2 g so ∆ ˜ f so ( θ ) , (22) ˜ f so ( θ ) ≡ (1 + q ) sin θ − (1 + q ) q sin θ − q + q . SUPPLEMENTARY NOTE 4: FREE ENERGY OFSOLITONS AND DOMAIN WALLS
As discussed in the main text, the symmetry in the planetransverse to the anisotropy axis is broken by the magneticfield and/or the presence of defects. In equilibrium, ˆ d ⊥ H and we take ˆ e = ˆ y along H . With this notation the KLSwall is a domain wall in q .We now describe the order-parameter textures in the PdBphase which are associated with the HQVs pinned to the nafenstrands. Similar, non-topological defects in the B-phase were3 ✓ big soliton soliton ⇡/ ⇡/ q < q > ± ⇡
The KLS domain wall is the change of sign in the trans-verse, in-plane gap components q ∆ PdB , q ∆ PdB determinedby the in-plane coherence length ξ ⊥ . Without loss of gener-ality, we fix the domain wall to act only on q ∆ PdB and thedirection normal to the domain wall to be ˆ x . Let us write thebulk free energy of the PdB phase in nafen, Eq. (16), as thesum of the polar phase free energy and the planar distortion f PdB = f P + f ⊥ , f P = α (cid:107) ∆ + β ∆ , (23) f ⊥ = ( α ⊥ + 2 β ∆ )( q ∆ + q ∆ )+ 2 β q ∆ q ∆ (24) + β ( q ∆ + q ∆ ) ,f grad [ q ∆ PdB ] = K (2) ij ( ∇ i q ∆ PdB )( ∇ j q ∆ PdB ) , (25)where K (2) ij = K ( δ ij − ˆ y i ˆ y j ) + ( K + K + K ) ˆ y i ˆ y j . From f P we obtain that ∆ = − α (cid:107) / β . For an infinite KLS wall along the y -axis, the order parameter is given by ξ ⊥ d dx q ∆ PdB = − q ∆ PdB + ( q ∆ PdB ) ( q ∆ ) , (26) q ( x )∆ PdB = q ∆ PdB tanh (cid:16) xξ ⊥ (cid:17) . (27)where ξ ⊥ /ξ (cid:107) ∼ q − (cid:29) and the KLS wall thickness is q − ξ (cid:107) . For the KLS wall to be stable, this should be (cid:29) ξ (cid:107) ,i.e. the distortion q should be small. However, on the lengthscale of the dipole length, ξ D ∼ K /g so , relevant in NMRexperiments, the KLS wall is thin, since g so (cid:28) − α ⊥ . Thefree energy of the domain wall per unit area is σ KLS ∼ ξ ⊥ ∆ f ∼ − ξ (cid:107) qf ⊥ , (28)where ∆ f ≈ f P − f PdB . This surface tension makes theisolated HQVs unstable in the PdB phase without the nafen-pinning.
Spin solitons
Spin solitons have thickness of the order of the dipolelength ξ D . The distribution of θ ( r ) in the presence of HQVspin solitons is found as a minimum of energy in Eq. (16), −∇ i δFδ ∇ i θ + δFδθ = − ξ D,ij ∇ i ∇ j θ ( r ) + δ ˜ f so ( θ ( r )) δθ = 0 . (29)In bulk the energy is minimized for a homogeneous θ = θ or π − θ , where θ = arcsin q q ) . (30)The minima for the spin-orbit potential f so ( θ ) depend on thesign of q , which changes across the KLS walls. In contrastto the polar phase with q = q = 0 , the potential is no longersymmetric under θ → θ + π , see Supplementary Figure 7.The equations can be solved analytically for the infinitesoliton uniform in the y - and z -directions. Integrating Eq. (29)over y and z we obtain ξ D ( θ (cid:48) ) = (1 + q ) (sin θ − sin θ ) + C, (31)where C = 0 by the bulk boundary conditions θ ( x → ±∞ ) = θ , θ (cid:48) ( x → ±∞ ) = 0 . The soliton solutions are θ ( ± ˜ x ) = ∓ π/ ∓ f ∓ (˜ x ; s ) , (32)where we scaled ˜ x ≡ (1 + q ) x/ξ D , abbreviated s ≡ sin θ and f ∓ (˜ x ; s ) = (cid:113) ∓ s ± s tanh( (cid:113) − s ˜ x/ (33)where there are two solutions corresponding to the two signsin Eq. (31) and we have used the boundary conditions θ (0) = {− π/ , + π/ } . The two soliton solutions in Eq. (32) have4windings π ∓ θ . Clearly the two solutions interchange as s → − s .When ξ (cid:28) ξ D , we can approximate the KLS domain wallas q ( x ) = q sign ( x ) . Across a KLS wall, θ → − θ and s → − s and we can respectively join the correspondingsolutions with boundary conditions θ (0) = 0 or ± π/ at theKLS wall, see Supplementary Figures 7 and 8. In particular,we can find a solution with ∆ θ = 2 θ that crosses θ (0) = 0 and a solution with ∆ θ = π , θ (0) = ± π/ composed of asmall and big soliton on the opposite sides of the domain wall.For q = sgn ( x ) | q | , the KLS soliton solution with ∆ θ = 2 θ is given by θ KLS (˜ x ) = 2 arctan (cid:18) sgn ( x ) | s | √ − s coth( √ − s | ˜ x | / (cid:19) . (34)The plots of the soliton solutions interpolating between thePdB spin-orbit energy minima in Supplementary Figure 7 arefound in Supplementary Figure 8: In summary, we find twosolitons (“soliton” and “big soliton”) without KLS wall andtwo solitons (the “ KLS soliton” and the “ π -soliton”) connect-ing the solutions with opposite sign of s . In terms of the morerealistic 2D HQV-pair structures depicted in SupplementaryFigure 6 or in the main text, the separate 1D small soliton andKLS soliton roughly corresponds to the case shown in (a) inSupplementary Figure 6 with the KLS wall outside the spinsoliton, whereas the π -soliton corresponds to that shown inSupplementary Figure 6 (b).The free energy per unit area of a 1D spin soliton is σ spin = LR HQV (cid:90) d r ( f [ θ ( r )] − f [ θ ]) (35) ≈ χ γ Ω (cid:90) dx
2( ˜ f so ( θ ( x )) − ˜ f so ( θ )) ∼ ξ D ∆ f soliton . where L is the linear size along z -direction (the height of thesample), and R HQV the the linear size along y -direction (thedistance between HQVs bounding the soliton). SUPPLEMENTARY NOTE 5: PINNING OF HQV BY ACOLUMNAR DEFECT
Let us consider what happens with HQVs, when the phasetransition is crossed between polar phase and the PdB phase.Our experiments demonstrate that if originally the polar phasecontains pinned HQVs, they survive the transitions to the PdBphase and back to the polar phase. From this one can concludethat the HQVs remained pinned even after the formation of aKLS domain wall formed between two HQVs, demonstratingthat HQVs are so strongly pinned that the tension of the KLSwall can not unpin vortices. Let us consider the pinning inmore detail (assuming (cid:126) = 1 and k B = 1 ).The radius of the columnar defect – the nafen strand – issmall compared to the coherence length in superfluid He.According to Ref. 78 the characteristic energy of the orderparameter distortion produced by a mesoscopic object of size
R < ξ ≡ ξ ( T = 0) is (per unit length of the cylinder): E P ∼ k R ∆ T c , R < ξ , (36) π + θ - π + θ - θ θ π - θ - π - θ - - x - - - θ - π - soliton π - soliton big solitonsoliton KLS soliton FIG. 8. The figure shows the 1D soliton solutions of Eq. (29) for q ( −∞ ) < . The ordinary soliton has ∆ θ = π + 2 | θ | and θ (0) = − π/ . The solution with θ (0) = π/ leads to the big soliton withwinding ∆ θ = π − | θ | . Across the KLS wall, one must join thesolutions with different signs of s with ∆ θ = 2 θ or π . The latter isa composite of a big soliton and an ordinary soliton across the KLSwall. where k F is the Fermi momentum, ∆ ∼ v F ξ − is the super-fluid gap (here we use general gap notation, since this is anorder-of magnitude estimation and ∆ ∼ ∆ PdB ), and v F is theFermi velocity. This equation was used in particular for theestimation of the orientational energy of the nafen strands onthe orbital vector ˆ l in He-A in relation to the Larkin-Imry-Ma effect. The Larkin-Imry-Ma effect due to the randomanisotropy produced by the random orientation of strands wasobserved later.
Eq. (36) can be applied for the pinning of the texture ∆( r ) by columnar defects – the nafen strands. The pinning forcecomes from the coordinate dependence of the energy of thecolumnar object in the texture: F P ∼ ∇ E P . For textures withcharacteristic length scale ξ , one has F P ∼ ∇ E P ∼ E P /ξ .For vortices, including the half-quantum vortices observed inRef. 21, the pinning force from the columnar defect of radius R is (assuming ∆ /T c ∼ ξ /ξ ): F P ∼ k v F Rξ ∆ T c ∼ k v F Rξ ξ , R < ξ . (37)Let us compare the pinning force with the tension of the KLSwall of thickness ξ W ∼ q − ξ (cid:29) ξ , given by F KLS ∼ k v F q ξ W ∆ T ∼ k v F q ξ ξ . (38)The tension from the KLS wall can not unpin the HQV if F KLS < F P , or if q < Rξ < . (39)Close to the transition from the polar to PdB phase, the HQVsremained pinned, while the KLS wall is pinned by the pinnedHQVs.5Let us consider the pinning force by the columnar defect fordifferent ranges of R . For R > ξ the pinning does not dependon R , but instead is given by the characteristic length scale ξ .The dependence of the pinning force on R is given by F P k v F ∼ Rξ ξ , R < ξ , (40) F P k v F ∼ R ξ , ξ < R < ξ , (41) F P k v F ∼ ξ , R > ξ . (42) SUPPLEMENTARY NOTE 6: SPIN WAVES AND NMR INTHE PDB PHASE
We study the HQVs and KLS domain walls in the PdBphase via their influence on the NMR spin-wave spectrumthrough the order parameter textures of solitons. The rele-vant Hamiltonian is given by the magnetic field energy andthe superfluid spin degrees of freedom in the London limit, H = 12 γ S χ − S − γ H · S + f grad + f so , (43)where S is the total spin density, γ is the gyromagnetic ratioof He and χ is the principal axis of the magnetic suscepti-bility tensor along ˆ d . The Leggett equations for the spin S and the order parameter spin-triad ˆ e I = { ˆ e , ˆ e , ˆ d } , where I = 1 , , , are ∂ t S = { S , H } = γ S × H + δ ( f grad + f so ) δ ˆ e I { S , ˆ e I } (44) ∂ t ˆ e I = { ˆ e I , H } = − γ χ ˆ e I × δ S , (45)with the semiclassical Poisson brackets { S α , S β } = (cid:15) αβγ S γ and { S α , e Iβ } = (cid:15) αβγ e Iγ . In this parametrization, the spin-orbit interaction Eq. (18) takes the form f so [ˆ e I ] = 2 g so ∆ (cid:0) ˆ e I · B IJ · ˆ e J (cid:1) (46)where B IJ = ( δ IM δ JN + δ IN δ JM )˜ r M ˜ r N is a matrix inorbital space ˜ r M = { q ˆ x , q ˆ y , ˆ z } defined by the orbital partof the order parameter and summation over repeated spin-triadand orbital indices I, J = 1 , , and M, N = 1 , , is im-plied, respectively.We look for solutions in small oscillations to linear orderaround an equilibrium state δHδ S = δHδ ˆ e I = 0 . Eliminating δ ˆ e I from the system of Leggett equations, we arrive to ω δ S = iωω L ( ˆ H × δ S ) + Ω Λ · δ S , (47)where we have defined the “Leggett frequency” of the PdBphase as the quantity Ω = 4 g so γ ∆ /χ PdB (48)which we stress is not equal to the longitudinal NMR fre-quency of the PdB phase for q , q (cid:54) = 0 , , see below. The matrix Λ = Λ grad + Λ so is defined by Λ grad αβ = ξ D,ij (cid:18) ( δ αβ − ˆ d α ˆ d β ) ∇ i ∇ j + ˆ d α ( ∇ i ∇ j ˆ d β ) − ˆ d β ( ∇ i ∇ j ˆ d α ) − d β ( ∇ i ˆ d α ) ∇ j (cid:19) (49) Λ so αβ = e I α B IJβδ e J δ − ˜ e I ν B IJνδ e J δ δ αβ + (cid:15) ανγ (cid:15) δµβ e I γ B IJνδ e J µ , (50)where B IJαβ is the matrix in Eq. (46) for each
I, J , ξ D,ij = K θij / g so and the gradient energy is taken to first order in q , q . In the limit q = q = 0 , the equations are those ofthe polar phase; in particular the lowest order Λ grad in Eq.(49) coincides with the expression for the polar phase given inRef. 14.With H (cid:107) ˆ y and within the approximation ω ≈ ω L tolowest order in Ω PdB ω L (cid:28) , Eq. (47) separates for transverse δS + = ( δS z + iδS x ) / √ and longitudinal spin waves as ω − ω L Ω δS + = (Λ xx + Λ zz ) δS + + i (Λ xz − Λ zx ) δS + , (51) ω Ω δS y = Λ yy δS y . (52) Transverse magnetic field
For the transverse orientation of the magnetic field to theuniaxial nafen anisotropy along ˆ z , the order parameter isgiven by Eq. (14). The transverse spin wave equation becomes − ω − ω Ω Ψ + = ξ D,ij (cid:18) ∇ i ∇ j + ( ∇ i θ )( ∇ j θ ) (cid:19) Ψ + (53) + (cid:18) ˜ f so ( θ ) − (1 + q ) q sin θ + q (cid:19) Ψ + where Ψ + = e iθ δS + , the dimensionless spin-orbit interaction ˜ f so ( θ ) is defined in Eq. (22) and the longitudinal spin waveequation is − ω Ω δS y = ξ D,ij ∇ i ∇ j δS y − (cid:18) ∂ ˜ f so ∂θ (cid:19) δS y . (54)The transverse frequency shift with uniform θ = θ (i.e. theresponse of the bulk) is given as ω ⊥ − ω Ω = q − q . (55)This frequency shift was reported also in Ref. 47. Axial field
In axial field, i.e. H along the uniaxial anistropy, the orderparameter is no longer given by Eq. (14). Since the magneticfield energy is dominating and ˆ d ⊥ H (cid:107) ˆ y which is also the6 FIG. 9. The dots represent the measured values for q . The solidred line is an estimation of q , calculated based on Ginzburg-Landautheory with strong-coupling corrections using two fitting parametersin the spirit of Ref. 47 and taking β -parameter values from Ref. 83.The PdB phase critical temperature is shown for warming transitionto the PdA phase. direction of the uniaxial anisotropy (and not along ˆ z as in thepreceding sections), we parametrize the orbital part of orderparameter as A µi = ∆ PdB e iφ (cid:16) ˆ d µ ˆ y i + q ˆ e µ ˆ x i + q ˆ e µ ˆ z i (cid:17) , (56)whereas the spin part is still given by Eq. (21) with the rotationangle in the plane perpendicular to the magnetic field. Thespin-orbit interaction takes the form ˜ f so , (cid:107) ( θ ) = q sin θ − q sin θ with sin θ , (cid:107) = 0 for | q | < / . The longitudinalspin wave equation in axial field follows from Eq. (54) withthis replacement.The transverse spin wave Ψ + = e iθ δS + equations aregiven as − ω (cid:107) − ω Ω Ψ + = ξ D,ij (cid:18) ∇ i ∇ j + ( ∇ i θ )( ∇ j θ ) (cid:19) Ψ + + (cid:18) − − q sin θ + q sin θ − q (cid:19) Ψ + . (57)The homogeneous transverse frequency shift in axial fieldwith uniform θ = θ , (cid:107) = sgn ( q ) π/ is given as ω (cid:107) − ω Ω = 1 + 52 | q | , (58)which is equal to the value reported in Ref. 47. SUPPLEMENTARY NOTE 7: DETERMINATION OF THEDISTORTION PARAMETER q The q -parameter value is determined from the frequencyshifts in Eqs.(55) and (58), following a method described in KLS solitonMain line q - q q λ solitonbig soliton π - soliton - - - q λ FIG. 10. The figures show the NMR resonance eigenvalue λ ( q ) forspin waves localized on infinite 1D solitons. The frequency shiftrelated the the KLS wall (upper figure) is indistinguishable from thefrequency shift of the main line in the experimental range of q . Thelower figure shows the frequency shifts for the other possible solitonsas a function of q . Ref. 47. In the experimental region of interest, the distortionfactor q = q = | q | is q = 2 − C − (cid:112) C − C + 4 , (59)where C = ( ω ⊥ − ω L ) / ( ω (cid:107) − ω L ) . The expression (59) isvalid in the range q ∈ [0 , ( √ − / . We carefully pre-pare the state by cooling the sample through the superfluidtransition temperature in zero rotation in transverse magneticfield to avoid creation of half-quantum vortices. Then we coolthe sample down to the lowest temperatures and start warm-ing it up slowly, continuously monitoring the NMR resonancespectrum either in axial or transverse field. This way we canmeasure the q -parameter in the coexistence region of the PdAand PdB phases. The results of our measurements are shownin Supplementary Figure 9.7 KLS solitonMain line q I M / A soliton π - solitonbig soliton q I M / A FIG. 11. The figures show the NMR oscillator intensities for spinwaves on infinite 1D solitons. All solutions show decrease in theoscillator intensity, which results in the decrease of NMR satelliteintensity – as observed in the experiments. However, the observeddecrease in the intensity is much larger than the calculated decreasefor realistic values of q . SUPPLEMENTARY NOTE 8: SPIN WAVES ON 1DSOLITONS AND KLS WALLS