Composite topological objects in topological superfluids
ЖЖЭТФ
COMPOSITE TOPOLOGICAL OBJECTS IN TOPOLOGICALSUPERFLUIDS
G. E. Volovik a,b * a Low Temperature Laboratory, Aalto University,P.O. Box 15100, FI-00076 Aalto, Finland b Landau Institute for Theoretical Physics,acad. Semyonov av., 1a, 142432, Chernogolovka, Russia
The spontaneous phase coherent precession of magnetization, discovered in 1984 by Borovik-Romanov, Bunkov,Dmitriev and Mukharskiy [1] in collaboration with Fomin, [2], became now an important experimental tool forstudy complicated topological objects in superfluid He.
1. INTRODUCTION
Superfluid phases of He discovered in 1972 [3]opened the new area of the application of topologi-cal methods to condensed matter systems. Due tothe multi-component order parameter which character-izes the broken SO (3) × SO (3) × U (1) symmetry inthese phases, there are many inhomogeneous objects– textures and defects in the order parameter field –which are protected by topology and are characterizedby topological quantum numbers. Among them thereare quantized vortices, skyrmions and merons, soli-tons and vortex sheets, monopoles and boojums, Alicestrings, Kibble-Lazarides-Shafi walls terminated by Al-ice strings, spin vortices with soliton tails, etc. [4] Mostof them have been experimentally identified and inves-tigated using nuclear magnetic resonance (NMR) tech-niquie, and in particular the phase coherent spin preces-sion discovered in 1984 in He-B by Borovik-Romanov,Bunkov, Dmitriev and Mukharskiy [1, 5] in collabo-ration with Fomin [2]. Such precessing state, whichhas got the name Homogeneously Precessing Domain(HPD), is the spontaneously emerging steady state ofprecession, which preserves the phase coherence acrossthe whole sample even in the absence of energy pump-ing and even in an inhomogeneous external magneticfield. This spontaneous coherent precession has all thesignatures of the coherent superfluid Bose-Einstein con-densate of magnons (see review paper [6]).The Bose condensation of magnons in superfluid He-B had many practical applications. In Helsinki,owing to the extreme sensitivity of the Bose condensate * E-mail: [email protected].fi to textural inhomogeneity, the phenomenon of Bosecondensation has been applied to studies of topolog-ical defects by the HPD spectroscopy.
2. SUPERFLUID PHASES OF LIQUID HE In bulk liquid He there are two topologically dif-ferent superfluid phases, He-A and He-B. [7] Oneis the chiral superfluid He-A with topologically pro-tected Weyl points in the quasiparticle spectrum. Inthe ground state of He-A the order parameter matrixhas the form A αi = ∆ A e i Φ ˆ d α (ˆ e i + i ˆ e i ) , ˆ l = ˆ e × ˆ e , (1)where ˆ d is the unit vector of the anisotropy in the spinspace due to spontaneous breaking of SO (3) S symme-try of spin rotations; ˆ e and ˆ e are mutually orthogonalunit vectors; and ˆ l is the unit vector of the anisotropy inthe orbital space due to spontaneous breaking of orbitalrotations SO (3) L symmetry. The ˆ l -vector also showsthe direction of the orbital angular momentum of thechiral superfluid, which emerges due to spontaneousbreaking of time reversal symmetry. The chirality of He-A has been probed in several experiments. [8–10]Another phase is the fully gapped time reversal in-variant superfluid He-B. In the ground state of He-Bthe order parameter matrix has the form A αi = ∆ B e i Φ R αi , (2)where R αi is the real matrix of rotation, R αi R αj = δ ij .This phase has topologically protected gapless Majo-rana fermions living on the surface (see reviews [11, 12]on the momentum space topology in superfluid He). a r X i v : . [ c ond - m a t . o t h e r] D ec . E. Volovik ЖЭТФIn He confined in the nematically ordered aerogel(nafen), new phase becomes stable – the polar phase of He [13, 14], whith the order parameter A αi = ∆ P e i Φ ˆ d α ˆ m i . (3)where orbital vector ˆ m is fixed by the nafen strands.The reason for the appearance of the polar phase innafen is the analog of the Anderson theorem appliedfor the polar phase in the presence of the columnar de-fects (nafen strands), see Refs. [15, 16]. While for allthe other phases of superfluid He the transition tem-perature is suppressed by these impurities.The polar phase is the time reversal invariant super-fluid, which contains Dirac nodal ring in the fermionicspectrum [16, 17].
3. STRINGS WITH SOLITONIC TAIL
There are different types of the topological defectsin the He-B. Among them there are the conventionalmass vortices with the N winding number of the phase Φ , and the Z spin vortex – the nontrivial winding ofthe matrix R αi . Due to spin-orbit coupling the spinvortex serves as the termination line of the topologicalsoliton wall. Because of the soliton tension the spinvortex moves to the wall of the vessel and escapes theobservation. However, the help comes from the massvortices. The mass and spin vortices are formed by dif-ferent fields. They do not interact since they "live indifferent worlds". The only instance, where the spinand mass vortices interact, arises when the cores of aspin and a mass vortex happen to get close to eachother and it becomes energetically preferable for themto form a common core. Thus by trapping the spinvortex on a mass vortex the combined core energy isreduced and a composite object Z -string + soliton +mass vortex, or spin-mass vortex is formed. This ob-ject is stabilized near the edge of the vortex cluster inthe rotating cryostat, see Fig. 1.These combined objects have been observed andstudied using HPD spectroscopy [18, 19]. The addi-tional absorption observed in the homogeneously pre-cessing domain (HPD) is proportional to the solitonarea A = lh , where h is the heght of the container, and l is the length of the cross-section of the soliton. In therotating container the length l is given by the widthof the counterflow vortex-free zone, which is regulated s o lit on cluster ofmassvorticesspin–massvortex spin–massvortex spin–massvortexsoliton Ω v vn = Ω r vs v n - v s c oun t e r f l o w r eg i on r suurennos=7y-skaala=0.47 vortexbundle Fig. 1. right : Vortex cluster in rotating container withthe vortex free region outside the cluster. Vortex clusteris formed when starting with the equilibrium vortex statein the rotating container the angular velocity of rotationis increased. The new vortices are not formed if the coun-terflow in the vortex region does not exceed the criticalvelocity for vortex formation. left : The spin-mass vor-tex finds its equibrium position on the periphery of thevortex cluster, where the soliton tension is compensatedby the Magnus force acting on the mass vortex part ofthe composite object. The size of the soliton is given byEq.(4), and this dependence on the angular velocity ofrotation is confirmed by the HPD spectroscopy. bottom :The combined object with N = 2 quanta of circulation:spin-mass vortex + soliton + spin-mass vortex. by changing the angular velocity of rotation Ω at fixednumber N of vortices in the cell: l (Ω) = R (cid:32) − (cid:114) Ω V ( N )Ω (cid:33) , (4)Here R is the radius of the cylindrical container, and Ω V ( N ) is the angular velocity in the state in the ro-tating container with equilibrium number of vortices N = 2 πR Ω V ( N ) /κ . The equilibtium state is obtainedby cooling through T c under rotation, and then we in-crease the angular velocity of rotation, Ω > Ω V ( N ) .The new vortices are not created because of high en-ergy barrier, and as a result the counterflow region ap-pears. The dependence of the attenuation of the HPDstate follows Eq.(4) [18, 19]. The half-quantum vortices (HQVs) were originallysuggested to exist in the chiral superfluid He-A [20].The half-quantum vortex represents the condensedmatter analog of the Alice string in particle physics. ЭТФ
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Fig. 2.
Illustration of the lattice of solitons emerging be-tween the Alice strings (half-quantum vortices) in the po-lar phase if He, when the magnetic field is tilted withrespect of the aerogel strands. The half quantum vor-tices survive the soliton tension because they are pinnedby the strands. The NMR measurements give informationon the total length of the soliton and thus on the numberof the Alice strings in the cell. [21] The HQV is the vortex with fractional circulationof superfluid velocity, N = 1 / . It is topologically con-fined with the fractional spin vortex, in which ˆ d changessign when circling around the vortex: ˆ d (ˆ r ) e i Φ(ˆ r ) = (cid:18) ˆ x cos φ y sin φ (cid:19) e iφ , (5)When the azimuthal coordinate φ changes from 0 to π along the circle around this object, the vector ˆ d (ˆ r ) changes sign and simultaneously the phase Φ changesby π , giving rise to N = 1 / . The order parameter (5)remains continuous along the circle. While a particlethat moves around an Alice string flips its charge, thequasiparticle moving around the half-quantum vortexflips its spin quantum number. This gives rise to theAharnov-Bohm effect for spin waves in NMR experi-ments. [22]However, before being experimentally observed in He-A, the HQVs were first observed in another topo-logical phase of He – the polar phase [23]. The rea-son for that is that in He-A the spin-orbit interac-tion chooses the preferrable orientation for the vector ˆ d describing the spin degrees of freedom of the orderparameter. This leads to formation of a soliton inter-polating between two degenerate vacua with ˆ d = ˆ l and ˆ d = − ˆ l . The energy of soliton prevents the nucleationof the Alice strings in He-A.In contrast, in the polar phase the spin-orbit inter-action can be controlled not prohibit the formation ofHQVs. In the absence of magnetic field, or if the field isalong the nafen strands the spin-orbit interaction doesnot lead to formation of the solitons attached to thespin vortices. As a result the half-quantum vorticesbecome emergetically favorable and appear in the ro-tating cryostat if the sample is cooled down from thenormal state under rotation.Nevertheless the solitons help to observe the Alicestring first in polar phase and after that in the polardistorted A-phase. In the polar phase, when the ori-entation of the magnetic field is tilted with respect toaerogel strands, the spin-orbit interaction generates thesolitons between the half-quantum vortices. But theAlice strings are strongly pinned by the nafen strands,and the soliton cannot shrink, see Fig. 2. The HQVsare identified due to peculiar dependence of the NMRfrequency shift on the tilting angle of magnetic field[23]. The NMR experiments also allow measure thedensity of the Alicie string by measuring the solitondensity.Due to the strong pinning, the Alice strings formedin the polar phase by rotation of the superfluid or bythe Kibble-Zurek mechanism, survive the transition tothe He-A (actually to the distorted A-phase) [24].
4. HPD AND KLS WALL BETWEEN ALICESTRINGS He-B as KLSwall bounded by Alice strings
The mass vortices in He-B are presented in severalforms. In particular, a pair of spin-mass vortices mayform a molecule, where the soliton serves as chemicalbond. As a result one obtains the doubly quantizedvortex, i.e. with N = 2 circulation quanta, see Fig.1. Such vortex molecules have been also identified inHPD spectroscopy [18, 19].The "conventional" N = 1 vortex has also an un-usual structure in He-B. Already in the first experi-ments with rotating He-B the first order phase transi-tion has been observed, which has been associated withthe transition inside the vortex core [25]. It was sug-gested that at the transition the vortex core becomesnon-axisymmetric, i.e. the axial symmetry of the vor-tex is spontaneously broken in the vortex core [26, 27].This was confirmed in the further experiments wherethe coherently precessing magnetization was used [28]. . E. Volovik ЖЭТФ
Fig. 3.
The vortex in He-B with the non-axisymmetriccore as a pair of Alice strings connected by Kibble-Lazarides-Shafi wall. The HPD with its coherent pre-cession of magnetization is used to twist the core. Thevortex with twisted core is analogous to Witten supercon-ducting string with the electric current along the stringcore [32].
In the weak coupling BCS theory, which is appli-cable at low pressure, such vortex can be consideredas splitted into two half-quantum vortices connectedby the domain wall [29, 30], which is the analog of theKibble-Lazarides-Shafi wall bounded by cosmic strings[31]. The separation between the half-quantum vorticesincreases with decreasing pressure.The phenomenon of the additional symmetry break-ing in the core of the topological defect has been alsodiscussed for cosmic strings [32]. The spontaneousbreaking of the electromagnetic U (1) symmetry in thecore of the cosmic string has been considered, due towhich the core becomes superconducting.For the He-B vortices, the spontaneous breakingof the SO (2) symmetry in the core leads to the Gold-stone bosons – the modes in which the degeneracy pa-rameter, the axis of anisotropy b of the vortex core,is oscillating. The homogeneous magnon condensate,the HPD state, has been used to study the structureand twisting dynamics of this non-axisymmetric core.The coherent precession of magnetization excites thevibrational Goldstone mode via spin-orbit interaction.Moreover, due to spin-orbit interaction the precessingmagnetization rotates the core around its axis with con-stant angular velocity. In addition, since the core waspinned on the top and the bottom of the container, itwas possible even to screw the core (see Figs. 3 and 4).Such a twisted core corresponds to the Witten super-conducting string with the electric supercurrent alongthe core. The rigidity of twisted core differs from thatof the straight core, which is clearly seen in HPD ex-periments, see Fig. 4.Oscillations of the vortex core under coherent spinprecession also lead to the observed radiation of acous- Fig. 4.
HPD absorbtion as the function of the tilting an-gle η of magnetic field in case of the Witten strings withtwisted core (filled circles) and strings with untwisted core(open circles). The estimated critical angle at which thetilted field prevents twisting by HPD is in agreement withexperiment. tic magnon modes [33]. In the vortices with asymmetric cores the equilib-rium distance between the Alice strings is rather small.The essentially larger KLS walls between the stringshave been observed in the B-phase in nafen [24], seeFig. 5. It appeared that the Alice strings formed inthe polar phase by rotation of the superfluid or by theKibble-Zurek mechanism, survive the transition to the He-B (actually to the distorted B-phase). They re-main pinned, in spite of the formation of the KLS wallsbetween them.This allows us to study the unque properties of theKLS wall. In particular, the KLS wall separates twodegenerate cvacua with different signs of the tetrad de-terminant, and thus between the "spacetime" and "an-tispacetime" [34]. ЭТФ
Landau Institute p ( b a r ) N =1/2 d - field : s o lit on K L S w a ll v s Fig. 5.
The Alice string terminating the Kibble-Lazarides-Shafi wall in the polar distorted B-phase in nematicalaerogel. Due to the pinning of Alice string by the are-ogel strands the KLS wall can be arbitrarily long: thewall tension is unable to unpin the string. In addition tothe KLS wall there is also the soliton tail of the string.As a result one has the triple object: KLS wall + Alicestring + soliton.
5. COMBINED OBJECTS TO BE OBSERVED
In the chiral superfluid, the superfluid velocity v s of the chiral condensate is determined not only by thecondensate phase Φ , but also by the orbital triad ˆ e , ˆ e and ˆ l : v s = (cid:126) m (cid:0) ∇ Φ + ˆ e i ∇ ˆ e i (cid:1) , (6)where m is the mass of the He atom. As distinct formthe non-chiral superfluids, where the vorticity is pre-sented in terms of the quantized singular vortices withthe phase winding ∆Φ = 2 π N around the vortex core,in He-A the vorticity can be continuous. The contin-uous vorticity is represented by the texture of the unitvector ˆ l according to the Mermi-Ho relation: [35] ∇ × v s = (cid:126) m e ijk ˆ l i ∇ ˆ l j × ∇ ˆ l k . (7)Experimentally the continuous vorticity is typicallyobserved in terms of skyrmions (or the Anderson-Toulouse-Chechetkin vortices [36, 37]), see the upperpart of Fig. 8. Each skyrmion has N = 2 quanta ofcirculation of superfluid velocity. The skyrmion can bealso presented as the combination of two merons with N = 1 each.In 1994 a new type of continuous vorticity has beenobserved in He-A – the vortex texture in the form ofthe vortex sheets [38–40], see Fig. 6 top with a singlevortex sheet in container. Vortex sheet is the topo-logical soliton with kinks, each kink representing the v s v s N = 1 N = 1 N = 1 N = 1 N = 1 Fig. 6. top : Typical vortex sheet in He-A in rotating con-tainer. It mimics the system of the equidistant cylindricalvortex sheets suggested by Landau and Lifshitz for thedescrption of the rotating superfluid [43]. bottom : Theelement of the vortex sheet in He-A. The vortex sheetis the soliton, which contains kinks in terms of merons.Each meron has circulation quantum N = 1 . There aredifferent scenarios in which the vortex sheets with differ-ent geometries are prepared in the experiments (see Ref.[40]). l - field: N =1 N =1 N =1 N =1 N =1 N =1 Fig. 7. left : The multi-quantized vortex can be stabilizedas the closed vortex sheet: cylindrical soliton with merons[41]. The tension of the soliton is compensated by repul-sion of vortices (merons). right : The cosmic analog ofthis composite object: cosmic necklace [53]. Monopolesand/or antimonopoles are joined together by flux tubes. . E. Volovik ЖЭТФ
Fig. 8.
Skyrmion in the A-phase splits into two merons.Each meron is terminated by boojum – the point topolog-ical objects, which lives at the interface between A-phaseand B-phase. Boojum also plays the role of the Nambumonopole, which terminates the string - the N = 1 vortexon the B-side of the interface. continuous Mermin-Ho vortex with N = 1 circulationof superfluid velocity, Fig. 6 bottom .In principle, using the vortex sheet one may con-struct the continuous vortices with arbitrary even num-ber N = 2 k circulation quanta. This is the solitonforming the closed cylindrical surface, which contains N "quarks" – merons, [41,42] see Fig. 7 left for N = 6 .However, such multi-quantum vortices are still waitingfor their observation. Another object which is waiting for its observationin He-A is the vortex terminated by hedgehog [44,45].This is the condensed matter analog of the electroweakmagnetic monopole and the other monopoles connectedby strings [46]. The hedgehog-monopole, which termi-nates the vortex, exists in particular at the interfacebetween He-A and He-B. The topological defects liv-ing on the surface of the condensed matter system or atthe interfaces are called boojums [47]. They are classi-fied in terms of relative homotopy groups [48]. Boojumsterminate the He-B vortex-strings with N = 1 . Theboojums do certainly exist on the surface of rotating He-A and at the interface between the rotating He-Aand He-B [49], see Fig. 8. However, at the momenttheir NMR signatures are too weak to be resolved inNMR experiments in He. But the vortex terminatedby the hedgehog-monopole was observed in cold gases[50].The HPD state has its own topological defects [51],and among them are the spin and orbital monopolesconnected by string in Fig. 9.
Fig. 9.
Spin and orbital hedgehogs in magnon BEC (HPD)connected by string from Ref. [51].
Fig. 10.
Two dimensional lattice of monopoles (hedge-hgogs in the ˆ l -field) joined together by Alice strings (half-quantum vortices). Each monopole is the source or sinkof 4 strings. In particle physics the monopoles terminatingstrings are called Nambu monopoles [52]. Severalmonopoles connected by strings may form the multi-monopole objects, such as necklace in Fig. 7 ( right )[53]. This is similar to the vortex sheet necklace in Fig.7 ( left ).In He-A the analogs of Nambu monopoles and Al-ice strings may form the more complex combinations.This is because the monopole serves as a source or sinkof N = 2 circulation quanta, and thus can be the termi-nation point of 4 Alice strings with N = 1 / each. Thisin particular allows construct the 2D and 3D lattices ofmonopoles, in Fig. 10 and in Fig. 11 correspondingly. ЭТФ
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Fig. 11.
Three dimensional lattice of monopoles (on sitesA) and anti-monopoles (on sites B), which are joined to-gether by Alice strings (half-quantum vortices).
6. CONCLUSION
Here we considered several types of the topologicalconfinement. The composite topological objects wereexperimentally observed in superfluid He by usingthe unique phenomenon of HPD – the spontaneouslyformed coherent precession of magnetization discoveredby the Borovik-Romanov group in Kapitza Institute.With HPD spectroscopy, two key objects have beenidentified in He-B: spin-mass vortex [ Z spin vortex+soliton+mass vortex] and non-axisymmetric vortex [Al-ice string + Kibble-Lazarides-Shafi wall + Alice string].One may expect the other more complicated examplesof the topological confinement of the objects of differentdimensions. The complicated composite objects, suchas nexus, live also in the momentum space of topolog-ical materials [54]. Acknowledgements . This work has been sup-ported by the European Research Council (ERC) underthe European Union’s Horizon 2020 research and inno-vation programme (Grant Agreement No. 694248). Ithank Q. Shafi for the discussions that ultimately led to this article.
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