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Volovik, G. E.Topological Superfluids

Published in:Journal of Experimental and Theoretical Physics

DOI:10.1134/S106377611910011X

Published: 01/10/2019

Document VersionPeer reviewed version

Please cite the original version:Volovik, G. E. (2019). Topological Superfluids. Journal of Experimental and Theoretical Physics, 129(4), 618-641. https://doi.org/10.1134/S106377611910011X

https://doi.org/10.1134/S106377611910011X

https://doi.org/10.1134/S106377611910011X

618

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2019, Vol. 129, No. 4, pp. 618–641. © Pleiades Publishing, Inc., 2019.Russian Text © The Author(s), 2019, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2019, Vol. 156, No. 4, pp. 700–706.

Topological SuperfluidsG. E. Volovika,b

aLow Temperature Laboratory, Aalto University, P.O. Box 15100, Aalto, FI-00076 FinlandbLandau Institute for Theoretical Physics, Chernogolovka, 142432 Russia

e-mail: [email protected] April 4, 2019; revised April 4, 2019; accepted April 4, 2019

Abstract—There are many topological faces of the superfluid phases of 3He. These superfluids contain vari-ous topological defects and textures. The momentum space topology of these superfluids is also nontrivial,as well as the topology in the combined (p, r) phase space, giving rise to topologically protected Dirac, Weyland Majorana fermions living in bulk, on the surface and within the topological objects. The nontrivial topol-ogy lead to different types of anomalies, which extended in many different directions the Landau-Khalat-ninkov theory of superfluidity.

DOI: 10.1134/S106377611910011X

1. INTRODUCTIONSuperfluid phases of 3He discovered in 1972 [1]

opened the new area of the application of topologicalmethods to condensed matter systems. Due to themulti-component order parameter which character-izes the broken symmetries in these phases, there aremany inhomogeneous objects—textures and defects inthe order parameter field—which are protected bytopology and are characterized by topological charges.Among them there are quantized vortices, skyrmionsand merons, solitons and vortex sheets, monopolesand boojums, Alice strings, Kibble walls terminated byAlice strings, spin vortices with soliton tails, etc. Someof them have been experimentally identified andinvestigated [2–5], the others are still waiting for theircreation and detection.

The real-space topology, which is responsible forthe topological stability of textures and defects, hasbeen later extended to the topology in momentumspace, which governs the topologically protectedproperties of the ground state of these systems. Thisincludes in particular the existence of the topologicallystable nodes in the fermionic spectrum in bulk and/oron the surface of superfluids [6, 7]. It appeared thatthe superfluid phases of liquid 3He serve as the cleanexamples of the topological matter, where themomentum-space topology plays an important role inthe properties of these phases [8–11]. The further nat-ural extension was to the combined phase-spacetopology [12], which in particular describes the robustproperties of the spectrum of fermionic states local-ized on topological defects.

In bulk liquid 3He there are two topologically dif-ferent superfluid phases, 3He-A and 3He-B [13]. One

is the chiral superfluid 3He-A with topologically pro-tected Weyl points in the quasiparticle spectrum. Inthe vicinity of the Weyl points, quasiparticles obey theWeyl equation and behave as Weyl fermions, with allthe accompanying effects such as chiral anomaly [14],chiral magnetic effect (CME), chiral vortical effect(CVE) [15], etc. The Adler-Bell-Jackiw equation,which describes the anomalous production of fermi-ons from the vacuum [16–18], has been verified inexperiments with skyrmions in 3He-A [19]. Weyl fer-mions have been reported to exist in the topologicalsemiconductors, which got the name Weyl semimetals[20–27], see reviews [28–30]. The possible manifesta-tion of the chiral anomaly in these materials is underdiscussion [31].

Another phase is the fully gapped time reversalinvariant superfluid 3He-B. It has topologically pro-tected gapless Majorana fermions living on the surface(see reviews [32, 33] on the momentum space topol-ogy in superfluid 3He).

The polar phase of 3He has been stabilized in 3Heconfined in the nematically ordered aerogel [34–37].It is the time reversal invariant superfluid, which con-tains Dirac nodal ring in the fermionic spectrum [38].

2. TOPOLOGICAL DEFECTS IN REAL SPACEThe classification of the topological objects in the

order parameter fields revealed the possibility of manyconfigurations with nontrivial topology, which aredescribed by the homotopy groups [39–41] and by therelative homotopy groups [42, 43]. Some of the topo-logical defects and topological textures are shown inFig. 1.

1 1

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 129 No. 4 2019

TOPOLOGICAL SUPERFLUIDS 619

All three superfluid phases discussed here are thespin-triplet p-wave superfluids, i.e. the Cooper pairhas spin S = 1 and orbital momentum L = 1. The orderparameter is given by 3 × 3 matrix Aαi (see Eq. (11)),which transforms as a vector under SO(3)S spin rota-tions (first index) and as a vector under SO(3)L orbitalrotations (second index). The order parameter Aαi

comes as the bilinear combination of the fermionicoperators, Aαi ∝ ⟨ψσα∇iψ⟩, where σα are Pauli matri-ces for the nuclear spin of 3He atom. Note that similarorder parameter appears in the so-called spinor quan-

tum gravity, where it describes the emergent tetradfield [44–48].

2.1. Chiral Superfluid 3He

In the ground state of 3He-A the order parametermatrix has the form

(1)

where is the unit vector of the anisotropy in the spinspace due to spontaneous breaking of SO(3)S symme-try; and are mutually orthogonal unit vectors; and is the unit vector of the anisotropy in the orbital

space due to spontaneous breaking of SO(3)L symme-try. The -vector also shows the direction of the orbitalangular momentum of the chiral superfluid, whichemerges due to spontaneous breaking of time reversalsymmetry. The chirality of 3He-A has been probed inseveral experiments [49–51].

In the chiral superfluid the superfluid velocity vs ofthe chiral condensate is determined not only by thecondensate phase Φ, but also by the orbital triad , and :

(2)

where m is the mass of the 3He atom. As distinct formthe non-chiral superfluids, where the vorticity is pre-sented in terms of the quantized singular vortices withthe phase winding ΔΦ = 2π around the vortex core,in 3He-A the vorticity can be continuous. The contin-uous vorticity is represented by the texture of the unitvector according to the Mermi-Ho relation [55]:

(3)

Vorticity is created in rotating cryostat. In 3He-A,the continuous textures are more easily created thanthe singular objects with the hard core of the coher-ence length size ξ, which formation requires overcom-ing of large energy barrier. That is why the typicalobjects which appear under rotation of cryostat with3He-A is the vortex-skyrmion. It is the continuous tex-ture of the orbital -vector in Fig. 1a without any sin-gularity in the order parameter fields. This texture rep-resents the vortex with doubly quantized ( = 2) cir-culation of superfluid velocity around the texture,

⋅ vs = κ, where κ = h/2m is the quantum of cir-culation [56, 57]. The vortex-skyrmions have beenidentified in rotating cryostat in 1983 [58, 59]. Theyare described by two topological invariants, in terms ofthe orbital vector and in terms of the spin nematicvector :

Φα α= Δ + = ×1 2 1 2

ˆ ˆ ˆ ˆˆ ˆ( ), ,i i ii AA e d e ie l e e

d

1e 2el

l

1e 2el

= ∇Φ + ∇s 1 2ˆ ˆ( ),

2i i

mv e e

1

l

∇ × = ∇ × ∇s

ˆ ˆ ˆ .4 ijk i j ke l l l

mv

l

1

∫ dr 1

1

ld

Fig. 1. (Color online) Some topological oblects in topo-logical superfluids. (a) Vortex-skyrmions with two quantaof circulation ( = 2) in the chiral superfluid 3He-A (seealso Fig. 2b). (b) Interface between 3He-A and 3He-B inrotation: lattice of skyrmions with = 2 in 3He-A trans-forms to the lattice of singly quantized vortices ( = 1) in3He-B. Each skyrmion with = 2 splits into two meronswith = 1 (Mermin-Ho vortices). The Mermin-Ho vor-tex terminates on the boojum—the surface singularitywhich is the analog of the Dirac monopole terminating the3He-B string (the =1 vortex). (c) Vortex sheet in 3He-A.(d) Elements of the vortex sheet: merons with = 1 withintopological soliton in 3He-A. (e) Hedgehog-monopole inthe -vector field in 3He-A. (f) Singly quantized vortex( = 1) with spontaneously broken axial symmetry in3He-B as a pair of half-quantum vortices ( = 1/2) con-nected by non-topological (dark) soliton [52, 53]. Suchvortex has been identified due to Goldstone mode associ-ated with the spontaneously broken axial symmetry of thevortex core—the twist oscillations of the vortex core prop-agating along the vortex line. This figure is also applicableto the = 2 vortex in 3He-B, which consists of two com-posite objects—spin-mass vortices connected by the topo-logical soliton [54], see Fig. 4g. Half-quantum vortices inthe polar phase of 3He, which in a tilted magnetic field areconnected by the topological spin soliton [4]. Red arrowsshow direction of magnetic field, and blue arrows—direc-tion of spin-nematic vector , which rotates by π aroundeach half-quantum vortex.

HQV

(a)

3He-A

3He-B

vs

(b)

(f)

(g)

(c)

(d)(e)

21

1

1 1

1

1

1

1

1

d1

1

1

d

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domain wall - analog of the Kibble wall bounded by cosmoc strings

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(4)

(5)

The last equality in Eq. (4) shows the connectionbetween the topological charge of the orbital textureand the circulation of superfluid velocity around it,which follows from the Mermin-Ho relation (3).

In a high magnetic field the vortex lattice consistsof isolated vortex-skyrmions with = 2, ml = 1 andmd = 0, see Fig. 2. In the low field, when the magneticenergy is smaller than the spin-orbit interaction, thevortex-skyrmion with md = ml = 1 becomes more pref-erable. The first order topological transition, at whichthe topological charge md of the skyrmion changesfrom 0 to 1, has been observed in acoustic experiments[60]. Finally, when magnetic field is close to zero, theskyrmions are not isolated. They form the periodicvortex texture represented in terms of merons—thecontinuous Mermin-Ho vortices with ml = 1/2 and

= 1 each. The elementary cell of the vortex struc-ture of rotating 3He-A contains four merons, seeFig. 3. It has topological charge ml = md = 2 and

= 4. The isolated skyrmion in the non-zero field inFig. 1a can be represented as the bound state of twomerons in Fig. 2.

In 1994 new type of continuous vorticity has beenobserved in 3He-A—the vortex texture in the form of

⎛ ⎞∂ ∂= ⋅ × =⎜ ⎟π ∂ ∂⎝ ⎠∫ˆ ˆ1 1ˆ ,

4 2lm dxdyx yl ll 1

⎛ ⎞∂ ∂= ⋅ ×⎜ ⎟π ∂ ∂⎝ ⎠∫ˆ ˆ1 ˆ .

4dm dxdyx yd dd

1 1

1

1

1

the vortex sheets, Fig. 1c [62, 63]. Vortex sheet is thetopological soliton with kinks, each kink representingthe continuous Mermin-Ho vortex with = 1,Fig. 1d.

In addition to continuous vortex textures, the rotat-ing state of 3He-A may consist of the singular vorticeswith = 1. They are observed in NMR experiments ifone starts rotation in the normal phase and then coolsdown to the A-phase [64]. In principle the samescheme can lead to the formation of the half-quantumvortices (HQVs), which have been suggested to exist inthin films of 3He-A [39]. The half-quantum vortexrepresents the condensed matter analog of the Alicestring in particle physics [65]. The half-quantum vor-tex is the vortex with fractional circulation of super-fluid velocity, = 1/2. It is topologically confinedwith the fractional spin vortex, in which d changes signwhen circling around the vortex:

(6)

When the azimuthal coordinate φ changes from 0 to 2πalong the circle around this object, the vector ( )changes sign and simultaneously the phase Φ changesby π, giving rise to = 1/2. The order parameter (6)

1

1

1

( )Φ φφ φ= +ˆ( ) /2ˆ ˆ ˆ ˆ( ) cos sin ,2 2

i ie erd r x y

d r

1

Fig. 2. (Color online) (a) Rotating state in 3He-A in highmagnetic field as an array of isolated skyrmions with ml =1 and = 2 in Fig. 1a. (b) Such skyrmion contains twomerons—circular and hyperbolic. (c) NMR signature ofvortex skyrmion—the satlellite peak in the NMR spec-trum. The type of the skyrmion is identified by the positionof the satellite peak, the peak amplitude reflects the num-ber of skyrmions in the sample. In the absence of magneticfield the skyrmions merge forming the periodic texture inFig. 3 with = 4 in the elementary cell.

(a)

(b)

hyperbolic meron circular meron

vortex lattice-lattice of skyrmions

0.05

00N

MR

Abs

orpt

ion

4 8 12Δν, kHz

16

vortex peakmain peak

l-field

(c)

1

1

11

1

Fig. 3. (Color online) Elementary cell of meron lattice inrotating superfluid 3He-A in the absence of magnetic field(adapted from [61]). The unit cell consists of four merons,two of which are circular and two are hyperbolic. In eachmeron the orbital vector covers half a sphere, and accord-ing to Eq. (3) each meron represents a vortex with a singlequantum of circulation, = 1. Thus the unit cell carriestopological charge ml = 2 and thus the circulation number

= 4. In the circular vortex-meron the orbital vector || Ωin the center, where Ω is the angular velocity of the rotatingcryostat. In the hyperbolic vortex-meron || –Ω in thecenter. When the magnetic field is switched on, this statetransforms to an array of isolated vortex-skyrmions with

= 2 each in Fig. 1a. Such skyrmion contains twomerons, see Fig. 2b.

^ ^, = m + n ^

,

^

m

n

l

1

1 l

l

11



remains continuous along the circle. While a particlethat moves around an Alice string f lips 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 [2].

In superfluid 3He the HQVs have been stabilizedonly recently and in a different phase—in the polarphase of 3He confined in aerogel [4], see Section 2.3.HQVs have been identified in NMR experiments dueto the topological soliton, which is attached to the spinvortex in tilted magnetic field because of spin-orbitinteraction, Fig. 1g.

Another object which is waiting for its observationin 3He-A is the vortex terminated by hedgehog [66,67]. This is the condensed matter analog of the elec-troweak magnetic monopole and the other monopolesconnected by strings [68]. The hedgehog-monopole,which terminates the vortex, exists in particular at theinterface between 3He-A and 3He-B, Fig. 1f. Thetopological defects living on the surface of the con-densed matter system or at the interfaces are calledboojums [69]. They are classified in terms of relativehomotopy groups [43]. Boojums in Fig. 1f terminatethe 3He-B vortex-strings with = 1. Though boojumsdo certainly exist on the surface of rotating 3He-A andat the interface between the rotating 3He-A and 3He-B, at the moment their NMR signatures are too weakto be resolved in NMR experiments. Experimentallythe vortex terminated by the hedgehog-monopole wasobserved in cold gases [70].

1

2.2. Superfluid 3He-B

In the ground state of 3He-B the order parametermatrix has the form

(7)where Rαi is the real matrix of rotation, RαiRαj = δij.

Vorticity in the non-chiral superfluid is always sin-gular, but in 3He-B it is also presented in several forms.Even the =1 vortex, where Φ( ) = φ, has an unusualstructure of the singular vortex core. Already in thefirst experiments with rotating 3He-B the first orderphase transition has been observed, which has beenassociated with the transition inside the vortex core[71]. It was suggested that at the transition the vortexcore becomes non-axisymmetric, i.e. the axial sym-metry of the vortex is spontaneously broken in the vor-tex core [72, 73]. This was confirmed in the furtherexperiments, where the Goldstone mode associatedwith the symmetry breaking was identified—the twistoscillations in Fig. 1f and Fig. 5 propagating along thevortex line [74]. In the weak coupling BCS theory,which is applicable at low pressure, such vortex can beconsidered as splitted into two half-quantum vorticesconnected by non-topological soliton [52, 53].

On the other side of the transition, at high pressure,the structure of the vortex core in 3He-B is axisym-metric, but it is also nontrivial. In the core the discretesymmetry is spontaneously broken, as a result the coredoes not contain the normal liquid, but is occupied bythe chiral superfluid—the A-phase of 3He [2]. The A-phase core in axisymmetric vortex results in theobserved large magnetization of the core comparedwith that in the non-axisymmetric vortex [75]. Laterwe shall discuss the Weyl fermions living within such acore in Sections 4.1 and 6.5.

The phenomenon of the additional symmetrybreaking in the core of the topological defect has been

Φα α= Δ ,i

i B iA e R

1 r

3

Fig. 4. Topological oblects observed in superfluid 3He-Bin rotating cryostat. Conventional mass current vorticeswith = 1 form a regular structure. If the number of vor-tices is less than equilibrium number for a given rotationvelocity, vortices are collected in the vortex cluster with thevortex free region outside the cluster, where the mass cur-rent is circulating. Spin vortices, which have the solitontail, are stabilized being pinned by the cores of the masscurrent vortices. They form the composite object—thespin-mass vortex with the soliton tail [54]. A single spin-mass vortex is stabilized at the periphery of the cluster bythe combined effect of soliton tension and the Magnusforce acting on the mass vortex from the super-flow in thevortex-free region. Pair of spin-mass vortices connected bysoliton forms the doubly quantized = 2 mass vortex.

vesselboundary

spin−massvortex

cluster ofmass current

N = 1 vortices

doubly quantizedN = 2 vortex as pair

of spin−mass vorticesconfined by soliton

spin−massvortex

superflowregion

spin−massvortex

soliton

soliton

vs

2

1

1

Fig. 5. (Color online) Twist of the asymmetric vortex corewith spontaneously broken SO(2) symmetry observed in3He-B [74] is analogous to the superconducting electriccurrent along the Witten cosmic string with spontaneouslybroken U(1) symmetry. The second Higgs field H2 =

|H2| , which spontaneously appears in the core of theWitten string, breaks the electromagnetic U(1) symmetrygiving rise to the 1D superconductivity with the current j ∝

∂zΦ2 concentrated in the string core [76].

Witten superconducting string

secondHiggs field

Higgs field

twist = superconducting currentalong Witten string core

H

r

H2

Φ2ie

z

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one can describe this object as the vortex, which splits into two half-quantum vortices (Alice strings) connected by the Kibble wall bounded by strings

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also discussed for cosmic strings [76]. The sponta-neous breaking of the electromagnetic U(1) symmetryin the core of the cosmic string has been considered,due to which the core becomes superconducting withsuper-current along the core. The string with thesuperconducting electric current is analogous to theasymmetric vortex with twisted core, see Fig. 5.

The topology of 3He-B also admits existence ofspin vortices, Z2 topological defects of the matrix Rαi.Due to spin-orbit interaction, which violates theinvariance under SO(3)S spin rotations, the spin vortexgives rise to the topological soliton attached to the vor-tex line, similar to that in Fig. 1 for the polar phase.That is why, if the spin vortex appears in the cell, it ispushed to the wall of the container by the soliton ten-sion and disappears at the wall. However, the spin vor-tex survives if it is pinned by the conventional vortex(mass current vortices) with = 1 and forms the com-posite object—the spin-mass vortex. Experimentallytwo types of composite objects have been identified in3He-B, see Fig. 4. (i) The other end of the soliton is atthe wall of container. (ii) The = 2 vortex is formedwhich consists of two spin-mass vortices connected bysoliton [54].

2.3. Polar Phase of Superfluid 3He

Polar phase of superfluid 3He has been stabilized ina nematically ordered aerogel with nearly parallelstrands (nafen) [34–36]. In the ground state of thepolar phase the order parameter matrix has the form

(8)where axis z is along the nafen strands. Topology ofpolar phase in nafen suggests existence of = 1 masscurrent vortices, Z2 spin vortices and the half-quantumvortices. The latter is the combination of the fractional

= 1/2 mass vortex and the fractional spin vortex inEq. (6). The spin-orbit interaction in the polar phaseis more preferable for the half-quantum vortices thanin 3He-A. In the absence of magnetic field, or if thefield is along the nafen strands the spin-orbit interac-tion does not lead to formation of the solitons attachedto the spin vortices. As a result the half-quantum vor-tices become emergetically favorable and appear in therotating cryostat if the sample is cooled down from thenormal state under rotation. The HQVs are identifieddue to peculiar dependence of the NMR frequencyshift on the tilting angle of magnetic field [4]. Figure1g shows a pair of half-quantum vortices in transversemagnetic field (red arrows). Blue arrows show the dis-tribution of the nematic vector of the spin part of theorder parameter in the polar phase.

Later it was found [5] that the half-quantumvorices survive the phase transition to 3He-B, wherethe half-quantum vortex is topologically unstable. Inthe B-phase the half-quantum vortices pinned by the

1

1

Φα α= Δ ˆ ˆ ,i i

i PA e d z

1

1

4

d

5

strands of nafen become the termination lines of thenon-topological domain walls - the analog of Kibblecosmic walls [77]. In 3He-B, the Kibble wall separatesthe states with different tetrad determinant, and thusbetween the “spacetime” and “antispacetime” [78].More on spacetime in cosmology and condensed mat-ter see [79–85].

3. MOMENTUM SPACE TOPOLOGY

The topological stability of the coordinate depen-dent objects—defects and textures - is determined bythe pattern of the symmetry breaking in these super-fluids. Now we shall discuss these three phases ofsuperfluid 3He from the point of view of momentum-space topology, which describes the topological prop-erties of the homogeneous ground state of the super-fluids. These superfluids represent three types of topo-logical materials, with different geometries of thetopologically protected nodes in the spectrum of fer-mionic quasiparticles: Weyl points in 3He-A, Diraclines in the polar phase and Majorana nodes on thesurface of 3He-B.

The properties of the fermionic spectrum in thebulk or/and on the surface of superfluids is deter-mined by the topological properties of the Bogoli-ubov-de Gennes Hamiltonian

(9)

or by the Green’s function

(10)

+

⎛ ⎞Δ −= =⎜ ⎟⎜ ⎟Δ −⎝ ⎠

2 2ˆ( ) ( )( ) , ( ) ,ˆ 2 *( ) ( )

Fp pHm

p pp p

p p

e

e

e

− ω = ω −1( , ) .G i Hp

Fig. 6. In 3He-B, the half-quantum vortex (analog of Alicestring) looses its topological stability and becomes the ter-mination line of a non-topological domain wall—the Kib-ble wall [5]. In terms of the tetrads, the Kibble wall sepa-rates the states with different tetrad determinant, and thusbetween the “spacetime” and “antispacetime” [78]. Thereare two roads to antispacetime: the “safe” route around theAlice string (along the contour C1) or “dangerous” routealong C2 across the Kibble wall.

Spacetime

Kibblewall

Antispacetime

Alice string1/2 vortex

C2

C1

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hedgehogs

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n

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t

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anti

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separates



For the spin triplet p-wave superfluid 3He the gapfunction is expressed in terms of the 3 × 3 orderparameter matrix Aαi:

(11)

The topologically stable singularities of the Hamil-tonian or of the Green’s function in the momentum ormomentum-frequency spaces look similar to the real-space topology of the defects and textures, see Fig. 7.The Fermi surface, which describes the normal liquid3He and metals, represents the topologically stable 2Dobject in the 4D frequency-momentum space (px, py,pz, ω), is analogous to the vortex ring. The Weyl pointsin 3He-A represents the topologically stable hedgehogin the 3D p-space, Fig. 7b. The Dirac nodal line in thepolar phase of 3He is the p-space analog of the spinvortex, Fig. 7c. The nodeless 2D systems (thin films of3He-A and the planar phase) and the nodeless 3D sys-tem—3He-B—are characterized by the topologicallynontrivial skyrmions in momentum space in Fig. 7e.

α αΔ = σˆ( ) .ii

F

pAp

p

6

6

1

3.1. Normal Liquid 3He

The normal state of liquid 3He belongs to the classof Fermi liquids, which properties at low energy aredetermined by quasiparticles living in the vicinity ofthe Fermi surface. The systems with Fermi surface,such as metals, are the most widespread topologicalmaterials in nature. The reason for that is that theFermi surface is topologically protected and thus isrobust to small perturbations. This can be seen on thesimple example of the Green’s function for the Fermigas at the imaginary frequency:

(12)

The Fermi surface at p = pF exists at positive chemical

potential, with /2m = μ. The topological protectionis demonstrated in Fig. 7a for the case of the 2D Fermigas, where the Fermi surface is the line p = pF in(px, py)-space. In the extended (ω, px, py)-space thisgives rise to singularity in the Green’s function on theline at which ω = 0 and p = pF, where the Green’sfunction is not determined. Such singular line inmomentum-frequency space looks similar to the vor-tex line in real space: the phase Φ(p, ω) of the Green’sfunction G(p, ω) = |G(p, ω)|eiΦ(p, ω) changes by 2πaround this line. In general, when the Green’s func-tion is the matrix with spin or/and band indices, theinteger valued topological invariant—the windingnumber of the Fermi surface—has the following form[15]:

(13)

Here the integral is taken over an arbitrary contour Caround the Green’s function singularity in the D + 1momentum-frequency space. Due to nontrivial topo-logical invariant, Fermi surface survives the perturba-tive interaction and exists in the Fermi liquid as well.Moreover the singularity in the Green’s functionremains if due to interaction the Green’s function hasno poles, and thus quasiparticles are not well defined.The systems without poles include the marginal Fermiliquid, Luttinger liquid, and the Mott pseudogap state[92].

It is possible that in the Mott pseudogap state, thepoles of the Green’s function transform to zeroes ofthe Green’s function. The topological invariantremains the same, which is the reason why the Lut-tinger theorem is still valid [93, 94]. The particle den-sity of interacting fermions is equal to the volume inthe momentum space enclosed by the singular surfacewith the topological charge N = 1, irrespective of therealization of the singularity. As distinct from the pole,the zero in the Green’s function is invisible, so that thepole region of the Fermi surface looks as the Fermiarc, see Fig.8b.

− ⎛ ⎞ω = ω − − μ⎜ ⎟⎝ ⎠

21( , ) .

2pG im

p

2Fp

−= ω ∂ ωπ∫

1( , ) ( , ).2 l

C

dlM G Gi

tr p p

Fig. 7. (Color online) Topological materials as configura-tions in momentum space. (a) Fermi surface in normal liq-uid 3He is topologically protected, since the Green’s func-tion has singularity in the form of the vortex ring in the (p,ω)-space. (b) Weyl point in 3He-A as the hedgehog in p-space. It can be also represented as the p-space Diracmonopole with the Berry magnetic f lux. (c) The polarphase has the Dirac nodal line in p-space—the counterpartof the spin vortex in real space. (d) The = ∞ singularvortex in chiral superfluid 3He-A has the one-dimensionalflat band terminated by the projections of the Weyl pointsto the vortex line [86–88]. In real space it has analogy withthe Dirac monopole terminating the Dirac string. (e) Sky-rmion configurations in p-space describe the fully gappedtopological superfluids. The 2D skyrmion describes thetopology of the 3He-A state in a thin film and the topologyof 2D planar phase of 3He [89–91]. The 3D p-space skyr-mion describes the superluid 3He-B [7, 10, 11, 32].

(a) (b) (c)

(d) pz

px pxpx

py(pz)py

py

pz pz

pF pF

Δθp = 2πΔΦ = 2π H = +cσ ⋅ p

ω

(e)

3He-B

p-space skyrmion

flat band on vortexin Weyl superfluid

Fermi surfaceas vortex ringin p-space

Weyl pointhedgehogin p-space

Diracnodal ring

normal 3He polar phaseof 3He

3He-A

3He-A

1

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It

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ring

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The Fermi surface may disappear in the topologi-cal quantum phase transition, when the chemicalpotential μ crosses zero: the vortex ring object shrinksto the point at μ = 0 and disappears at μ < 0 if the pointis not protected by another topological invariant dis-cussed in Section 3.2. The Fermi surface may disap-pear also due to non-perturbative process of the sym-metry breaking phase transition, when the fermionicspectrum of the system is drastically reconstructed, asit happens under the transition from the normal to thesuperfluid state.

3.2. Weyl Superfluid 3He-A

Under the superfluid transition the 2 × 2 matrix ofthe normal liquid Green’s function with spin indicestransforms to the 4 × 4 Gor’kov Green’s function. Thesimplified Green’s function, which describes thetopology of the chiral superfluid 3He-A, has the form:

(14)

The Pauli matrices τ1, 2, 3 and σx, y, z correspond to theBogoliubov-Nambu spin and ordinary spin of 3Heatom respectively; μ = /2m as before; and theparameter c = ΔA/pF.

Instead of the Fermi surface, now there are twopoints in the fermionic spectrum, at K± = ±pF , wherethe energy spectrum is nullified and the Green’s func-tion is not determined at ω = 0. There are several waysof how to describe the topological protection of thesetwo points. In terms of the Green’s function there isthe following topological invariant expressed via inte-ger valued integral over the 3-dimensional surface σ

− ⎛ ⎞= ω+ τ − μ + ⋅ τ ⋅ + τ ⋅⎜ ⎟⎝ ⎠

21

3 1 1 2 2ˆ ˆ ˆ( )( ).

2pG i cm

σ d e p e p

2Fp

l

around the singular point in the 4-momentum spacepμ = (ω, p) [12, 15]:

(15)

If the invariant (15) is nonzero, the Green’s functionhas a singularity inside the surface σ, and this meansthat fermions are gapless. The typical singularitieshave topological charges N = +1 or N = –1. Close tosuch points the quasiparticles behave as right-handedand left-handed Weyl fermions [95] respectively, thatis why such point node in the spectrum is called theWeyl point. The isolated Weyl point is protected bytopological invariant (15) and survives when the inter-action between quasiparticles is taken into account.The Weyl points with the opposite charge N may can-cel each other, when they merge together at the quan-tum phase transition, if some continuous or discretesymmetry does not prohibit the annihilation.

In 3He-A, the topological invariants of the points atK(a) = ±kF are correspondingly N = +2 or N = –2: theWeyl points are degenerate over the spin of the 3Heatoms. Considering only single spin projection onecomes to the 2 × 2 Bogoluibov-Nambu Hamiltonianfor the spinless fermions:

(16)

where the vector function g(p) has the following com-ponents in 3He-A:

(17)

The Hamiltonian (16) is nullified at two pointsK(a) = ±pF , where p = pF and p ⋅ = p ⋅ =0. At thesepoints the unit vector (p) = g(p)/|g(p)| has the singu-larity of the hedgehog-monopole type in Fig. 7b,which is described by the dimensional reduction of theinvariant (15):

(18)

β μ ν

α − − −αβμν

σ

= ∂ ∂ ∂π ∫

1 1 12 .

24p p p

eN dS G G G G G Gtr

l

= ⋅ ( ),H τ g p

= ⋅ = ⋅ = − μ2

1 1 2 2 3ˆ ˆ, , .2pg g gm

e p e p

l 1e 2eg

σ

⎛ ⎞∂ ∂= ⋅ ⋅⎜ ⎟π ∂ ∂⎝ ⎠∫

ˆ ˆ1 ˆ ,8

iikl

k l

N e dSp pg gg

Fig. 8. (Color online) Due to interaction, in some parts ofthe Fermi surface, the poles in the Green’s function maytransform to zeroes. The topological charge of these partsof Fermi surface remain the same, but they become invisi-ble, while the other parts of the Fermi surface look asFermi arcs.

px

py(pz)

pF

ω Fermi surface:vortex ringin p-space

From pole of Green’s function to zero

non-interacting fermions

G =

N1 =

G = Z(iω + ε(p))

1

1 tr[ dlG∇lG−1] = 1

iω − ε(p)

4πi

ε < 0

strongly interacting fermions

pole

zero zeroes

zeroes

poles

occupiedlevels:

Fermi sea

poles

poles Fig. 9. (Color online) Weyl point as magnetic monopole inthe Berry phase field.

Berry phaseγ(C) = Φ0 = 2π

Dirac stringC

(unobservable)

Dirac2p2

pmonopole

inmomentum

space

Berry ‘magnetic’ field

H(p) = Φ0

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where σ now is the 2D spherical surface around thehedhehog.

The hedgehog has N = ±1 and it represents theBerry phase magnetic monopole [6, 15]. In the vicin-ity of the monopole the Hamiltonian can be expandedin terms of the deviation of the momentum p from theWeyl point at K(a) [14, 96, 97]:

(19)The emergent linear relativistic spectrum of Weyl fer-mions leads to the observed T4 behavior of the thermo-dynamic quantities [98]. Introducing the effectiveelectromagnetic field A(r, t) = pF (r, t) and effectiveelectric charge q(a) = ±1, one obtains

(20)Such Hamiltonian decsribes the Weyl fermions mov-ing in the effective electric and magnetic fields

(21)

and also in the effective gravitational field representedby the triad field (r, t). The effective quantum elec-trodynamics emerging in the vicinity of the Weyl pointleads to many analogs in relativistic quantum field the-ories, including the zero charge effect—the famous“Moscow zero” by Abrikosov, Khalatnikov and Lan-dau [99].

In the presence of the superfluid velocity andspacetime dependent chemical potential, the effectivespin connection emerges. It enters the long derivative

Dμ = ∂μ + (τατβ – τβτα) – iqAμ with the following

non-zero components: = m (r, t) and = μ(r,t). The continuous vorticity in Eq. (3) gives rise to thenonzero components of the curvature tensor: R12ik =m(∂i – ∂k ).

The Weyl points as the topologically protectedtouching point of two bands [100, 101] have been dis-cussed in semimetals [20, 22–27, 102–104]. Accord-ing to the bulk-edge and bulk-defect correspondence,the Weyl points in bulk may produce the Fermi arc onthe surface of the material [103, 104], and the f latband of fermionic excitations in the vortex core in Sec-tion 6.3.

In a different form the topological invariant for theWeyl point has been introduced for the massless neu-trino in 1981 [105]. The evidence of neutrino oscilla-tions does not exclude the possibility that in neutrinosector of particle physics instead of formation of theDirac mass there is a splitting of two Weyl points withbreaking of CPT symmetry [106].

In 1982 the topological invariant for the Weylpoints in 3He-A have been described in terms of theboojum living on the Fermi surface [107], the analogof the real-space boojum in Fig. 1b. As follows from

7

αα= τ − +( ) ( )( ) ....a i a

i iH e p K

l

αα= τ − +( ) ( )( ) ....a i a

i iH e p q A

= − ∂ = ∇ ×eff effˆ ˆ, ,F t Fp pE l B l

αie

αβμ

14

C

12iC vsi

120C

vsk vsi

Eq. (9) the node in the spectrum occurs when (p) =0 and the determinant of the gap function is zero,det (p) = 0. That is why the node represents thecrossing point of the surface (p) = 0 (the Fermi sur-face of the original normal state of liquid 3He) and theline where det (p) = 0. The latter is described by theinteger winding number of the phase Φ(p) of the deter-minant:

(22)around the nodal line of the determinant. The topo-logical charge N in Eq. (15) is nothing but the windingnumber of the phase Φ(p) when viewed from theregion outside the Fermi surface [107]. Such defini-tion of the topological charge N also coincides withthe Berry monopole charge, giving N = ±2 for theWeyl points at K(a) = ±pF .

In general, the state with the topological chargelarger than unity, |N| > 1, is unstable towards splittingto the states with nodes described elementary chargesN = ±1. In 3He-A the nodes with topological chargesN = ±2 are protected by the discrete Z2 symmetryrelated to spins.

3.3. Nodal Line Polar PhaseThe topology of the fermionic quasiparticles in the

polar phase of 3He is described by the following sim-plified Hamiltonian:

(23)

where c = ΔP/pF. This Hamiltonian is nullified when

pz = 0 and + = , i.e. the spectrum of quasi-particles has the nodal line in Fig. 7c. The nodal line isprotected by topology due to the discrete symmetry:the Hamiltonian (23) anticommutes with τ2, whichallows us to write the topological charge, see e.g.review [108]:

(24)

Here C is an infinitesimal contour in momentumspace around the line, which is called the Dirac line.The topological charge N in Eq. (24) is integer and isequal to 2 for the nodal line in the polar phase due tospin degeneracy. In a different form the invariant canbe found in [109].

Dirac lines exist in cuprate superconductors [110]and also in semimetals [111–118]. According to thebulk-edge and bulk-defect correspondence, the Diracline in bulk may produce the f lat band on the surfaceof the material [119–121] and the condensation of lev-els in the vortex core [122], which we discuss in Sec-tion 6.4. The flat band physics is important for the

e

Δe

Δ

ΦΔ = Δ ( )ˆ ˆdet ( ) |det ( )| ,ie pp p

l

⎛ ⎞= τ − μ + ⋅ τ⎜ ⎟⎝ ⎠

2

3 1ˆ( ) ,

2 zpH cpm

σ d

2xp 2

yp 2Fp

−= τ ∂π∫

12 ( ) ( ).

4 lC

dlH H Hi

tr p p

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construction of materials experiencing superconduc-tivity at room temperature [123].

3.4. Time Reversal Invariant Fully Gapped 3He-B3He-B belongs to the same topological class as the

vacuum of Standard Model in its present insulatingphase [124]. The topological classes of the 3He-Bstates can be represented by the following simplifiedBogoliubov–de Gennes Hamiltonian:

(25)

Here Rαi is the matrix of rotation; the phase of theorder parameter in Eq. (7) is chosen either Φ = 0 orΦ = π; we also included the effective mass m* to dis-cuss the possible topological quantum phase transi-tions. In the limit of heavy effective mass, 1/m* → 0,this model 3He-B Hamiltonian transforms to theHamiltonian for massive relativistic Dirac particleswith speed of light cB = ΔB/pF, and the mass parameterM = –μ. There are no nodes in the spectrum of fermi-ons: the system is fully gapped. Nevertheless 3He-B isthe topological superfluid. This can be seen from theinteger valued integral over the former Fermi surface[7]:

(26)

where Nd = ± depending on the sign in Eq. (25).In terms of the Hamiltonian the topological invari-

ant can be written as integral over the whole momen-tum space (or over the Brillouin zone in solids)

(27)

where K = τ2 is the matrix which anticommutes withthe Hamiltonian. One has NK = 2Nd due to spindegrees of freedom. For the interacting systems theGreen’s function formalism can be used. In the fullygapped systems, the Green’s function has no singular-ities in the whole (3+1)-dimensional space (ω, p).That is why we are able to use in Eq. (27) the Green’sfunction at ω = 0, which corresponds to the effectiveHamiltonian, Heff(p) = –G–1(0, p) [125].

In 3He-B, the K = τ2 symmetry is the combinationof time reversal and particle-hole symmetries. ForStandard Model the corresponding matrix K, whichanticommutes with the effective Hamiltonian andenters the invariant, is constructed from the γ-matri-ces: it is K = γ5γ0.

Figure 10 shows the phase diagram of topologicalstates of 3He-B in the plane (μ, 1/m*). The line

α α

⎛ ⎞= τ − μ + τ Δ ⋅⎜ ⎟⎝ ⎠

= ±

2

3 1ˆ( ),

2 *

ˆ ( ) .

B

ii

F

pHm

pd Rp

σ d p

p

⎛ ⎞∂ ∂= ⋅ ⋅⎜ ⎟π ∂ ∂⎝ ⎠∫

2

ˆ ˆ1 ˆ ,8

id ikl

k lS

N e dSp pd dd

− − −⎡ ⎤= ∂ ∂ ∂⎣ ⎦π ∫3 1 1 1

2 ,24 i j k

ijkK p p p

eN d pKH HH HH Htr

1/m* = 0 corresponds to the Dirac vacuum of massivefermions, whose topological charge is determined bythe sign of mass parameter M = –μ:

(28)

Finally, the point μ = 1/m* = 0 corresponds to themassless Dirac particle, where the Dirac node consistsof two Weyl nodes with opposite chirality, Na = ±Nd.

The real superfluid 3He-B lives in the corner of thephase diagram μ > 0, m* > 0, μ ≫ m* , which alsocorresponds to the limit ΔB ≪ μ of the weakly interact-ing gas of 3He atoms, where the superfluid state isdescribed by Bardeen-Cooper-Schrieffer (BCS) the-ory. However, in the ultracold Fermi gases with tripletpairing the strong coupling limit is possible near theFeshbach resonance [126]. When μ crosses zero thetopological quantum phase transition occurs, at whichthe topological charge NK changes from NK = 2 toNK = 0. The latter regime with trivial topology alsoincludes the Bose–Einstein condensate (BEC) oftwo-atomic molecules. In other words, the BCS-BECcrossover in this system is always accompanied by thetopological quantum phase transition, at which thetopological invariant changes.

There is an important difference between 3He-Band Dirac vacuum. The space of the Green’s functionof free Dirac fermions is non-compact: G has differentasymptotes at |p| → ∞ for different directions ofmomentum p. As a result, the topological charge ofthe interacting Dirac fermions depends on the regular-ization at large momentum. 3He-B can serve as regu-larization of the Dirac vacuum, which can be made inthe Lorentz invariant way [124]. One can see fromFig. 10, that the topological charge of free Dirac vac-uum has intermediate value between the charges of the3He-B vacua with compact Green’s function. On themarginal behavior of free Dirac fermions see [10, 15,127, 128].

The vertical axis separates the states with the sameasymptote of the Green’s function at infinity. Theabrupt change of the topological charge across theline, ΔNK = 2, with fixed asymptote shows that onecannot cross the transition line adiabatically. Thismeans that all the intermediate states on the line ofthis QPT are necessarily gapless. For the intermediatestate between the free Dirac vacua with opposite massparameter M this is well known. But this is applicableto the general case with or without relativistic invari-ance: the gaplessness is protected by the difference oftopological invariants on two sides of transition. Thegaplessness of the intermediate state leads also to thefermion zero modes at the interface between the bulkstates with different topological invariants, see Section6.1. For electronic materials this was discussed in[129].

= sign( ).KN M

2Bc

8

8

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4. COMBINED TOPOLOGY.EVOLUTION OF WEYL POINTS

According to Eq. (21), the effective (synthetic)electric and magnetic fields acting on the Weyl quasi-particles emerge if the position Ka of the ath Weylpoints depends on coordinate and time. The topolog-ical protection of the Weyl points together with topol-ogy of the spatial distribution of the Weyl points in thecoordinate space gives rise to the more complicatedcombined topology in the extended phase space (p, r)[107]. This combined topology connects the effect ofchiral anomaly and the dynamics of skyrmions, whichallowed us to observe experimentally the consequenceof the chiral anomaly and to verify the Adler-Bell-Jackiw equation [19]. We consider two examples ofsuch dependence: the skyrmion in 3He-A and the corestructure of the 3He-B vortex with = 1, which existsat high pressure.

1

1

4.1. From A to B. Topologyof Weyl Nodes in 3He-B Vortex

In the core of the 3He-B axisymmetric vortex withthe spontaneously broken parity [75] one has the 3He-A order parameter on the vortex axis, at r = 0, whichcontinuously transforms to the 3He-B order parameterfar from the core. Figure 11 demonstrates the evolu-tion of the Weyl points on the way from 3He-A to the3He-B [2, 107]. At r > 0 the spin degeneracy of theWeyl points is lifted, and the nodes with N = ±2 splitinto the elementary Weyl nodes with N = ±1. At r =rcore the Weyl points with opposite N merge to form theDirac points with trivial topological charge, N = 0. Atr > rcore, the Dirac points disappear, because they arenot protected by topology, and the fully gapped stateemerges. Far from the vortex core one obtains the3He-B order parameter with the 2π phase windingaround the vortex line.

In this evolution of Weyl points the chirality of3He-A, which is the property of topology in momen-tum space, continuously transforms to the integer val-ued circulation of superfluid velocity around the vor-tex, which is described by the real space topology. Thetopological connection of the real-space and momen-tum-space properties is encoded in Eq. (4.7) of [107]:

(29)

Here is the real-space topological invariant—thewinding number of the vortex; Na is the momentum-space topological invariant describing the ath Weylpoint. Finally the index νa connects the two spaces: itshows how many times the Weyl point Ka coverssphere, when the coordinates r = (x, y) run over thecross-section of the vortex core:

(30)

For the discussed 3He-B vortex the Weyl nodeswith Na = ±1 cover the half a sphere, νa = ±1/2, which

gives = (1/2 + 1/2 + 1/2 + 1/2) = 1.

4.2. Topology of Evolution of the Weyl Nodesin the 3He-A Skyrmion

Equation (29) is also applicable to the continuoustextures in 3He-A. For example, in the skyrmion inFig. 2 the Weyl nodes with Na = ±2 cover the whole

sphere once, νa = ±1. This gives = (2 × 1 + (–2) ×

(–1)) = 2, which means that the skyrmion representsthe continuous doubly quantized vortex. The steps inthe NMR spectrum corresponding to the = 2 vorti-ces were observed in the NMR experiments on rotat-ing 3He-A [130]. The meron—the Mermin-Ho vortex

= ν∑1 .2 a a

a

N1

1

ν = ⋅ ∂ × ∂π ∫ 3

1 1 ( ).4 | |

a a aa x yadxdy K K K

K

112

112

1

Fig. 10. (Color online) Phase diagram of topological statesof 3He-B in Eq. (25) in the plane (μ, 1/2m*). States on theline 1/m* = 0 correspond to the Dirac vacua, which Ham-iltonian is non-compact. Topological charge of the Diracfermions is intermediate between charges of compact 3He-B states. The line 1/m* = 0 separates the states with differ-ent asymptotic behavior of the Green’s function at infinity:G–1(ω = 0, p) → ±τ3p2/2m*. The line μ = 0 marks topo-logical quantum phase transition, which occurs betweenthe weak coupling 3He-B (with μ > 0, m* > 0 and topolog-ical charge NK = 2) and the strong coupling 3He-B (withμ < 0, m* > 0 and NK = 0). This transition is topologicallyequivalent to quantum phase transition between Diracvacua with opposite mass parameter M = ±|μ|, whichoccurs when μ crosses zero along the line 1/m* = 0. Theinterface which separates two states contains single Majo-rana fermion in case of 3He-B, and single chiral fermion incase of relativistic quantum fields. Difference in the natureof the fermions is that in Fermi superfluids and in super-conductors the components of the Bogoliubov-Nambuspinor are related by complex conjugation. This reducesthe number of degrees of freedom compared to Dirac case.

Dirac Dirac

0

1/m*

NK = 0

NK = 0

NK = +1

NK = +2

NK = −1

NK = −2

μ

topological superfluid non-topological superfluid

topological 3He-Bnon-topological superfluid

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with Na = ±2 and νa = ±1/2—represents the singly

quantized vortex, = 2 × + (–2) × = 1.

The elementary cell of the skyrmion lattice in the lowmagnetic field contains 4 merons, see Fig. 3, and thushas = 4. Merons are also the building blocks of thevortex sheet observed in 3He-A [62, 63].

4.3. Topology of the Phaseof the Gap Function in (p, r)-Space

The close connection between topologies in realand momentum space can be also seen when Eq. (22)for the gap function is extended to the inhomogeneouscase

(31)Then the phase Φ(p, r) of the determinant of the

gap function may include the phase winding in realspace (the state with quantized vortex) and the wind-ing in momentum space, which gives rise to the line ofzeroes in the determinant of the gap function and thusto the Weyl points in spectrum. In general, the wind-ing number of the phase protects the 4-dimensionalvortex singularity of the phase Φ(p, r) in the (3 + 3)-dimensional (p, r)-space. By changing the orientationof this 4-D manifold in the (3 + 3)-space, one cantransform the 3He-B state with the = 1 vortex, to thehomogeneous 3He-A state with Weyl points. In thevortex state of 3He-B, outside of the vortex core thephase depends only on the coordinates: Φ(p, r) =Φ(φ) = 2φ. The corresponding 4D manifold is the(3 + 1)-subspace (1D vortex line times the 3Dmomentum space). In the vortex-free 3He-A state thephase depends only on the momentum, Φ(p, r) =Φ(p), and it has the winding number N in momentumspace. In this case the corresponding 4D manifold isthe (1 + 3)-subspace (the 1D line of determinantnodes in momentum space times the 3D coordinatespace).

For the inhomogeneous 3He-A with (r), the non-zero winding of the phase Φ(p, r) gives in particularthe following 4D generalization of the singular vortic-ity: [107, 131]

(32)

(33)

5. CHIRAL ANOMALY5.1. Hydrodynamic Anomalies in Chiral Superfluids

The singularity in Eq. (32) leads to the anomalies inthe equations for the mass current (linear momentumdensity) and angular momentum density of the chiral

1 (12

12 ( ))− 1

2

1

ΦΔ = Δ ( , )ˆ ˆdet ( , ) |det ( , )| .ie p rp r p r

1

l

⊥

⎛ ⎞∂ ∂ ∂ ∂⋅ − ⋅ Φ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

= − π ⋅ ⋅ ∇ × δ

( , )

ˆ ˆ ˆ2 ( )( ) ( ( )),

p rp r r p

l p l l p r

⊥ = − ⋅ˆ ˆ( ) ( )( ( ) ).p r p l r l r p liquid, since these quantities can be expressed via thegradients of the generalized phase Φ(p, r) [107, 131].Later it became clear, that these anomalies are themanifestation of the chiral anomaly in 3He-A relatedto the Weyl points, which led to the modification ofthe hydrodynamic equations derived by Khalatnikovand Lebedev [132]. The effect of the chiral anomalyhas been observed in experiments with dynamics of

Fig. 11. Combined topology of the Weyl points in the coreof the axisymmetric 3He-B vortex according to [107] (fromreview paper [2]). Here r is the distance from the vortexaxis. Weyl points are topologically stable nodes in the qua-siparticle particle spectrum, which have integer topologi-cal charge N in momentum space. The Weyl points are sit-uated at momenta Ka in momentum space, which dependon the position r in the real space. In superfluid 3He, theWeyl points live at p = pF, i.e. on the former Fermi surface,

Ka = ±pF , where are unit vectors. That is why origi-nally the Weyl point was called “boojum on Fermi sur-face”. On the vortex axis, at r = 0, one has two pairs ofWeyl points with = = . Each pair forms the Weylpoint with double topological charges, N = +2 on thenorth pole and N = –2 on the south pole. This correspondsto the chiral 3He-A on the axis without any vorticity. Forr > 0, the multiple nodes split into pairs of Weyl points,each carrying unit topological charges N = +1 or N = –1.For increasing r, the Weyl points move continuouslytowards the equatorial plane, where they annihilate eachother (+1 – 1 = 0). For larger r the fully gapped state isformed, which becomes the isotropic 3He-B far from thevortex. The coordinate dependence of the Weyl point givesrise to vorticity concentrated in the vortex core, as a resultthe vortex in the B-phase acquires the winding number. Inother words, according to [107] the vortex—the topologi-cal defects in r-space—flows out into p-space due to evo-lution of the Weyl points. The topology of the evolution isgoverned by Eq. (29), which connects three topologicalinvariants: real-space winding number of the vortex ,momentum-space invariant of the Weyl point N and theinvariant ν, which describes the evolution of the Weylpoint in real space.

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the vortex-skyrmions [19], which revealed the exis-tence of the anomalous spectral-flow force acting onskyrmions.

Originally the anomalies in the dynamics of 3He-Ahave been obtained from the calculations of theresponse functions and from the hydrodynamic equa-tions, which take into account the singularity of thephase Φ in Eq. (32) [107, 131]. In these calculationsthe main contribution to the anomalous behaviorcomes from the momenta p far from the Weyl points,where the spectrum is highly nonrelativistic. Theresults of calculations coincide with the later resultsobtained using the relativistic spectrum of chiral fer-mions emerging in the vicinity of the Weyl points. Thisis because the spectral f low through the Weyl nodes,which is in the origin of anomalies, does not dependon energy and is the same far from and close to thenodes.

The original approach uses the semiclassicalapproximation, which takes into accouns that thephase Φ(p, r) of the determinant can be considered asthe action in the quasiparticle dynamics,

(34)

The mass current j = f(p, r), where f is the distri-bution function of bare particles (atoms of 3He), seedetails in [107, 131]. In the inhomogeneous state the

momentum and coordinate are shifted by ∇S and –

respectively, and one has at T = 0 (we take = 1):

(35)

(36)

(37)

The first term in Eq. (37) is the superfluid mass cur-rent with velocity vs and mass density ρ = mn, where nis particle density. The second term is the mass currentproduced by the inhomogeneity of the orbital angularmomentum density L = ( n/2) in the chiral liquid.This is what one would naturally expect for the angularmomentum of the chiral liquid with particle density nand the angular momentum for each pair of atoms.The last term in Eq. (37) is anomalous, it is nonzerodue to the 4D vortex singularity in the gap function

1

1

9

= Φ( , ) ( , ).4

S p r p r

∑pp

∂∂Sp

⎛ ⎞∂= − ∇ +⎜ ⎟∂⎝ ⎠

∂⎛ ⎞∂Φ= ∇ ⋅ − ∇Φ ⋅⎜ ⎟∂ ∂⎝ ⎠

∑

∑

,

14

Sf S

ff

p

p

j p p rp

pp p

⎛ ⎞∂Φ= ∇Φ + ∇ ⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂− ⋅ − ⋅ Φ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∑ ∑

∑

1 14 4

1 ( , )4

ii

f f

f

p p

p

pp

p p rp r r p

= ρ + ∇ × − ⋅ ∇ ×01 1 ˆ ˆ ˆ( ).2 2s Cv L l l l

l

l

determinant in Eq. (32). The parameter C0, whichcharacterizes this hydrodynamic anomaly, equals:

(38)

The same parameter enters the other hydrodynamicanomalies in the chiral superfluid: in the conservationlaws for the linear momentum and the angularmomentum in chiral superfluids. At T = 0 one has[131]:

(39)

(40)

The nonzero value of the rhs of Eq. (39) manifests thenon-conservation of the vacuum current, whichmeans that the linear momentum is carried away bythe fermionic quasiparticles created from the super-fluid vacuum. Equation (40) shows that the angularmomentum of the superfluid vacuum is also not con-served. This equation can be also represented in termsof the nonlocal variation of the angular momentum:

∂tδL = –∂E/∂θ with ∂L = (n – C0)δ – δn. In other

words, there is the dynamical reduction of the angular

momentum from its static value to the dynamic

value (n – C0) . Note that in 3He-A the Cooper pair-

ing is in the weak coupling regime, which means thatthe gap amplitude Δ is much smaller that the Fermienergy μ. As a result the particle density n in the super-fluid state is very close to the parameter C0, which isequal to the particle density in the normal state, C0 =n (Δ = 0). One has (n – C0)/C0 = (n(Δ) – n(Δ =0))/n(Δ = 0) ~ 10–5, so that the reduction of thedynamical angular momentum is crucial.

5.2. Hydrodynamic Anomalies from Chiral AnomalyTo connect the hydrodynamic anomalies in Eqs.

(39), (40) and the chiral anomaly in relativistic theo-ries, let us take into account that the parameter pFwhich enters C0 marks the position of the Weyl pointsin 3He-A: Ka = ±pF . When μ → 0 and correspond-ingly Ka → 0, the Weyl points merge and annihilate. Atthe same time C0 → 0, and all the hydrodynamicanomalies disappear. They do not exist in the strongcoupling regime at μ < 0, where the chiral superfluidhas no Weyl points and C0 = 0. All the dynamic anom-alies experienced by 3He-A come from the Weylpoints: the existence of the Weyl nodes in the spec-trum allows the spectral f low of the fermionic levelsthrough the nodes, which carry the linear and angularmomentum from the vacuum to the quasiparticle

=π

2

0 2 2 .3

FpC

∂ − ∇ π = − ∂ ⋅ ∇ ×03 ˆ ˆ ˆ( ),2t i k ik i tj C l l l

∂ δ ∂+ =∂ δ ∂0

ˆ1 .2

E Ct tL l

θ

12

l 1ˆ2

l

ˆ2

nl

2l

l

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world. The state of the chiral superfluid with anoma-lies at μ > 0 and the anomaly-free state of the chiralsuperfluid at μ < 0 are separated by the topologicalquantum phase transition at μ = 0.

Close to the Weyl point the spectral f low can beconsidered in terms of the relativistic fermions. Thechiral fermions experience the effect of chiral anomalyin the presence of the synthetic electric and magneticfields in Eq. (21). The left-handed or right-handedfermions are created according to the Adler-Bell-Jackiw equation for chiral anomaly:

(41)

The fermionic quasiparticles created from the super-fluid vacuum carry the fermionic charge from the vac-uum to the “matter”—the normal component of theliquid, which at low temperatures consists of thermalWeyl fermions. For us the important fermionic chargeis the quasiparticle momentum: each fermion createdfrom the vacuum carry with it the momentum K(a) =±pF . According to the Adler-Bell-Jackiw equationsfor chiral anomaly, this gives the following momentumcreation from the vacuum per unit time per unit vol-ume:

(42)

Here as before Na is the topological charge of the athWeyl point, which sign determines the chirality of theWeyl quasiparticles near the Weyl node; and qa = ±1 isthe effective electric charge. The rhs of Eq. (42) isnon-zero, because the quasiparticles with oppositechirality Na carry opposite momentum K(a). Since inthe supefluids the momentum density equals the masscurrent density, P = j, Eq. (42) reproduces theEq. (39). This demonstrates that nonconservation ofthe linear momentum of the superfluid vacuum is theconsequence of the chiral anomaly.

The angular momentum anomaly is also related tothe spectral f low through the nodes in bulk or on thesurface [133–137].

According to the Newton law, the creation of thelinear momentum from the vacuum per unit time dueto the spectral f low through the Weyl nodes under theeffective electric and magnetic fields produced by thetime dependent texture of the vector (r, t), is equiva-lent to an extra force acting on the texture. In experi-ment, the relevant time dependent texture is the vor-tex-skyrmion moving with velocity vL, where (r, t) =(r – vLt). This together with the effective magnetic

field Beff = pF∇ × gives also the effective electric fieldEeff = –pF∂t = pF(vL ⋅ ∇) .

= − = ⋅π

2eff eff2

1 ( , ) ( , ).4

R Ln n q t tB r E r

l

∂ − ∇ π = ⋅π ∑ ( ) 2

eff eff21 ( , ) ( , ) .

4a

t i k ik i a aa

P t t K N qB r E r

l

ll

ll l

The anomalous spectral-flow force acting on thistopological object is obtained after integration of therhs of Eq. (42) over the cross-section of the skyrmion:

(43)

Here is the vortex winding number of the texture,which via Eq. (29) is expressed in terms of the Weylpoint charges Na and the topological π2 charges oftheir spatial distributions in the texture. This demon-strates that the spectral f low force acting on the vor-tex-skyrmion, which has = 2, is the result of thecombined effect of real-space and momentum-spacetopologies.

5.3. Experimental Observation of Chiral Anomalyin Chiral Superfluid 3He-A

The force acting on the vortex-skyrmions has beenmeasured experimentally [19]. There are several forcesacting on vortices, including the Magnus force,Iodanski force and the anomalous spectral-f low force,which is called the Kopnin force. For the steady statemotion of vortices the sum of all forces acting on thevortex must be zero. This gives the following equationconnecting velocity of the superfluid vacuum vs, thevelocity of the vortex line vL and the velocity vn of“matter”—the velocity of the normal component ofthe liquid:

(44)

For the continuous vortex-skyrmion in 3He-A withthe spectral f low force in Eq. (43) the reactive param-eter d⊥ is expressed in terms of the anomaly parameterC0:

(45)

Here ns(T) = n – nn(T) is the density of the superfluidcomponent.

Since in 3He-A the anomaly parameter C0 is veryclose to the particle density n, the chiral anomaly in3He-A should lead to equation d⊥ – 1 = 0 for practi-cally all temperatures. This is what has been observedin Manchester experiment on skyrmions in 3He-A, seeFig. 12 (right) which experimentally confirms the gen-eralized Adler-Bell-Jackiw Eq. (42).

In conclusion, the chiral anomaly related to theWeyl fermionic quasiparticles, whose gapless spec-trum is protected by the topological invariant in p-space, has been observed in the experiments with sky-rmions—objects, which are protected by the topologi-cal invariant in the r-space. The effect of chiral anom-aly observed in 3He-A incorporates several topologicalcharges described by the combined topology in the

= ∇ × ⋅ ⋅ ∇π

= −π ×∫

32

spectral flow 2

0 L

ˆ ˆ ˆ(( ) ( ) )2

ˆ .

FL

p d x

C

F l l v l

z v1

1

1

1

⊥× − + × − + − =L s n L || n Lˆ ˆ( ) ( ) ( ) 0.d dz v v z v v v v

⊥−− = 0

s

1 .( )

C ndn T

1

1



extended (p, r)-space, which is beyond the conven-tional anomalies in the relativistic systems.

5.4. Chiral Magnetic and Chiral Vortical Effectsin 3He-A

Another combination of the fermionic charges ofthe Weyl fermions gives rise to the chiral magneticeffect (CME)—the topological mass current along themagnetic field. The effect can be written in terms ofthe following contribution to the free energy:

(46)

where μ(a) are chemical potentials of the right and leftWeyl quasiparticles. The variation of the energy overthe effective vector potential Aeff = pF gives the effec-tive current along the effective magnetic field. TheCME is nonzero if there is a disbalance of left andright quasiparticles. In 3He-A this disbalance isachieved by application of supercurrent, which pro-duces the chemical potentials for chiral quasiparticleswith opposite sign: μ(a) = ±pF ⋅ vs, see [15, 138] fordetails. The supercurrent is created in the rotatingcryostat. For us the most important property of theCME term is that it is linear in the gradient of . Itssign thus can be negative, which leads to the observedhelical instability of the superflow towards formation

= ⋅ μπ ∑∫

3 ( ) 2CME eff eff2

1 ( , ) ( , ) ,8

aa a

a

F d x t t N qA r B r

l

l

l

of the inhomogeneous -field in the form of skyrmi-ons, Fig. 13 [139]. Since the texture ∇ × inside theskyrmion plays the role of magnetic field, the processof formation of skyrmions in the superflow is analo-gous to the formation of the (hyper)magnetic fields inthe early Universe [140, 141]. Now the CME is studiedin relativistic heavy ion collisions where strong mag-netic fields are created by the colliding ions [142].

The related effect is the chiral vortical effect(CVE), which is described by the following term in thefree energy [15]:

(47)

Here Bg is the effective gravimagnetic field, which in3He-A is produced for example by rotation, Bg =2Ω/c2, where Ω is the angular velocity of rotation. Thevariation of the energy over the effective vector poten-tial Aeff = pF gives the effective current along the rota-tion axis. The real mass current along the rotation axishas been discussed in [2], see Fig. 14.

In general, the total current along the magneticfield or along the rotation axis in the ground state ofthe condensed matter system is prohibited by theBloch theorem [143]. In our case of CME the field A =pF is effective, and the corresponding current J =δF/δA is also effective. It does not coincide with thereal mass current j = δF/δvs. The imbalance betweenthe chiral chemical potentials of left-handed andright-handed Weyl fermions is also effective: it is pro-vided by the superflow due to the Doppler shift. Forthe effective fields and currents the no-go theorem isnot applicable. In the case of the chiral vortical effectwith the real mass current along the rotation in Fig. 14the Bloch theorem is obeyed. Such configuration cor-responds to the ground state in the given topologicalclass of rotating states in 3He-A. In equilibrium, thetotal mass current along the rotation axis is absent: thecurrents concentrated in the soft cores of the vortex-skyrmions are compensated by the opposite superfluidcurrent in bulk.

6. FERMION ZERO MODES

The nontrivial topology of 3He superfluids leads tothe topologically protected massless (gapless) Majo-rana fermions living on the surface of the superfluidor/and inside the vortex core. Let us start with the fer-mion zero modes on the surface of 3He-B.

6.1. Majorana Edge States in 3He-BFigure 15 reproduces Fig. 12 from [7] for the fer-

mion zero modes living at the interface between twobulk states of 3He-B with opposite topological chargesNK in Eqs. (26), (27). There are two branches E(py, pz)

l 1

l

1

= ⋅ μπ ∑∫

3 ( ) 2CVE eff2

1 ( , ) ( , ) ( ) .8

ag a a

a

F d x t t N qA r B r

l

l

1

Fig. 12. (Color online) Experimental verification of theAdler-Bell-Jackiw anomaly equation in 3He-A (from[19]). Left: A uniform array of vortices is produced byrotating the whole cryostat, and oscillatory superflow per-pendicular to the rotation axis is produced by a vibratingdiaphragm, while the normal f luid (thermal excitations) isclamped by viscosity, vn = 0. The velocity vL of the vortexarray is determined by the overall balance of forces actingon the vortices in Eq. (44). The vortices produce the dissi-pation d|| and also the coupling between two orthogonalmodes of membrane oscillations, which is proportional tod⊥. Right: The parameters d|| and d⊥ measured for the con-tinuous vortices-skyrmions in 3He-A. As distinct from the3He-B vortices, for skyrmions the measured parameter d⊥is close to unity. According to Eq. (45), the experimentdemonstrates that the anomaly parameter C0 is close to theparticle density n and thus it verifies the Adler-Bell-Jackiwanomaly Eq. (42).

anomalyparameter

vs

vs

anomaly parameter0.3

electrodes

vortices

l - field:

oscillatingdiaphragm

200

μm 0.2

0.1d⊥ − 1

d||

00.85 0.90 1.000.95

T/Tc

11

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of fermion zero modes with different directions ofspin, which form the Majorana cones. They aredescribed by the effective 2 + 1 theory with the follow-ing Hamiltonian:

(48)

Here is along the normal to the interface, and c is the“speed of light” of the emergent relativistic spectrumof these surface fermions. The same Hamiltoniandescribes the surface states on the boundary of 3He-B[127, 144], which we consider here.

The parameter c depends on the structure of theanisotropic order parameter Aαi(x) in the interface, ornear the wall of the container. Let us consider theorder parameter near the wall, with the bulk 3He-B atx > 0:

(49)

Δ⊥(x = ∞) = Δ||(x = ∞) = ΔB, where ΔB is the gap inbulk 3He-B. The corresponding 3D Hamiltonian is:

(50)

where τα and σi are as before the Pauli matrices ofBogoliubov-Nambu spin and nuclear spin corre-spondingly; and px now is the operator. Let us find thebound surface states of this Hamiltonian.

= ⋅ ×ˆ ( ).H cx σ p

x

⊥

α

Δ⎛ ⎞⎜ ⎟= Δ⎜ ⎟⎜ ⎟Δ⎝ ⎠

||

||

( ) 0 00 ( ) 0 .0 0 ( )

i

xA x

x

⊥⎛−= τ + τ Δ σ⎜⎝

⎞+Δ σ + Δ σ ⎟⎠

2 2

3 1

|| ||

( )2 *

( ) ( ) ,

xFx

F

y zy z

F F

pp pH xm p

p px x

p p

We can use the method of trajectories. In 3Hesuperfluids the Fermi momentum pF ≫ 1/ξ, where ξ isthe coherence length, which determines the character-istic thickness of the surface layer, where the orderparameter evolves. That is why with a good accuracythe classical trajectories are the straight lines, and wecan consider the Hamiltonian along the trajectory. Weassume here the specular reflection of quasiparticles atthe boundary, then the momentum component px

which is normal to the wall of the container, changes

Fig. 13. (Color online) Demonstration of the chiral magnetic effect in 3He-A [139]. In NMR experiments the height of the sat-ellite peak is measured, which comes from the vortex skyrmions, see Fig. 2c. Initially no vortices are present in the vessel. Whenthe velocity of the superflow vs, which corresponds to the chiral chemical potential in Eq. (46), exceeds a critical value determinedby spin-orbit interaction, the helical instability takes place, and the container becomes filled with the skyrmions. Skyrmions carrythe analog of a hypermagnetic field, as a result the process of their creation, which is governed by chiral anomaly, is analogous tothe process of formation of magnetic field in early Universe [140, 141].

1

1 1

Fig. 14. Schematic illustration of the chiral vortical effectin rotating 3He-A from [2]. The mass currents along therotation axis, which are concentrated in the soft cores ofthe vortex skyrmions, are fully compensated by the bulkcurrent in the equilibrium state. This is the realization ofthe Bloch theorem which forbids the total current in thecondensed matter system in equilibrium.

1



sign after reflection. This means that after ref lectionfrom the wall the quasiparticle moving along the tra-jectory feels the change from Δ⊥(x) to –Δ⊥(x). Thenthe problem transforms to that discussed in [7] of find-ing the spectrum of the fermion bound states at theinterface separating two bulk 3He-B states with theopposite values of the topological charge NK = ±2 inEq.(15):

(51)α

Δ⎛ ⎞⎜ ⎟+∞ = Δ = +⎜ ⎟⎜ ⎟Δ⎝ ⎠

0 0( ) 0 0 , 2,

0 0

B

i B K

B

A N

and

(52)

Introducing px = pF – i∂x and neglecting the termsquadratic in py, pz and ∂x one obtains the Hamiltoinian

(53)

(54)

(55)

where = pF/m* is the Fermi velocity.For p|| = (py, pz) = 0 one has the Hamiltonian (54).

This Hamiltonian is supersymmetric, where Δ⊥(x)serves as superpotential, since it changes sign acrossthe interface. That is why (54) has eigenstates withexactly zero energy. There are two solutions withE(p|| = 0) = 0, which correspond to different orienta-tions of spin:

(56)

(57)

For nonzero p|| one may use the perturbation theorywith H1 as perturbation, if |p||| ≪ pF. The second ordersecular equation for the matrix elements of H1 gives

the relativistic spectrum E2 = c2( + ) of the gaplessedge states in (48) with the “speed of light”:

(58)

The speed of light of surface fermions is sensitive to thestructure of the surface layer, and only in the limit ofthe infinitely thin surface layer, when Δ||(x) = Δ⊥(x) =ΔBΘ(x), where Θ(x) is the Heaviside step function, itapproaches the value determined by the bulk orderparameter, c → cB = ΔB/pF.

In applied magnetic field the Pauli term violatesthe K symmetry of the Hamiltonian, the topologicalinvariant NK ceazes to exist, and the edge states acquiremass [145] (see also [32, 33]):

(59)

The topologically protected Majorana edge statesare under intensive investigations in superfluid 3He-B

α

−Δ⎛ ⎞⎜ ⎟−∞ = −Δ = −⎜ ⎟⎜ ⎟−Δ⎝ ⎠

0 0( ) 0 0 , 2.

0 0

B

i B K

B

A N

= +0 1,H H H

⊥= − τ ∂ + τ σ Δv0 3 1 ( ),F z xH i x

Δ= τ σ + σ||

1 1( )

( ),y y z zF

xH p p

p

vF

+ ⊥τ σ

⎛ ⎞⎛ ⎞ ⎛ ⎞Ψ ∝ − Δ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠∫

v 0

1 1 1( ) exp ' ( ') ,0

x

F

x dx xi

− ⊥τ σ

⎛ ⎞⎛ ⎞ ⎛ ⎞Ψ ∝ − Δ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠∫

v 0

1 0 1( ) exp ' ( ') .1

x

F

x dx xi

2xp 2

yp

∞

⊥

∞

⊥

⎛ ⎞Δ− Δ⎜ ⎟⎜ ⎟⎝ ⎠=

⎛ ⎞− Δ⎜ ⎟⎜ ⎟⎝ ⎠

∫ ∫

∫ ∫

v

v

||

0 0

0 0

( ) 2exp ' ( ')

.2exp ' ( ')

x

F F

x

F

xdx dx x

pc

dx dx x

= + + = γ2 2 2 2 2 1( ) , .2x yE c p p M M H

Fig. 15. Schematic illustration of the spectrum of Majo-rana fermions at the Kibble wall in 3He-B—the nontopo-logical domain wall which separates the regions with theopposite topological charges NK = +2 and NK = –2 (from[7]). The spectrum satisfies Eq. (48).

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The spectrum remains galpess on the lines on the boundary, which separate boundary states with opposite sign of the mass. This phenomenon is now called the higher order topology.

634


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experiments. They are probed through anomaloustransverse sound attenuation [146–149], in measure-ments of the surface contribution to specific heat[150–152], and by magnon BEC technique in NMRexperiments [153].

6.2. Caroli-de Gennes-Matricon Bound Statesin the Vortex Core

The topological properties of the fermionic boundstates in the core of the topological objects is deter-mined both by the real space topology of the objectand by the momentum space topology of the environ-ment.

Let us start with the spectrum of the low-energybound states in the core of the vortex with windingnumber = 1 in the isotropic model of s-wave super-conductor. The spectrum is characterized by twoquantum numbers, the linear momentum pz along thevortex line, and the integer quantum number n, whichdetermines the angular momentum Lz. This spectrumhas been analytically obtained in microscopic theoryby Caroli, de Gennes and Matricon [154], see Fig. 16.The low energy branch has the form:

(60)

This spectrum is two-fold degenerate due to spindegrees of freedom. For the nonrelativistic vortex in s-wave superconductors the exact zero energy states areabsent, see Fig. 16. However, typically the level spac-ing—the so called minigap—is very small compared tothe energy gap of the quasiparticles outside the core,ω0 ~ Δ2/μ ≪ Δ. In the approximation, when the min-igap is neglected, which is valid in many applications,the angular momentum quantum number n can beconsidered as continuous variable. Then Eq. (60) sug-gests that there is a branch En, which as a function ofcontinuous n crosses zero energy level in Fig. 16(right). And, indeed, there is the topological indextheorem, which relates the number of branches, whichcross zero as a function of Lz and the winding number

of the vortex [155]. This is the analog of index the-orem for relativistic cosmic strings, where the theoremrelates the number of branches, which cross zero as afunction of 2pz, and the string winding number [156].

6.3. Flat Band of Majorana Fermionsin the Singular Vortex in 3He-A

Now let us consider fermions in the core of singular = 1 in the chiral superfluid 3He-A. The anomalous

branch, which crosses zero as function of continuousangular momentum quantum number n, remains thesame, see Fig. 17 (right). The existence of the anoma-lous branch E(n) depends only on the winding number

of the vortex, and is not sensitive to the topology ofthe bulk superfluid. The latter determines the fine

1

( )= − + ω01 ( ).2n zE n p

1

1

1

structure of the spectrum. Now the spectrum containsthe exact zero energy states [15, 157]:

(61)

This is the result of the bulk-defect correspondence:the odd winding number of the phase of the gap func-tion, Δ(p) ∝ (px + ipy), in bulk is responsible for thezero-energy states in the core. In the two-dimensionalcase the n = 0 levels represent two spin-denenerateMajorana modes [158, 159]. The 2D half-quantumvortex, which is the vortex in one spin component,contains single Majorana mode.

In the 3D case Eq. (61) at n = 0 describes the f latband [86]: all the states in the interval –pF < pz < pF

have zero energy, where K(a] = ±pF mark the posi-tions of two Weyl points in the bulk material [88]. Thereason for that can be explained using the general case,when the vortex is along the z-axis and the Weyl pointsare at K(a] = ±pF with = cosλ + sinλ, see Fig. 18.At each pz except for |pz| = pFcosλ, i.e. away from theWeyl nodes, the spectrum is gapped, and thus the sys-tem represents the set of the fully gapped (2+1)-dimensional chiral superfluids. Such superfluids arecharacterized of the topological invariant N(pz),obtained by dimensional reduction from the invariantN in Eq. (15), which describes the Weyl points:

(62)

Here pμ = (ω, px, py). Such invariant is responsible forthe quantized value of the Hall conductivity in theabsence of external magnetic field in the (2+1) topo-logical materials with broken time reversal symmetry[90, 91, 128, 160–165]. It is the generalization of the

= − ω0( ).n zE n p

z

l l z x

β μ ν

− − −βμν= ω ∂ ∂ ∂π ∫

1 1 12

( )

.24

z

x y p p p

N pe

dp dp d G G G G G Gtr

Fig. 16. Illustration of the spectrum of fermion boundstates on symmetric singly-quantized ( = 1) vortex in thenon-topological s-wave superconductors. Left: There areno states with exactly zero energy. Right: However, if oneconsiders the angular momentum quantum number n ascontinuous variable, there is a branch which crosses zero asa function of n. This is the consequence of the specialindex theorem [155], which becomes applicable in thesemiclassical limit pF ≫ 1/ξ (or which is the same Δ ≪ μ).

pz n

E(pz, n)E(pz, n)

1



topological invariant introduced for quantum Halleffect [166].

This invariant, which is applicable both to interact-ing and non-interacting systems, gives

(63)

(64)

Since the vortex core of the topologically nontrivial(2+1) superfluid contains the zero energy Majoranamode, one obtains the zero energy states in the wholeinterval –pFcosλ < pz < pFcosλ:

(65)

This is the topological origin of the f lat band in thecore of the singular 3He-A vortex, first calculated in[86].

Flat band has been also discussed for vortices in d-wave superconductors [168].

6.4. Fermion Condensation on Vortices in Polar Phase

Here we consider vortices, in which the minigapω0(pz) vanishes at pz =0. This leads to the enhanceddensity of states of the fermions in the vortex core, andas a consequence to the non-analytic behavior of theDoS as a function of magnetic field in superconductoror of rotation velocity in superfluid.

Examples are provided by vortices [4] in therecently discovered [34] non-chiral (N = 0) spin-trip-let polar phase of superfluid 3He. In general, the min-igap for the Caroli-de Gennes-Matricon bound statesin the core of symmetric vortices in which the gap

= < λ3( ) 1, | | cos ,z z FN p p p

= > λ3( ) 0, | | cos .,z z FN p p p

= − λ < < λ( ) 0, cos cos .z F z FE p p p p

function behaves as Δ(pz)f(r) is given by the followingequation (see e.g. [157]):

(66)

where

(67)

is the wave function of the bound state.For s-wave superconductors one has isotropic gap,

Δ(pz) = ΔS; for 3He-A the gap has point nodes, Δ(pz) =

ΔA|p⊥|/pF = ΔA /pF; and for the polar phase inEq. (23) the gap has nodal line, Δ(pz) = ΔP|pz|/pF. The

∞

∞

Δψω = = −

ψ

∫

∫

20

2 200

20

0

( ) ( )| ( )|

( ) , ,

| ( )|

z

z F z

p f rdr r

qrp q p p

dr r

⎛ ⎞Δψ = −⎜ ⎟⎜ ⎟⎝ ⎠∫

v

00

( ) ( )( ) exp ' ,

rz

F

p f rr dr

−2 2F zp p

Fig. 17. Illustration of the spectrum of fermion boundstates on the most symmetric = 1 vortex in the chiralWeyl superfluid 3He-A. As distinct from Fig. 16, the spec-trum in this topological superfluid contains the states withexactly zero energy at n = 0 in Eq. (61). These states formthe f lat band, which terminates on the projections of twoWeyl points to the vortex line, see Fig. 18.

pz n

E(pz, n)E(pz, n)

1

Fig. 18. (Color online) Projections of Weyl points on thedirection of the vortex axis (the z-axis) determine theboundaries of the f lat band in the vortex core in Fig. 17.Weyl point in 3D systems represents the hedgehog (Berryphase monopole) in momentum space. For each planepz = const one has the effective 2D system with the fullygapped energy spectrum (px, py), except for the planeswith pz± = ±pFcosλ, where the energy (px, py) has anode due to the presence of the hedgehogs in these planes.Topological invariant N(pz) in (62) is non-zero for |pz| <pF|cosλ|, which means that for any value of the parameterpz in this interval the system behaves as 2D fully gappedtopological superfluid. Point vortex in such 2D superfluidshas fermionic state with exactly zero energy. For the vortexline in the original 3D system with two Weyl points thiscorresponds to the dispersion-less spectrum of fermionzero modes in the whole interval |pz| < pF|cosλ| (thick line).This consideration can be extended to the boundary states.Due to the bulk-boundary correspondence, the 1Dboundary of the 2D topological insulator with N(pz) = 1contains the branch of the spectrum of edge states, whichcrosses zero energy [167]. In the 3D system these zeroesform the 1D line of the edge states with zero energy—theFermi arc, which is terminated by the projections of theWeyl points to the surface [103, 104].

N(pz) = 0

pF cosλ

2D trivial chiralsuperfluid

2D topological chiralsuperfluid

2D trivial chiralsuperfluid

flatband

Weylpoint

Weylpoint

−pF cosλ

N(pz) = 1

N(pz) = 0

pz

z

vortices in r-space

zpE

zpE

grigoryvolovik

Cross-Out

grigoryvolovik

Cross-Out

636


VOLOVIK

nodal line in the polar phase leads to the large suppres-sion of the minigap at small pz ≪ pF:

(68)

where ω00 has an order of the minigap in the conven-tional s-wave superconductors. The spectrum isshown in Fig. 19. All the branches with different ntouch the zero energy level. It looks as the f lat band interms of n for pz = 0. However, at pz → 0 the size of thebound state wave function diverges, the state mergeswith the bulk spectrum and disappears.

The effect of squeezing of all energy levels ntowards the zero energy at pz → 0 can be called thecondensation of Andreev-Majorana fermions in thevortex core. The condensation leads to the divergentdensity of states (DoS) at small energy. In the vortexcluster with the vortex density nV the DoS is

(69)

In calculation of Eq. (69) we assume that the relevantvalues of n are large, and instead of summation over none can use the integration over dn:

(70)

According to Eq. (68) the integral in Eq. (70) divergesat small pz. The infrared cut-off is provided by the

intervortex distance rV = : the size of the wavefunction of the bound state ξpF/|pz| approaches theintervortex distance when |pz| ~ pFξ/rV. This cut-offleads to the following dependence of DoS on the inter-vortex distance:

(71)

The result in Eq. (71) is by the factor rV/ξ larger thanthe DoS of fermions bound to conventional vortices.Since in the vortex array rV ∝ Ω–1/2, the DoS has thenon-analytic dependence on rotation velocity, NV ∝Ω1/2. Similar effect leads to the dependence of theDoS on magnetic field in cuprate superconductors[110, 169].

6.5. Vortices with Multiple Branches Crossing Zeroin 3He-B

Finally let us mention, that vortices with brokensymmetry in the vortex core may contain a large num-ber of the non-topological branches of spectrum,which cross zero as function of pz [170–173], see e.g.Fig. 20 for the spectrum in the 3He-B vortex with theA-phase core.

Δω = ω ω2 22

00 00 002 2( ) ln , ~ ,z F

zFF z

p ppEp p

= δ ω − + ωπ∑ ∫ 0( ( 1/2) ( )).

2z

V V zn

dpN n n p

=π ω∫

0

1 .2 ( )

zV V

z

dpN n

p

−1/2Vn

Δ

2

0

~ .FV

V

pNr

B

7. CONCLUSIONS

At the moment the known phases of liquid 3Hebelong to 4 different topological classes.

(i) The normal liquid 3He belongs to the class ofsystems with topologically protected Fermi surfaces.The Fermi surface is described by the first odd Chernnumber in terms of the Green’s function in Eq. (13)[15].

(ii) Superfluid 3He-A and 3He-A1 are chiral super-fluids with the Majorana-Weyl fermions in bulk,which are protected by the Chern number in Eq. (15).In the relativistic quantum field theories, Weyl fermi-ons give rise to the effect of chiral (axial) anomaly. Thedirect analog of this effect has been experimentallydemonstrated in 3He-A [19]. It is the first condensedmatter, where the chiral anomaly effect has beenobserved.

The singly quantized vortices in superfluids withWeyl points contain dispersionless band (flat band) ofAndreev-Majorana fermions in their cores [86].

New phases of the chiral superfluids have beenobserved in aerogel. These are the so-called Larkin-Imry-Ma states—the glass states of the -field [174],which can be represented as disordered tangle of vor-tex skyrmions. These are the first representatives of thein homogeneous disordered ground state of the topo-logical material. The spin glass state in the -field hasbeen also observed. The recently observed chiralsuperfluid with polar distortion [175] also has Weylpoints in the spectrum.

(iii) 3He-B is the purest example of a fully gappedsuperfluid with topologically protected gapless Majo-rana fermions on the surface [32].

l

1

d

Fig. 19. Illustration of the spectrum of fermion zero modesat |pz| ≪ pF on vortices in the polar phase of superfluid 3He.The branches with different n approach zero-energy levelat pz → 0. This is the consequence of the Dirac nodal ringin the spectrum of the polar phase.

pz

E(pz, n)



(iv) The most recently discovered polar phase ofsuperfluid 3He belongs to the class of fermionic mate-rials with topologically protected lines of nodes, andthus contains two-dimensional f lat band of Andreev-Majorana fermions on the surface of the sample [120],see also recent reviews on superconductors with topo-logically protected nodes [176].

It is possible that with the properly engineerednanostructural confinement one may reach also newtopological phases of liquid 3He including:

(i) The planar phase, which is the non-chiralsuper-fluid with Dirac nodes in the bulk and withFermi arc of Anfreev-Majorana fermions on the sur-face [177].

(ii) The two-dimensional topological states in theultra-thin film, including inhomogeneous phases ofsuperfluid 3He films [178]. The films with the 3He-Aand the planar phase order parameters belong to the2D fully gapped topological materials, which experi-ence the quantum Hall effect and the spin quantumHall effect in the absence of magnetic field [90].

(iii) α-state, which contains 4 left and 4 right Weyl

points in the vertices of a cube, Ka = (± ± ± )

[106, 179]. This is close to the high energy physicsmodel with 8 left-handed and 8 right-handed Weylfermions in the vertices of a four-dimensional cube[180, 181]. This is one of many examples when thetopologically protected nodes in the spectrum serve asan inspiration for the construction of the relativisticquantum field theories.

3Fp x y z

See also the other proposals in [182].We did not touch the wide area of the bosonic col-

lective modes in the topological superfluids. Super-fluid phases of 3He are the objects of the quantumfield theory, in which the fermionic quantum fieldsinteract with propagating bosonic modes, some ofwhich have the relativistic spectrum. Among thesemodes there are analogs of gravitational and electro-magnetic fields, W and Z gauge bosons and Higgsfields. Higgs bosons—the amplitude modes—havebeen experimentally investigated in superfluid 3He formany years. For example, among 18 collective modesof 3 × 3 complex order parameter in 3He-B, four aregapless Nambu-Goldstone modes: oscillations of thephase Φ represent the sound waves, while and oscilla-tions of the rotation matrix Rαi are spin waves. Theother 14 modes are the Higgs modes with energy gapsof the order of ΔB. These heavy Higgs modes in 3He-Bhave been investigated both theoretically [183–186]and experimentally [187–191]. Due to spin-orbitinteraction one of the spin-wave Goldstone modeacquires a small mass and becomes the light Higgsboson. The properties of the heavy and light Higgsmodes in 3He-A, 3He-B and in the polar phase suggestdifferent scenarios for the formation of the compositeHiggs bosons in particle physics [192–197].

One may expect the topological confinement oftopological objects of different dimensions in realspace, momentum space and in the combined phasespace. Examples are: Fermi surface with Berry phaseflux in 3He-A in the presence of the superflow [15];the confinement of the point defect (monopole) andthe line defect (string) in real space [198]; the nexus inmomentum space [111, 199], which combines theDirac lines, Fermi surfaces and the crossing points;the so-called type II Weyl point, which lead to forma-tion of the analog of the black hole horizon [200–204]; etc.

Topology also gives rise to different types of super-fluid glass states, including skyrmion glass, Weyl glass,analogs of spin foam, etc. [205].

ACKNOWLEDGMENTS

This work has been supported by the EuropeanResearch Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (GrantAgreement no. 694248).

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SPELL: 1. skyrmions, 2. oblects, 3. splitted, 4. emergetically, 5. vorices, 6. nodeless, 7. hedhehog, 8.gaplessness, 9. accouns

this is an electronic reprint of the original article. this reprint ......properties of the spectrum...

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