ЖЭТФ, 2020, том 158, вып. 1 (7), стр. 17-23

COMPOSITE TOPOLOGICAL OBJECTS

IN TOPOLOGICAL SUPERFLUIDS

G. E. Volovik^*

Low Temperature Laboratory, Aalto University

FI-00076, Aalto, Finland

Landau Institute for Theoretical Physics, Russian Academy of Sciences

142432, Chernogolovka, Moscow Region, Russia

Received December 15, 2019,

revised version December 15, 2019

Accepted for publication March 3, 2020

Contribution for the JETP special issue in honor of A. S. Borovik-Pomanov’s 100th anniversary

DOI: 10.31857/S0044451020070020

all the signatures of the Bose-Einstein condensate of

magnons (see review [6]).

In bulk liquid³He there are two topologically dif-

The spontaneous phase coherent precession of mag-

ferent superfluid phases,³He-A and³He-B [7]. One

netization, discovered in 1984 by Borovik-Romanov,

is the chiral superfluid³He-A with topologically pro-

Bunkov, Dmitriev and Mukharskiy [1] in collaboration

tected Weyl points in the quasiparticle spectrum. In

with Fomin [2] became now an important experimental

the ground state of³He-A the order parameter matrix

tool for study complicated topological objects in super-

has the form A_αi = Δ_AeiΦ

d_α(êi1 + iêi2), where

d is the

fluid³He.

unit vector of the anisotropy in the spin space due to

Superfluid phases of 3He discovered in 1972 [3]

spontaneous breaking of SO(3)_S symmetry of spin ro-

opened the new area for the application of topologi-

tations; ê₁ and ê₂ are mutually orthogonal unit vectors;

cal methods to condensed matter systems. Due to the

andˆl = ê1×ê2 is the unit vector of the anisotropy in the

multi-component order parameter which characterises

orbital space due to spontaneous breaking of orbital ro-

the broken SO(3) × SO(3) × U(1) symmetry in these

tations SO(3)_L symmetry. Theˆl-vector also shows the

phases, there are many textures and defects in the or-

direction of the orbital angular momentum of the chiral

der parameter field, which are protected by topology.

superfluid, which emerges due to spontaneous breaking

Among them there are quantized vortices, skyrmions

of time reversal symmetry. The chirality of³He-A has

and merons, solitons and vortex sheets, monopoles and

been probed in several experiments. [8-10]

boojums, Alice strings, Kibble-Lazarides-Shafi (KLS)

Another phase is the fully gapped time reversal in-

walls terminated by Alice strings, spin vortices with

variant³He-B. In the ground state of³He-B the or-

soliton tails, etc [4]. Most of them have been experi-

der parameter matrix has the form A_αi = Δ_B eiΦR_αi,

mentally identified and investigated using nuclear mag-

where R_αi is the real matrix of rotation, R_αiR_αj = δ_ij .

netic resonance (NMR) technique, and in particular the

This phase has topologically protected gapless Majo-

phase coherent spin precession discovered in 1984 in

rana fermions living on the surface, see reviews [11,12].

³He-B [1, 2, 5]. Such precessing state, which has got

In³He confined in the nematically ordered aerogel

the name homogeneously precessing domain (HPD), is

(nafen), new phase becomes stable — the polar phase

the spontaneously emerging steady state of precession,

of³He [13,14], whith the order parameter

which preserves the phase coherence across the whole

sample even in the absence of energy pumping and even

A_αi = Δ_P eiΦ

d_α m_i .

(1)

in an inhomogeneous external magnetic field. It has

where orbital vector m is fixed by the nafen strands.

The polar phase in nafen obeys the analog of the Ander-

^* E-mail: volovik@boojum.hut.fi

son theorem for the columnar defects (nafen strands),

ЖЭТФ, вып. 1 (7)

G. E. Volovik

ЖЭТФ, том 158, вып. 1 (7), 2020

cluster of

spin-mass

mass

vortex

vortices

vortex

bundle

soliton

v_s

spin-mass

vortex

Fig. 1. a) Vortex cluster in rotating container with the vortex free region outside the cluster. Vortex cluster is formed when

starting with the equilibrium vortex state in the rotating container the angular velocity of rotation is increased. The new vortices

are not formed if the counterflow in the vortex region does not exceed the critical velocity for vortex formation. b) The spin-mass

vortex finds its equibrium position on the periphery of the vortex cluster, where the soliton tension is compensated by the Magnus

force acting on the mass vortex part of the composite object. The size of the soliton is given by Eq. (2), and this dependence on

the angular velocity of rotation is confirmed by the HPD spectroscopy. c) The combined object with N = 2 quanta of circulation:

spin-mass vortex + soliton + spin-mass vortex

see Refs. [15,16]. The polar phase is the time reversal

These combined objects have been observed and

invariant superfluid, which contains Dirac nodal ring in

studied using HPD spectroscopy [18, 19]. The addi-

the fermionic spectrum [16, 17].

tional absorption observed in the HPD is proportional

to the soliton area A = lh, where h is the height of

the container, and l is the distance between the spin

Among the topological defects in³He-B, there are

vortex and the wall of container, i. e. the width of the

the conventional mass vortices with the integer N

counterflow vortex-free zone. The latter is regulated

winding number of the phase Φ, and the Z₂ spin vor-

by changing the angular velocity of rotation Ω at fixed

tex — the nontrivial winding of the matrix R_αi. Due

number N of vortices in the cell:

to spin-orbit coupling the spin vortex serves as the ter-

(

√

)

mination line of the topological soliton wall. Because

Ω_V (N)

l(Ω) = R

(2)

of the soliton tension the spin vortex moves to the wall

of the vessel and escapes the observation. However,

the help comes from the mass vortices. The mass and

Here R is the radius of the cylindrical container, and

spin vortices are formed by different fields. They do

Ω_V (N) is the angular velocity in the state in the rota-

not interact since they “live in different worlds”. The

ting container with equilibrium number of vortices N =

only instance, where the spin and mass vortices inter-

= 2πR²Ω_V (N)/κ. The equilibtium state is obtained by

act, arises when the cores of a spin and a mass vortex

cooling through T_c under rotation, and then we increase

happen to get close to each other and it becomes en-

the angular velocity of rotation, Ω > Ω_V (N). The new

ergetically preferable for them to form a common core.

vortices are not created because of high energy barrier,

Thus by trapping the spin vortex on a mass vortex the

and as a result the counterflow region appears. The

combined core energy is reduced and a composite ob-

dependence of the relaxation of the HPD state follows

ject Z₂-string + soliton + mass vortex, or spin-mass

Eq. (2) [18, 19].

vortex is formed. This object is stabilized near the

Among the topological defects in³He-A, is the

edge of the vortex cluster in the rotating cryostat, see

half-quantum vortex (HQV) [20] — the condensed mat-

Fig. 1.

ter analog of the Alice string in particle physics [21].

ЖЭТФ, том 158, вып. 1 (7), 2020

Composite topological objects in topological superfluids

field is tilted with respect to aerogel strands, the spin

solitons are formed between the HQVs, as in A-phase.

But HQVs are remained pinned by the nafen strands,

see Fig. 2. The NMR absorption from excitation of

spin waves localized on solitons allows to identify the

HQVs and measure their density [23]. Due to the strong

pinning, the Alice strings formed in the polar phase,

survive after transition to the A-phase [24].

In³He-B, spin solitons give rise to doubly quantized

vortices (N = 2) — a pair of spin-mass vortices forms

a molecule, where the soliton serves as chemical bond,

see Fig. 1c. Such vortex molecules have been identified

in HPD spectroscopy [18, 19].

The “conventional” N = 1 vortex has also an un-

usual structure in³He-B. Already in the first experi-

ments with rotating³He-B the first order phase transi-

tion has been observed, which has been associated with

the transition inside the vortex core [25]. At the transi-

Fig. 2. Illustration of the lattice of solitons emerging between

tion the vortex core becomes non-axisymmetric [26,27].

the Alice strings (HQVs) in the polar phase of³He, when the

Spontaniously broken axial symmetry of the vortex was

magnetic field is tilted with respect of the aerogel strands. The

confirmed in the further HPD experiments [28]. Such

HQVs survive the soliton tension because they are pinned by

the strands. The NMR measurements give information on the

vortex can be considered as two half-quantum vortices

total length of the soliton and thus on the number of the Alice

in Fig. 3 connected by the domain wall [29, 30] — the

strings in the cell

analog of the Kibble-Lazarides-Shafi wall bounded by

cosmic strings [31]. The topological classification of

such combined objects in terms of relative homotopy

The HQV is the vortex with fractional circulation of

groups in the polar distorted B-phase see in Ref. [32]

superfluid velocity. Its order parameter is

The phenomenon of the symmetry breaking in the core

(

)

of the topological defect has been discussed for cosmic

d(r)eiΦ(r) =

xcos

+ ŷ sin

e^iφ .

(3)

strings [33]. For the³He-B vortices, the broken SO(2)

symmetry leads to the Goldstone bosons — the modes

When the azimuthal coordinate φ changes from 0 to

in which the axis of anisotropy b of the core is oscilla-

2π along the circle around this object, the vector

d(r)

ting.

changes sign and simultaneously the phase Φ changes

The HPD state, has been used to study the structu-

by π, giving rise to N

= 1/2. The order param-

re and twisting dynamics of this non-axisymmetric

eter (3) remains continuous along the circle. While

core. The coherent precession of magnetization excites

a particle that moves around an Alice string flips

the vibrational Goldstone mode via spin-orbit interac-

its charge, the quasiparticle moving around the HQV

tion. Moreover, due to spin-orbit interaction the pre-

flips its spin quantum number. This gives rise to the

cessing magnetization rotates the core around its axis

Aharonov-Bohm effect for spin waves in NMR experi-

with constant angular velocity. In addition, since the

ments [22].

core was pinned on the top and the bottom of the con-

The HQVs were first observed in the polar phase

tainer, it was possible even to screw the core, see Fig. 3.

[23]. In³He-A the spin-orbit interaction leads to for-

The twisted core corresponds to the Witten supercon-

mation of spin soliton interpolating between two de-

ducting string with the electric supercurrent along the

generate vacua with

l and

d= -ˆl. The energy of

core. The rigidity of twisted core differs from that of

soliton prevents the nucleation of the Alice strings in

the straight core, which is clearly seen in HPD exper-

³He-A. In contrast, in the polar phase in the absence of

iments in Fig. 3. Oscillations of the vortex core under

magnetic field, or if the field is along the nafen strands,

coherent spin precession lead to the observed radiation

the solitons are not formed, and HQVs are energet-

of acoustic magnon modes [34].

ically most favourable vortices in rotating container.

In the vortices with asymmetric cores the equilib-

Thus they appear if the sample is cooled down from

rium distance between the HQVs (Alice strings) is

the normal state under rotation. Then, if magnetic

rather small. The essentially larger KLS walls between

G. E. Volovik

ЖЭТФ, том 158, вып. 1 (7), 2020

100

29.3 bar

14.2 mT

v_s

V₂

N = 1

v_s

0.2

0.4

0.6

0.8

1.0

Fig. 4. a) Typical vortex sheet in³He-A in rotating container.

cos²

It mimics the system of the equidistant cylindrical vortex sheets

suggested by Landau and Lifshitz for the descrption of the ro-

Fig. 3. The vortex in³He-B with the non-axisymmetric core

tating superfluid [44]. b) The element of the vortex sheet in

represents the pair of Alice strings connected by Kibble-La-

³He-A. The vortex sheet is the soliton, which contains kinks in

zarides-Shafi wall. The core can be twisted by applying HPD

terms of merons. Each meron has circulation quantum N = 1.

with its coherent precession of magnetization. The vortex with

There are different scenarios in which vortex sheets with dif-

twisted core is analogous to Witten superconducting string

ferent geometries are prepared in the experiments [41]

with the electric current along the string core [33]. Figure

shows HPD absorption as the function of the tilting angle η of

magnetic field in case of the Witten strings with twisted core

(filled circles) and strings with untwisted core (open circles).

in³He-A the vorticity can be continuous. The contin-

The estimated critical angle at which the tilted field prevents

uous vorticity is represented by the texture of the unit

twisting by HPD is in agreement with experiment

vectorˆl according to the Mermi-Ho relation [36]:

ℏ

∇×v_s =

e_ijkl_i∇lj × ∇lk .

(5)

the strings have been observed in the B-phase in nafen

[24]. It appeared that the Alice strings formed in the

polar phase survive the transition to the B-phase. They

Experimentally the continuous vorticity is typically ob-

remain pinned, in spite of the formation of the KLS

served in terms of skyrmions (or the Anderson-Toulou-

walls between them. This allows us to study the unique

se-Chechetkin vortices [37,38]), see the upper part of

properties of the KLS wall. In particular, the KLS wall

Fig. 5. Each skyrmion has N = 2 quanta of circulation

separates two degenerate vacua with different signs of

of superfluid velocity. The skyrmion can be also pre-

the tetrad determinant, and thus between the “space-

sented as the combination of two merons with N = 1

time” and “antispacetime” [35].

each.

In the A-phase, the superfluid velocity v_s of the chi-

In 1994 a new type of continuous vorticity has been

ral condensate is determined not only by the phase Φ,

observed in³He-A — the vortex texture in the form of

but also by the orbital triad ê₁, ê₂ andl:

the vortex sheets [39-41], see Fig. 4a with a single vor-

tex sheet in container. Vortex sheet is the topological

ℏ

(

)

v_s =

∇Φ + êi1∇êi2

(4)

soliton with kinks, each kink representing the meron —

the continuous Mermin-Ho vortex with N = 1 circu-

where m is the mass of the³He atom. As distinct from

lation of superfluid velocity, Fig. 4b. In principle, us-

the non-chiral superfluids, where the vorticity is pre-

ing the vortex sheet one may construct the continu-

sented in terms of the quantized singular vortices with

ous vortices with arbitrary even number N = 2k cir-

the phase winding ΔΦ = 2πN around the vortex core,

culation quanta. This is the soliton in the form of

ЖЭТФ, том 158, вып. 1 (7), 2020

Composite topological objects in topological superfluids

Fig. 5. Skyrmion in the A-phase splits into two merons. Each

meron is terminated by boojum — the point topological ob-

jects, which lives at the interface between A-phase and B-pha-

se. Boojum also plays the role of the Nambu monopole, which

terminates the string — the N = 1 vortex on the B-side of

the interface

closed cylindrical surface, which contains N “quarks” —

merons [42, 43].

A B

Another object which is waiting for its observa-

tion in³He-A is the vortex terminated by hedgehog

Fig. 6. (Color online) Three dimensional lattice of monopoles

[45,46]. This is the condensed matter analog of the elec-

(on sites A) and anti-monopoles (on sites B), which are

troweak magnetic monopole (Nambu monopoles [47])

joined together by Alice strings (half-quantum vortices). Each

or the other monopoles connected by strings [48]. The

monopole is the source or sink of 4 strings

hedgehog-monopole, which terminates the vortex, ex-

ists in particular at the interface between³He-A and

vortex [Z₂ spin vortex + soliton+mass vortex] and

³He-B [49], see Fig. 5. In general, the topological de-

non-axisymmetric vortex [Alice string + Kibble-Laza-

fects living on the surface of the condensed matter sys-

rides-Shafi wall + Alice string]. One may expect the

tem or at the interfaces are called boojums [50]. Boo-

other more complicated examples of the topological

jums are classified in terms of relative homotopy groups

confinement of the objects of different dimensions. The

[51]. The vortex terminated by the hedgehog-monopole

complicated composite objects, such as nexus, live also

was observed in cold gases [52]. The HPD state has

in the momentum space of topological materials [55].

its own topological defects [53], and among them are

the spin and orbital monopoles connected by string.

Finding. This work has been supported by the

Several monopoles connected by strings may form the

European Research Council (ERC) under the Euro-

multi-monopole objects, such as necklace [54].

pean Union’s Horizon 2020 research and innovation

In³He-A the analogs of Nambu monopoles and Al-

programme (Grant Agreement № 694248).

ice strings may form the more complex combinations.

Acknowledgements. I thank Q. Shafi for the

This is because the monopole serves as a source or sink

discussions that ultimately led to this article.

of N = 2 circulation quanta, and thus can be the termi-

nation point of 4 Alice strings with N = 1/2 each. This

The full text of this paper is published in the English

in particular allows construct lattices of monopoles, in

version of JETP.

Fig. 6.

Here we considered several types of the topological

confinement. The composite topological objects were

REFERENCES

experimentally observed in superfluid 3He by using

the unique phenomenon of HPD — the spontaneously

1. A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmit-

formed coherent precession of magnetization discovered

riev, and Yu. M. Mukharskiy, JETP Lett. 40, 1033,

by the Borovik-Romanov group in Kapitza Institute

(1984).

for Physical Problems. With HPD spectroscopy, two

key objects have been identified in³He-B: spin-mass

2. I. A. Fomin, JETP Lett. 40, 1037 (1984).

G. E. Volovik

ЖЭТФ, том 158, вып. 1 (7), 2020

D. D. Osheroff, R. C. Richardson, and D. M. Lee,

23.

S. Autti, V. V. Dmitriev, V. B. Eltsov, J. Mäkinen,

Phys. Rev. Lett. 28, 885 (1972).

G. E. Volovik, A. N. Yudin, and V.V. Zavjalov, Phys.

Rev. Lett. 117, 255301 (2016).

G. E. Volovik, The Universe in a Helium Droplet,

Clarendon Press, Oxford (2003).

24.

J. T. Mäkinen, V. V. Dmitriev, J. Nissinen, J. Rysti,

G. E. Volovik, A. N. Yudin, K. Zhang, and V. B. El-

A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmit-

tsov, Nature Comm. 10, 237 (2019).

riev, and Yu. M. Mukharskiy, JETP 61, 1199 (1985).

25.

O. T. Ikkala, G. E. Volovik, P. J. Hakonen,

Yu. M. Bunkov and G. E. Volovik, in Novel Super-

Yu. M. Bunkov, S. T. Islander, and G. A. Kharadze,

fluids, ed. by K. H. Bennemann and J. B. Ketterson,

JETP Lett. 35, 416 (1982).

Int. Ser. of Monographs on Physics 156, Vol. 1, ch. 4,

pp. 253-311 (2013).

26.

E. V. Thuneberg, Phys. Rev. Lett. 56, 359 (1986).

D. Vollhardt and P. Wölfle, The Superfluid Phases of

27.

G. E. Volovik and M. M. Salomaa, JETP Lett. 42,

Helium 3, Dover Publ. (2013).

521 (1985).

P. M. Walmsley and A. I. Golov, Phys. Rev. Lett.

28.

Y. Kondo, J. S. Korhonen, M. Krusius, V. V. Dmit-

109, 215301 (2012).

riev, Yu. M. Mukharskiy, E. B. Sonin, and G. E. Vo-

lovik, Phys. Rev. Lett. 67, 81 (1991).

H. Ikegami, Y. Tsutsumi, and K. Kono, Science 341,

59 (2013).

29.

G. E. Volovik, JETP Lett. 52, 358 (1990).

10.

H. Ikegami, Y. Tsutsumi, and K. Kono, J. Phys. Soc.

30.

M. A. Silaev, E. V. Thuneberg, and M. Fogelström,

Jpn. 84, 044602 (2015).

Phys. Rev. Lett. 115, 235301 (2015).

11.

T. Mizushima, Ya. Tsutsumi, M. Sato, and K. Ma-

31.

T. W. B. Kibble, G. Lazarides, and Q. Shafi, Phys.

chida, J. Phys.: Condens. Matter 27, 113203 (2015).

Rev. D 26, 435 (1982).

12.

T. Mizushima, Ya. Tsutsumi, T. Kawakami, M. Sato,

32.

G. E. Volovik and K. Zhang, arXiv:2002.07578.

M. Ichioka, and K. Machida, J. Phys. Soc. Jpn. 85,

022001 (2016).

33.

E. Witten, Nucl. Phys. B 249, 557 (1985).

13.

V. V. Dmitriev, A. A. Senin, A. A. Soldatov, and

34.

V. V. Zavjalov, S. Autti, V. B. Eltsov, P. Heikkinen,

A. N. Yudin, Phys. Rev. Lett. 115, 165304 (2015).

and G. E. Volovik, Nature Comm. 7, 10294 (2016).

14.

W. P. Halperin, J. M. Parpia, and J. A. Sauls, Phys.

35.

G. E. Volovik, Pis’ma v ZhETF 109, 509 (2019)

Today 71, 11, 30 (2018).

[JETP Lett. 109, 499 (2019)].

15.

И. А. Фомин, ЖЭТФ 154, 1034 (2018) [I. A. Fomin,

36.

N. D. Mermin and T.-L. Ho, Phys. Rev. Lett. 36, 594

JETP 127, 933 (2018)].

(1976).

16.

V. B. Eltsov, T. Kamppinen, J. Rysti, and G. E. Vo-

37.

V. R. Chechetkin, JETP 44, 766 (1976).

lovik, arXiv:1908.01645.

38.

P. W. Anderson and G. Toulouse, Phys. Rev. Lett.

17.

G. E. Volovik, Pis’ma v ZhETF 107, 340 (2018)

38, 508 (1977).

[JETP Lett. 107, 324 (2018)].

39.

Ü. Parts, E. V. Thuneberg, G. E. Volovik, J. H. Koi-

18.

Y. Kondo, J. S. Korhonen, M. Krusius, V. V. Dmit-

vuniemi, V. M. H. Ruutu, M. Heinilä, J. M. Karimä-

riev, E. V. Thuneberg, and G. E. Volovik, Phys. Rev.

ki, and M. Krusius, Phys. Rev. Lett. 72, 3839 (1994).

Lett. 68, 3331 (1992).

40.

Ü. Parts, M. Krusius, J. H. Koivuniemi,

19.

J. S. Korhonen, Y. Kondo, M. Krusius, E. V. Thu-

neberg, and G. E. Volovik, Phys. Rev. B 47, 8868

V. M. H. Ruutu, E. V. Thuneberg, and G. E. Volovik,

JETP Lett. 59, 851 (1994).

(1993).

20.

G. E. Volovik and V. P. Mineev, JETP Lett. 24, 561

41.

V. B. Eltsov, R. Blaauwgeers, N. B. Kopnin, M. Kru-

(1976).

sius, J. J. Ruohio, R. Schanen, and E. V. Thuneberg,

Phys. Rev. Lett. 88, 065301 (2002).

21.

A. S. Schwarz, Nucl. Phys. B 208, 141 (1982).

42.

G. E. Volovik and M. Krusius, Priroda 4, 56 (1994).

22.

M. M. Salomaa and G. E. Volovik, Rev. Mod. Phys.

59, 533 (1987).

43.

Г. Е. Воловик, УФН 185, 970 (2015).

ЖЭТФ, том 158, вып. 1 (7), 2020

Composite topological objects in topological superfluids

44. L. D. Landau and E. M. Lifshitz, Doklady Akademii

50. N. D. Mermin, in Quantum Fluids and Solids, ed. by

Nauk SSSR 100, 669 (1955).

S. B. Trickey, E. D. Adams, and J. W. Dufty, Plenum,

New York (1977), pp. 3-22.

45. S. Blaha, Phys. Rev. Lett. 36, 874 (1976).

51. G. E. Volovik, JETP Lett. 28 59 (1978).

46. G. E. Volovik and V. P. Mineev, JETP Lett. 23, 593

(1976).

52. M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen,

and D. S. Hall, Nature 505, 657 (2014).

47. Y. Nambu, Nucl. Phys. B 130, 505 (1977).

53. T. Sh. Misirpashaev and G. E. Volovik, ZhETF 101,

48. Yifung Ng, T. W. B. Kibble, and Tanmay Vachaspati,

1197 (1992); JETP 75, 650 (1992).

Phys. Rev. D 78, 046001 (2008).

54. G. Lazarides and Q. Shafi, arXiv:1904.06880.

49. M. Krusius, A. P. Finne, R. Blaauwgeers, V. B. El-

tsov, and G. E. Volovik, Physica B 329-333,

55. T. T. Heikkilä and G. E. Volovik, New J. Phys. 17,

(2003).

093019 (2015).