ЖЭТФ, 2019, том 156, вып. 4 (10), стр. 700-706

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 April 4, 2019,

revised version April 4, 2019

Accepted for publication April 5, 2019

Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

DOI: 10.1134/S0044451019100110

the ground state of these systems. This includes in par-

ticular the existence of the topologically stable nodes in

There are many topological faces of the superfluid

the fermionic spectrum in bulk and/or on the surface of

phases of³He. These superfluids contain various topo-

superfluids. It appeared that the superfluid phases of

logical defects and textures. The momentum space

liquid³He serve as the clean examples of the topologi-

topology of these superfluids is also nontrivial, as well

cal matter, where the momentum-space topology plays

as the topology in the combined (p, r) phase space,

an important role in the properties of these phases.

giving rise to topologically protected Dirac, Weyl, and

In bulk liquid³He, there are two topologically dif-

Majorana fermions living in bulk, on the surface, and

ferent superfluid phases,³He-A and³He-B [6]. One

within the topological objects. The nontrivial topology

is the chiral superfluid³He-A with topologically pro-

leads to different types of anomalies, which extend the

tected Weyl points in the quasiparticle spectrum. In

Landau-Khalatnikov theory of superfluidity in many

the vicinity of the Weyl points, quasiparticles obey the

different directions.

Weyl equation and behave as Weyl fermions, with all

Superfluid phases of 3He discovered in 1972 [1]

the accompanying effects such as chiral anomaly, chiral

opened the new area of the application of topologi-

magnetic effect, chiral vortical effect, etc. The Adler-

cal methods to condensed matter systems. Due to

Bell-Jackiw equation, which describes the anomalous

the multi-component order parameter which character-

production of fermions from the vacuum, has been ver-

izes the broken symmetries in these phases, there are

ified in experiments with skyrmions in³He-A [7]. An-

many inhomogeneous objects — textures and defects

other phase is the fully gapped time reversal invariant

in the order parameter field — which are protected by

superfluid³He-B. It has topologically protected gapless

topology and are characterized by topological charges.

Majorana fermions living on the surface.

Among them there are quantized vortices, skyrmions

The polar phase of³He stabilized in³He confined

and merons, solitons and vortex sheets, monopoles and

in the nematically ordered aerogel [8] contains Dirac

boojums, Alice strings, Kibble walls terminated by Ali-

nodal ring in the fermionic spectrum.

ce strings, spin vortices with soliton tails, etc. Some

The classification of the topological objects in the

of them have been experimentally identified [2-5], the

order parameter fields revealed the possibility of many

others are still waiting for their creation and detection.

configurations with nontrivial topology, which are de-

The real-space topology, which is responsible for the

scribed by the homotopy groups and by the relative

topological stability of textures and defects, has been

homotopy groups. Some of the topological defects and

later extended to the topology in momentum space,

topological textures are shown in Fig. 1.

which governs the topologically protected properties of

All three superfluid phases discussed here are the

spin-triplet p-wave superfluids, i.e., the Cooper pair

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

has spin S = 1 and orbital momentum L = 1. The

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ЖЭТФ, том 156, вып. 4 (10), 2019

Topological superfluids

³He-A

HQV

3He-B

Fig. 1. (Color online) Some topological objects in topological superfluids. (a) Vortex-skyrmions with two quanta of circulation

(N = 2) in the chiral superfluid³He-A. (b) Interface between³He-A and³He-B in rotation: lattice of skyrmions with N = 2 in

³He-A transforms to the lattice of singly quantized vortices (N = 1) in³He-B. Each skyrmion with N = 2 splits into two merons

with N = 1 (Mermin - Ho vortices). The Mermin-Ho vortex terminates on the boojum — the surface singularity which is the

analog of the Dirac monopole terminating the³He-B string (the N = 1 vortex). (c) Vortex sheet in³He-A. (d) Elements of the

vortex sheet: merons with N = 1 within topological soliton in³He-A. (e) Hedgehog-monopole in the

d-vector field in³He-A.

(f) Singly quantized vortex (N = 1) with spontaneously broken axial symmetry in3He-B as a pair of half-quantum vortices

(N = 1/2) connected by non-topological (dark) soliton. Such vortex has been identified due to Goldstone mode associated with

the spontaneously broken axial symmetry of the vortex core — the twist oscillations of the vortex core propagating along the

vortex line. This Figure is also applicable to the N = 2 vortex in³He-B, which consists of two composite objects — spin-mass

vortices connected by the topological soliton. (g) Half-quantum vortices (HQV) in the polar phase of³He, which in a tilted

magnetic field are connected by the topological spin soliton. Red arrows show direction of magnetic field, and blue arrows show

direction of spin-nematic vector

d, which rotates by π around each half-quantum vortex

order parameter is given by 3 × 3 matrix A_αi (see

A_αi = Δ_AeiΦ

d_α(êi1 + iêi2),

l=ê₁ ×ê₂,

(1)

below Eq. (11)), which transforms as a vector under

where

d is the unit vector of the anisotropy in the spin

SO(3)_S spin rotations (first index) and as a vector un-

space due to spontaneous breaking of SO(3)_S symme-

der SO(3)_L orbital rotations (second index). The order

try;

ê₁ and ê₂ are mutually orthogonal unit vectors;

parameter A_αi comes as the bilinear combination of the

andˆl is the unit vector of the anisotropy in the orbital

fermionic opertators, A_αi ∝ 〈ψσ_α∇_iψ〉, where σ_α are

space due to spontaneous breaking of SO(3)_L symme-

Pauli matrices for the nuclear spin of³He atom.

ˆ-vector also shows the direction of the or-

try. The

bital angular momentum of the chiral superfluid, which

In the ground state of³He-A, the order parameter

emerges due to spontaneous breaking of time reversal

matrix has the form

symmetry.

701

G. E. Volovik

ЖЭТФ, том 156, вып. 4 (10), 2019

In the chiral superfluid, the superfluid velocity v_s

N = 1/2 — the condensed matter analogs of the Alice

of the chiral condensate is determined not only by the

string in particle physics, in which

d changes sign when

condensate phase Φ, but also by the orbital triad ê₁,

circling around the vortex:

ê₂, andl:

(

)

ℏ

(

)

v_s =

∇Φ + êi1∇êi2

(2)

d(r)eiΦ(r) =

x cos

+ ŷ sin

eiφ/2.

(6)

where m is the mass of the³He atom. In non-chiral

superfluids, the vorticity is presented in terms of the

When the azimuthal coordinate φ changes from 0 to

quantized singular vortices with the phase winding

2π along the circle around this object, the vector

d(r)

ΔΦ = 2πN around the vortex core. In chiral superfluid

changes sign and simultaneously the phase Φ changes

³He-A, the vorticity can be continuous. The continu-

by π, giving rise to N = 1/2. The half-quantum vor-

ous vorticity is represented by the texture of the unit

tices have been stabilized only recently and in a dif-

vectorˆl according to the Mermi- Ho relation

ferent phase — in the polar phase of³He confined in

ℏ

aerogel [4].

∇×v_s =

e_ijkl_i∇lj × ∇lk .

(3)

In the ground state of³He-B, the order parameter

matrix has the form

Vorticity is created in the rotating cryostat. In

³He-A, the continuous textures are more easily cre-

A_αi = Δ_BeiΦR_αi ,

(7)

ated than the singular objects with the hard core of

the coherence length size ξ, which formation requires

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

overcoming of large energy barrier. That is why the

Vorticity in the non-chiral superfluid is always singu-

typical object which appears under rotation of cryostat

lar, but it is also presented in several forms. Even the

with³He-A is the vortex-skyrmion. It is the continuous

N = 1 vortex, where Φ(r) = φ, has an unusual struc-

texture of the orbital

l-vector in Fig. 1a without any

ture of the singular vortex core. The first order phase

singularity in the order parameter fields. This texture

transition has been observed, at which the vortex core

represents the vortex with doubly quantized (N = 2)

becomes non-axisymmetric, i. e., the axial symmetry of

circulation of superfluid velocity around the texture,

∮

the vortex is spontaneously broken in the vortex core.

dr · v_s = N κ, where κ = h/2m is the quantum of

The Goldstone mode associated with this symmetry

circulation. The vortex-skyrmions have been identified

breaking was experimentally identified. On the other

in rotating cryostat in 1983. They are described by two

side of the transition, at high pressure, the structure

topological invariants, in terms of the orbital vector

of the vortex core in³He-B is axisymmetric, but it is

and in terms of the spin nematic vector

also nontrivial. In the core, the discrete symmetry is

(

)

∫

∂l

spontaneously broken. As a result, the core does not

m_l =

dx dyl·

(4)

4π

∂x

∂y

contain the normal liquid, but is occupied by the chi-

ral superfluid — the A-phase of³He. The phenomenon

(

)

∫

of the additional symmetry breaking in the core of the

∂^d

m_d =

dx dy

^d·

(5)

topological defect has been also discussed for cosmic

4π

∂x

∂y

strings — the so-called Witten string.

In a high magnetic field, the vortex lattice consists

In the ground state of the polar phase, the order

of isolated vortex-skyrmions with N = 2, m_l = 1, and

parameter matrix has the form

m_d = 0. In the low field, when the magnetic energy

is smaller than the spin-orbit interaction, the vortex-

A_αi = Δ_P eiΦ

d_α zⁱ ,

(8)

skyrmion with m_d = m_l = 1 becomes more prefer-

able. The first order topological transition, at which

where axis z is along the nafen strands. Topology of po-

the topological charge m_d of the skyrmion changes from

lar phase in nafen suggests existence of the half-quan-

0 to 1, has been observed in acoustic experiments. In

tum vortices, which have been observed in the NMR

1994, new type of continuous vorticity has been ob-

experiments. Fig. 1g shows a pair of half-quantum vor-

served in³He-A: the vortex texture in the form of the

tices in transverse magnetic field (red arrows).

vortex sheet, the topological soliton with kinks, the

Later it was found that the half-quantum vor-

Mermin-Ho vortices with N = 1 (Fig. 1d).

tices survive the phase transition to 3He-B, where

In addition to continuous vortex textures, the ro-

the half-quantum vortex is topologically unstable. In

tating state of³He-A may consist of the singular vor-

the B-phase, the half-quantum vortices pinned by the

tices with N = 1 and the half-quantum vortices with

strands of nafen become the termination lines of the

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ЖЭТФ, том 156, вып. 4 (10), 2019

Topological superfluids

C₂

The topologically stable singularities of the Hamil-

Spacetime

tonian or of the Green’s function in the momentum

or momentum-frequency spaces look similar to the

Alice string

real-space topology of the defects and textures, see

1/2 vortex

Kibble

Fig. 3. The Fermi surface, which describes the normal

wall

liquid³He and metals, represents the topologically sta-

ble 2D object in the 4D frequency-momentum space

Antispacetime

(p_x, p_y, p_z, ω), is analogous to the vortex ring. The

Weyl point in³He-A represents the topologically stable

hedgehog in the 3D p-space, Fig. 3b. The Dirac nodal

Fig. 2. (Color online) In³He-B, the half-quantum vortex (ana-

line in the polar phase of³He is the p-space analog of

log of Alice string) looses its topological stability and becomes

the spin vortex, Fig. 3c. The nodeless 2D systems (thin

the termination line of a non-topological domain wall — the

Kibble wall [5]. In terms of the tetrads, the Kibble wall sep-

films of³He-A and the planar phase) and the nodeless

arates the states with different tetrad determinant, and thus

3D system —³He-B — are characterized by the topo-

between the “spacetime” and “antispacetime”. There are two

logically nontrivial skyrmions in momentum space in

roads to antispacetime: the “safe” route around the Alice string

Fig. 3e.

(along the contour C1) or “dangerous” route along C2 across

The normal state of liquid³He belongs to the class

the Kibble wall

of Fermi liquids, which properties at low energy are de-

termined by quasiparticles living in the vicinity of the

Fermi surface. The systems with Fermi surface, such

non-topological domain walls — the analog of Kibb-

as metals, are the most widespread topological mate-

le cosmic walls [5]. In³He-B, the Kibble wall sepa-

rials in nature. The reason for that is that the Fermi

rates the states with different tetrad determinant, and

surface is topologically protected and thus is robust to

thus separates the “spacetime” and “antispacetime”, see

small perturbations. This can be seen on the simple

Fig.2.

example of the Green’s function for the Fermi gas at

The topological stability of the coordinate depen-

the imaginary frequency:

dent objects — defects and textures — is determined

)

by the pattern of the symmetry breaking in these su-

⁽ p²

G-1(ω, p) = iω -

-μ

(12)

perfluids. Now we shall discuss these three phases of

superfluid³He from the point of view of momentum-

The Fermi surface at p = p_F exists at positive chem-

space topology, which describes the topological proper-

ical potential, with p2F /2m = μ. The topological pro-

ties of the homogeneous ground state of the superfluids.

tection is demonstrated in Fig. 3a for the case of the

These superfluids represent three types of topological

2D Fermi gas, where the Fermi surface is the line p =

materials, with different geometries of the topologically

= p_F in (p_x,p_y)-space. In the extended (ω,p_x,p_y)-

protected nodes in the spectrum of fermionic quasipar-

space, this gives rise to a singularity in the Green’s

ticles: Weyl points in³He-A, Dirac lines in the polar

function on the line at which ω = 0 and p = p_F , where

phase, and Majorana nodes on the surface of³He-B.

the Green’s function is not determined. Such singu-

The properties of the fermionic spectrum in the bulk

lar line in momentum-frequency space looks similar to

or/and on the surface of superfluids are determined by

the vortex line in real space: the phase Φ(p, ω) of the

the topological properties of the Bogoliubov - de Gen-

Green’s function

nes Hamiltonian

(

)

G(p, ω) = |G(p, ω)|eiΦ(p,ω)

ϵ(p)

Δ(p)

p² - p2F

H (p) =

ϵ(p) =

(9)

changes by 2π around this line. In general, when the

Δ⁺(p)

-ϵ(p)

2m^∗

Green’s function is the matrix with spin or/and band

indices, the integer valued topological invariant — the

or by the Green’s function

winding number of the Fermi surface — has the follo-

G-1(ω, p) = iω - H .

(10)

wing form:

∮

For the spin triplet p-wave superfluid³He, the gap func-

N = Tr

G(ω, p) ∂_lG-1(ω, p).

(13)

2πi

tion is expressed in terms of the 3 × 3 order parameter

matrix A_αi:

p_i

Here, the integral is taken over an arbitrary contour C

Δ(p) = A_αiσ_α

(11)

around the Green’s function singularity in the D + 1

p_F

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G. E. Volovik

ЖЭТФ, том 156, вып. 4 (10), 2019

Fermi surface

Weyl point

p_z

Dirac

p_z

Normal³He

³He-A

Polar phase

as vortex ring

hedgehog

nodal ring

of³He

in p-space

p (p_yz)

p_x

p_F

= 2

H = +c . p

= 2

³He-B

Flat band on vortex

³He-A

in Weyl superfluid

p_z

p-space skyrmion

Fig. 3. Topological materials as configurations in momentum space. (a) Fermi surface in normal liquid³He is topologically

protected, since the Green’s function has singularity in the form of the vortex ring in the (p, ω)-space. (b) Weyl point in³He-A

as the hedgehog in p-space. It can be also reperesented as the p-space Dirac monopole with the Berry magnetic flux. (c) The

polar phase has the Dirac nodal line in p-space - the counterpart of the spin vortex in real space. (d) The N = ∞ singular vortex

in chiral superfluid³He-A has the 1D flat band terminated by the projections of the Weyl points to the vortex line. In real space

it has analogy with the Dirac monopole terminating the Dirac string. (e) Skyrmion configurations in p-space describe the fully

gapped topological superfluids. The 2D skyrmion describes the topology of the³He-A state in a thin film and the topology of

2D planar phase of³He. The 3D p-space skyrmion describes the superluid³He-B

momentum-frequency space. Due to nontrivial topo-

The Pauli matrices τ1,2,3 and σ_x,y,z correspond to

logical invariant, Fermi surface survives the perturba-

the Bogoliubov-Nambu spin and ordinary spin of³He

tive interaction and exists in the Fermi liquid as well.

atom, respectively; and the parameter c = Δ_A/p_F .

Moreover, the singularity in the Green’s function re-

Now, there are two points in the fermionic spec-

mains if due to interaction the Green’s function has

trum, at K_± = ±p_Fˆl, where the energy spectrum is

no poles, and thus quasiparticles are not well defined.

nullified and the Green’s function is not determined at

The systems without poles include the marginal Fermi

ω = 0. There are several ways of how to describe the

liquid, Luttinger liquid, and the Mott pesudogap state.

topological protection of these two points. In terms of

Under the superfluid transition, the 2 × 2 matrix

the Green’s function, there is the following topological

of the normal liquid Green’s function with spin indices

invariant expressed via integer valued integral over the

transforms to the 4 × 4 Gor’kov Green’s function. The

3D surface σ around the singular point in the 4-mo-

simplified Green’s function, which describes the topo-

mentum space p_μ = (ω, p):

logy of the chiral superfluid³He-A, has the form

e_αβμν

(

)

N =

24π²

∫

G-1

= iω + τ₃

-μ

× Tr

dS^α G∂_p

G-1G∂pμG-1G∂pνG-1 .

(15)

+ c(σ ·

)(τ₁ê₁ · p + τ₂ê₂ · p).

(14)

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ЖЭТФ, том 156, вып. 4 (10), 2019

Topological superfluids

If the invariant (15) is nonzero, the Green’s function has

electrodynamics emerging in the vicinity of the Weyl

a singularity inside the surface σ, and this means that

point leads to many analogs in relativistic quantum

fermions are gapless. The typical singularities have

field theories, including the zero charge effect — the

topological charge N = +1 or N = -1. Close to such

“Moscow zero” by Abrikosov, Khalatnikov, and Lan-

points, the quasiparticles behave as right-handed and

dau [9].

left-handed Weyl fermions, respectively, that is why

In the polar phase of³He, the simplified Hamilto-

such point node in the spectrum is called the Weyl

nian is

)

point. The isolated Weyl point is protected by topo-

⁽ p²

logical invariant (15) and survives when the interaction

H =τ₃

-μ

+ cp_z(σ ·

d)τ₁ .

(22)

between quasiparticles is taken into account.

In³He-A, the topological invariants of the points

It is nullified when p_z = 0 and p2x + p2y = p2F , i. e., the

at K(a) = ±k_Fˆl are correspondingly N = +2 and N =

spectrum of quasiparticles has the nodal line in Fig. 3c.

= -2: the Weyl points are degenerate over the spin of

The nodal line is protected by topology due to the dis-

the³He atoms. Considering only single spin projection

crete symmetry: the Hamiltonian (22) anticommutes

one comes to the 2×2 Bogoluibov-Nambu Hamiltonian

with τ₂, which allows us to write the topological charge:

∮

for the spinless fermions:

N = Tr

τ₂H-1(p)∂_lH(p).

(23)

4πi

H = τ · g(p),

(16)

Here, C is an infinitesimal contour in momentum space

where the vector function g(p) has the following com-

around the line, which is called the Dirac line. The

ponents in³He-A:

topological charge N in Eq. (23) is integer and is equal

to 2 for the nodal line in the polar phase due to spin

g₁ = ê₁ · p, g₂ = ê₂ · p, g₃ =

-μ.

(17)

degeneracy.

³He-B belongs to the same topological class as the

The Hamiltonian (16) is nullified at two points

vacuum of Standard Model in its present insulating

K(a) = ±p_Fˆl, where p = pF and p · ê1 = p · ê2 = 0. At

phase. The topological classes of the 3He-B states

these points the unit vector ĝ(p) = g(p)/|g(p)| has the

can be represented by the simplified Bogoliubov - de

singularity of the hedgehog-monopole type in Fig. 3b,

Gennes Hamiltonian

which is described by the dimensional reduction of the

)

invariant (15):

⁽ p²

H =τ₃

-μ

+τ₁Δ_Bσ ·

d(p),

∫

)

(24)

⁽∂ĝ

∂ĝ

p_i

N =

e_ikl

dSⁱ ĝ ·

(18)

d_α(p) = ±R_αi

8π

∂p_k

∂p_l

p_F

Here, R_αi is the matrix of rotation; the phase of the

where σ now is the 2D spherical surface around the

order parameter in Eq. (7) is chosen either Φ = 0 or

hedghehog.

Φ = π. There are no nodes in the spectrum of fermions:

The hedgehog has N = ±1 and it represents the

the system is fully gapped. Nevertheless,³He-B is the

Berry phase magnetic monopole in terms of the devia-

topological superfluid, which can be seen from the in-

tion of the momentum p from the Weyl point at K(a):

teger valued integral over the former Fermi surface:

(

)

H(a) = e^iατ^α(p_i - K(a)i) + . . .

(19)

∫

∂^d

N_d =

e_ikl

dSⁱ d ·

(25)

8π

∂p_k

∂p_l

Introducing the effective electromagnetic field A(r, t) =

S²

=p_Fˆl(r, t) and effective electric charge q(a) = ±1, one

where N_d = ± depending on the sign in Eq. (24).

obtains

In terms of the Hamiltonian, the topological invari-

H(a) = e^iατ^α(p_i - q(a)A_i) + . . .

(20)

ant can be written as integral over the whole momen-

Such Hamiltonian decsribes the Weyl fermions moving

tum space (or over the Brillouin zone in solids)

in the effective electric and magnetic fields

e_ijk

N_K =

E_eff = -p_F ∂_tl,

B_eff = p_F ∇ ×l,

(21)

24π²

[∫

]

× Tr d³p KH-1∂piHH-1∂pj HH-1∂_p

H ,

(26)

and also in the effective gravitational field represented

by the triad field e^iα(r, t). The effective quantum

705

ЖЭТФ, вып. 4 (10)

G. E. Volovik

ЖЭТФ, том 156, вып. 4 (10), 2019

where K = τ₂ is the matrix which anticommutes with

effect and the spin quantum Hall effect in the absence

the Hamiltonian. One has N_K = 2N_d due to spin de-

of magnetic field.

grees of freedom.

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

The nontrivial topology of³He superfluids leads to

points in the vertices of a cube [12]. This is close

the topologically protected massless (gapless) Majo-

to the high energy physics model with 8 left-handed

rana fermions living on the surface of the superfluid

and 8 right-handed Weyl fermions in the vertices of

or/and inside the vortex core.

a 4D cube. This is one of many examples when the

In conclusion, at the moment the known phases of

topologically protected nodes in the spectrum serve as

liquid³He belong to 4 different topological classes.

an inspiration for the construction of the relativistic

(i) The normal liquid³He belongs to the class of sys-

quantum field theories.

tems with topologically protected Fermi surfaces. The

Fermi surface is described by the first odd Chern num-

Acknowledgements. This work has been sup-

ber in terms of the Green’s function in Eq. (13).

ported by the European Research Council (ERC) under

(ii) Superfluid³He-A and³He-A₁ are chiral super-

the European Union’s Horizon 2020 research and inno-

fluids with the Majorana-Weyl fermions in bulk, which

vation programme (Grant Agreement No. 694248).

are protected by the Chern number in Eq. (15). In the

The full text of this paper is published in the English

relativistic quantum field theories, Weyl fermions give

version of JETP.

rise to the effect of chiral (axial) anomaly. The direct

analog of this effect has been experimentally demon-

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superfluid with topologically protected gapless Majo-

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

rana fermions on the surface. In magnetic field, this

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

phase becomes the higher order topological superfluid.

Helium 3, Dover Publ., New York (2013).

(iv) The most recently discovered polar phase of su-

perfluid³He belongs to the class of fermionic materials

T. D. C. Bevan, A. J. Manninen, J. B. Cook, J. R. Ho-

with topologically protected lines of nodes, and thus

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contains two-dimensional flat band of Andreev-Majo-

rana fermions on the surface of the sample.

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

It is possible that with the properly engineered

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

nanostructural confinement one may reach also new

L. D. Landau, A. A. Abrikosov, and I. M. Khalat-

topological phases of liquid³He including the follow-

nikov, Dokl. Akad. Nauk SSSR 95, 497, 773, 1177

ing.

(1954).

(i) The planar phase, which is the non-chiral super-

10.

N. B. Kopnin and M. M. Salomaa, Phys. Rev. B 44,

fluid with Dirac nodes in the bulk and with Fermi arc

9667 (1991).

of Andreev-Majorana fermions on the surface.

(ii) The 2D topological states in the ultra-thin

11.

G. E. Volovik, J. Rysti, J. T. Makinen, and V. B. El-

film, including inhomogeneous phases of superfluid³He

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films. The films with the³He-A and the planar phase

12.

G. E. Volovik and L. P. Gor’kov, Zh. Eksp. Teor. Fiz.

order parameters belong to the 2D fully gapped topo-

88, 1412 (1985).

logical materials, which experience the quantum Hall

706