Pis’ma v ZhETF, vol. 111, iss. 10, pp. 689 - 690
© 2020
May 25
Spin vortex lattice in the Landau vortex-free state of rotating
superfluids
G. E. Volovik1)
Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
Submitted 11 April 2020
Resubmitted 11 April 2020
Accepted 11
April 2020
DOI: 10.31857/S1234567820100079
In the rotating vessel the lattice of mass vortices rep-
superfluid velocity of the background mass superfluid
resents the ground state or the thermal equilibrium state
and superfluid velocity of magnon BEC, is another real-
of the rotating superfluid. For spin vortices the situation
ization of the Andreev-Bashkin effect in superfluid mix-
is different. Orbital rotations do not act on the spin vor-
tures, when the superfluid current of one component de-
tices, and the other external or internal fields are needed
pends on the superfluid velocity of another component
for the formation of the lattice, such as Dzyaloshinskii-
[7]. The counterflow vs - vn together with spin density
Moriya interaction [1, 2], which leads to the formation
plays the similar role as Dzyaloshinskii-Moriya interac-
of skyrmion lattice. Here we show that the lattice of
tion in magnets, which violates the space inversion sym-
spin vortices can be created in rotating vessel, if the
metry and leads to formation of skyrmion lattices, see,
formation of the mass vortices is suppressed. The Lan-
e.g., [8]. The mixed term modifies the kinetic energy:
dau vortex-free state in the rotating vessel (the analog of
1
Fgrad =
mMnMv2M + mMnMvM · (vs - vn).
(4)
Meissner state in superconductors) acts on spin vortices
2
as rotation acts on mass vortices.
We consider the Landau state in the container ro-
We start with the Bose-Einstein condensation of
tating with angular velocity Ω, when the normal com-
magnons (magnon BEC), which is realized in superfluid
ponent of the liquid has the solid body rotation with
3He-B as the homogeneously precessing domain (HPD)
velocity vn = Ω × r, while the superfluid component of
[3-5]. Magnon BEC is characterized by the density of
the B-phase is vortex-free, vs = 0, and thus
magnons nM = S - Sz, where S = χH is spin density
1
in magnetic field; Sz is the projection of the precessing
Fgrad =
mMnM (vM - Ω × r)2 .
(5)
2
spin on magnetic field, see review [6]. The magnon BEC
This means that the Landau state in the rotating ves-
has the superfluid velocity
sel acts on spin superfluid (magnon BEC) in the same
ℏ
way as rotation acts on mass superfluid, i.e., it should
vM =
∇α,
(1)
mM
lead to formation of spin vortices, in which the phase α
has 2π winding (single spin vortex has been constructed
where α is the angle of the precession, which plays the
and identified [9]). So, if the creation of mass vortices is
role of the phase of magnon BEC; and mM is magnon
suppressed, but the creation of spin vortices is allowed,
mass. In spin dynamics, the magnon density nM and the
one obtains the state with the lattice of spin vortices.
phase α are canonically conjugate variables, and thus
The number of these spin vortices in the Landau
P = nM∇α = mMnMvM,
(2)
state in rotating vessel is determined by the circulation
quantum of spin vortex κM = 2πℏ/mM and by angular
represents the momentum density of the magnon field.
velocity. That is why the number of spin vortices in the
In the moving superfluid the magnon BEC acquires
lattice in the Landau state can be expressed in terms
the Doppler shift energy term:
of the equilibrium number of quantized mass vortices in
the fully equilibrium rotating state:
Fmix = P · (vs - vn) = mMnMvM · (vs - vn).
(3)
Nspin
κ3
Here vs and vn are correspondingly superfluid and nor-
=
=
mM ,
(6)
Nmass
κM
2m3
mal velocities of the B-phase. This term, which mixes
where κ3 = 2πℏ/2m3 is the quantum of circulation in
1)e-mail: grigori.volovik@aalto.fi
superfluid3He-B; m3 is the mass of3He atom; magnon
8
Письма в ЖЭТФ том 111 вып. 9 - 10
2020
689
690
G. E. Volovik
mass is mM = ωL/2c2s, where ωL is Larmor frequency,
3.
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and cs is the speed of spin waves in3He-B [6]. So we
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4.
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5.
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6.
Yu. M. Bunkov and G. E. Volovik, in Novel Superfluids,
absent, while spin vortices form the lattice.
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Similar effect of the formation of the vortex lattice
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7.
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8.
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914 (2019).
this superfluid with Dirac nodal line [22]. According to
9.
A. S. Borovik-Romanov, Yu.M. Bunkov, V. V. Dmitriev,
Brauner and Moroz [23], in the presence of both the
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10.
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(2006).
term in Eq. (3):
11.
R. Sh. Askhadullin, V. V. Dmitriev, D. A. Krasnikhin,
Fmix = S∇α · (vs - vn),
(7)
P. N. Martynov, A. A. Osipov, A. A. Senin, and
A. N. Yudin, JETP Lett. 95, 326 (2012).
where α is the angle of the unit
d-vector, which describes
12.
V. V. Dmitriev, A. A. Senin, A.A. Soldatov, and
the spin part of the order parameter, and S = χH is spin
A. N. Yudin, Phys. Rev. Lett. 115, 165304 (2015).
density in magnetic field. In the Landau state of the po-
13.
V. V. Dmitriev, A. A. Soldatov, and A.N. Yudin, Phys.
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Rev. Lett. 120, 075301 (2018).
vs = 0, the gradient energy for spin textures is:
14.
V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, and
(
)2
A. N. Yudin, JETP Lett. 110, 734 (2019).
1
S
Fgrad =
ρspin
∇α -
Ω×r
,
(8)
15.
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2
ρspin
day 71, 30 (2018).
which should lead to the lattice of spin vortices. The
16.
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number of equilibrium spin vortices in the Landau state
17.
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in the vessel of radius R is:
(2017).
18.
T. Hisamitsu, M. Tange, and R. Ikeda, Phys. Rev. B
χH
Nspin =
ΩR2.
(9)
101, 100502 (2020).
ρspin
19.
S. Autti, V. V. Dmitriev, J. T. Mäkinen, A. A. Solda-
So, in the spin-triplet superfluids, such as superfluid
tov, G. E. Volovik, A. N. Yudin, V. V. Zavjalov, and
V. B. Eltsov, Phys. Rev. Lett. 117, 255301 (2016).
3He [24, 25], the Landau state of the superfluid in the
20.
J. T. Mäkinen, V. V. Dmitriev, J. Nissinen, J. Rysti,
rotating container can be the source of the formation of
G. E. Volovik, A. N. Yudin, K. Zhang, and V. B. Eltsov,
the vortex lattice of spin vortices. For the experimen-
Nat. Commun. 10, 237 (2019).
tal realization of spin-vortex lattice the sufficently large
21.
G. E. Volovik and K. Zhang, arXiv:2002.07578.
magnetic field is required. Similar phenomenon may oc-
22.
S. Autti, J. T. Mäkinen, J. Rysti, G. E. Volovik,
cur in rotating neutron stars (review on superfluidity
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and superconductivity in neutron stars see in [26]).
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This work has been supported by the European
(2019).
Research Council (ERC) under the European Union’s
24.
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2020
research and innovation programme
Helium 3, Taylor and Francis, London (1990).
(Grant Agreement # 694248).
25.
T. Mizushima, Ya. Tsutsumi, T. Kawakami, M. Sato,
I thank Vladimir Eltsov for discussions.
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Full text of the paper is published in JETP Letters
022001 (2016).
journal. DOI: 10.1134/S0021364020100045
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2020
