Classification of spatially-nonuniform equilibrium states of superfluids
The classification of equilibrium states of superfluid with scalar, vector and tensor order parameters is carried out on the basis of the quasiaveragues concept. The generalization of a requirement of the residual symmetry for nonuniform equilibrium states is given. The admissible requirements of a...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | Classification of spatially-nonuniform equilibrium states of superfluids / М.Y. Kovalevsky, N.N. Chekanova, А.А. Rоzhkov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 351-355. — Бібліогр.: 13 назв. — англ. |
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| author | Kovalevsky, М.Y. Chekanova, N.N. Rоzhkov, A.A. |
| author_facet | Kovalevsky, М.Y. Chekanova, N.N. Rоzhkov, A.A. |
| citation_txt | Classification of spatially-nonuniform equilibrium states of superfluids / М.Y. Kovalevsky, N.N. Chekanova, А.А. Rоzhkov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 351-355. — Бібліогр.: 13 назв. — англ. |
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| description | The classification of equilibrium states of superfluid with scalar, vector and tensor order parameters is carried out on the basis of the quasiaveragues concept. The generalization of a requirement of the residual symmetry for nonuniform equilibrium states is given. The admissible requirements of a spatial symmetry in the terms of integrals of motion are found. The connection of these requirements with helicoidal structure of vectors of a spin and spatial anisotropy is established. At some restrictions is shown, that the equilibrium structure of an order parameter can be represented as product of a nonuniform part, depending on spatial coordinates, and homogeneous part of an order parameter.
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CLASSIFICATION OF SPATIALLY-NONUNIFORM EQUILIBRIUM
STATES OF SUPERFLUIDS
М.Yu. Kovalevsky, N.N. Chekanova, А.А. Rоzhkov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: mik@kipt.kharkov.ua
The classification of equilibrium states of superfluid with scalar, vector and tensor order parameters is carried
out on the basis of the quasiaveragues concept. The generalization of a requirement of the residual symmetry for
nonuniform equilibrium states is given. The admissible requirements of a spatial symmetry in the terms of integrals
of motion are found. The connection of these requirements with helicoidal structure of vectors of a spin and spatial
anisotropy is established. At some restrictions is shown, that the equilibrium structure of an order parameter can be
represented as product of a nonuniform part, depending on spatial coordinates, and homogeneous part of an order
parameter.
PACS: 67.57.-z, 05.30.Ch
1. INTRODUCTION
Investigations into the phenomenon of superfluidity
in 3Не have resulted in the prediction and discovery of a
number of superfluid phases. Among the discovered
phases we mention the isotropic B-phase, anisotropic A-
and А1-phases in the presence of a magnetic field. The
other phase states have not been found in experiment.
The classification of homogeneous equilibrium
superfluid states in 3He was carried out in [1-5] on the
basis of the Ginzburg-Lаndau theory or with the use of
group-theoretical methods.
The papers [6-8] dealt with nonuniform equilibrium
states in superfluid 3Не. Within the framework of model
expressions for a free energy, those authors have
elucidated the stability conditions of helicoidal
structures. In papers [9,10] the stability boundaries of
the mentioned states were extended to a wider range of
temperatures. The interest in this problem has increased
due to its close connection with the problem of critical
velocities in superfluid 3Не. However, no classification
of equilibrium inhomogeneous states was performed.
The purpose of the present study has been to classify
superfluid phases for singlet or triplet pairing on the
basis of the quasiaverages concept [11,12], taking into
account possible nonhomogeneous equilibrium
structures. We have formulated the condition of residual
symmetry of the equilibrium state and the simultaneous
condition of spatial symmetry. Nonhomogeneous
equilibrium structures of order parameter are found.
2. NOPMAL EQUILIBRIUM STATE OF
FERMI LIQUID
The Gibbs statistical operator is given by the
standard expression
( )NYYw ˆ
4ˆ0expˆ −−Ω= H , (2.1)
where H ˆ - Hamiltonian, N̂ - particle number operator,
TY ≡− 1
0 - temperature, kYY µ≡− 0/4 - chemical
potential. For simplicity we assume that the condensed
medium as a whole is at rest, and the effective magnetic
field is equal to zero. The statistical operator has the
following symmetry properties:
[ ] 0ˆ,ˆ =kw P , [ ] 0ˆ,ˆ =Hw , [ ] 0ˆ,ˆ =Nw ,
[ ] 0ˆ,ˆ =αSw , [ ] 0ˆ,ˆ =kw L , (2.2)
kPˆ - operator of impulse, αŜ and iLˆ - operators of spin
and orbital momenta. The first three relations represent
the space-time translational invariance and the phase
invariance. The symmetry conditions relative to the
rotations in spin and configurational spaces imply the
neglect of weak dipole and spin-orbit interactions at
characterization of equilibrium state.
The unitary transformations gGiU ˆexp≡ , where
H,PL ˆˆ,ˆ,ˆ,ˆˆ
NSG ∈ are the integrals of motion and g
denotes the parameters of transformation, leave the
Gibbs statistical operator invariant
wUwU ˆˆ =+ . (2.3)
Note that the averages of the form ( )[ ] 0ˆ,ˆˆ ≡xbGwSp at
an arbitrary operator ( )xb̂ . If ( ) ( )xaxb ∆≡ ˆˆ , then,
taking into account that the commutators ( )[ ]xaG ∆̂,ˆ
are linear and homogeneous in the order parameter
operators ( )xa∆̂ , we come that the average normal-
state equilibrium order parameters vanish
( ) 0ˆˆ =∆ xawSp .
3. EQUILIBRIUM. SINGLET PAIRING OF
FERMI SUPERFLUID
The quasiaverage ( ) ( )xaxa ˆ= in the equilibrium
state with broken symmetry is given by the formula [11]
( ) ( ),ˆˆlim
0
limˆ xawSp
V
xa νν ∞→→
≡ (3.1)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 351-355. 95
( )FYNYYw ˆ
0
ˆ
4ˆ0expˆ ννν −−−Ω≡ H .
Here the operator F̂ is a linear functional of the order
parameter operator
( ) ( )( )..ˆ3ˆ chxxfxdF +∆∫= ,
)(xf is the function of coordinates, conjugate of the
order parameter, which sets the equilibrium values of
the latter. The Fermi superfluid with singlet pairing is
characterized by the scalar order parameter
( ) ( ) ( ) ( )xxix ψσψ ˆ2ˆ2/ˆ ≡∆ . (3.2)
Here ( )xψˆ is the Fermi operator of particle annihilation
at the point x , and −2σ is the Pauli matrix. The order
parameter operator satisfies the commutation relations
( )[ ] ( )xxN ∆−=∆ ˆ2ˆ,ˆ , ( )[ ] ,0ˆ,ˆ =∆ xSi α
( )[ ] ( )xkxki ∆− ∇=∆ ˆˆ,ˆP ,
( )[ ] ( )xlkxiklxii ∆∇−=∆ ˆˆ,ˆ εL . (3.3)
We shall consider homogeneous equilibrium states
to establish possible structures of the scalar order
parameter. The homogeneous equilibrium statistical
operator satisfies the relation
[ ] 0ˆ,ˆ =kw P . (3.4)
The analysis of homogeneous subgroups of the residual
symmetry of equilibrium states is realizable on the basis
of the relation
[ ] 0ˆ,ˆ =Tw , (3.5)
where the residual symmetry generator T̂ presents a
linear combination of integrals of motion
( )ξαα TNcSbiiaT ˆˆˆˆˆ ≡++≡ L , (3.6)
with real numerical parameters ( ξα ≡cbia ,, ).
According to relations (3.4), (3.5), we have
[ ] ( ) 0ˆˆ,ˆ =∆ xTwiSp ,
[ ] ( ) 0ˆˆ,ˆ =∆ xiwiSp P .
Whence, in view of (3.3), we have 0=c , as ( ) 0≠∆ x .
The residual symmetry generator takes on the form
αα SbiiaT ˆˆˆ +≡ L . (3.7)
The order parameter in the equilibrium state for this
case looks like
( ) ( ) ϕη iYx expˆ =∆ .
Here η is the modulus of the order parameter, and ϕ is
the superfluid phase.
Let us now consider the states of equilibrium, which
have no translational invariance (3.4). The physical
possibilities of violation of this equilibrium state
invariance are as follows: (i) violation of phase
invariance (the superfluid impulse is nonzero), (ii)
violation of symmetry relative to spin rotations (the
vector of a magnetic spiral is distinct from zero), (iii)
violation of rotation symmetry in the configurational
space (the vector of a cholesteric spiral is not equal to
zero). Such symmetry of the equilibrium states can be
given by the following relation:
[ ] 0ˆ,ˆ =kPw ,
jkjtSkqNkpkkP LP ˆˆˆˆˆ −−−≡ αα , (3.8)
here kjkk tqp ,, α are certain real parameters. The
residual symmetry generator for these states now
includes the operator of impulse
iidNcSbiiaT PL ˆˆˆˆˆ +++≡ αα . (3.9)
According to the relations
[ ] ( ) 0ˆˆ,ˆ =∆ xTwiSp ,
[ ] ( ) 0ˆˆ,ˆ =∆ xiPwiSp
we come to the equalities that relate the parameters of
spatial symmetry generators to the parameters of
residual symmetry generator as
0=vpjuvkjt ε , ( ) ( )xkipxk ∆=∆∇ 2 , (3.10)
0=lpiklia ε , 0=+ ipidc .
We shall find additional relationships between these
parameters using the Jacobi identity for the operators
PTw ˆ,ˆ,ˆ
and kPiPw ˆ,ˆ,ˆ . Taking into account the
properties of residual and spatial symmetries we have
[ ][ ] ( ) 0ˆˆ,ˆ,ˆ =∆ xkPTwSp ,
[ ][ ] ( ) 0ˆˆ,ˆ,ˆ =∆ xkPiPwSp .
These formulas lead to an admissible structure of
residual and spatial symmetry generators
( )NipliidSbiialT ˆˆˆˆˆ −++≡ PL αα , (3.11)
jjlkAlSkqNkplkkP LP ˆˆˆˆˆ −−−≡ αα .
Let us consider a special case of 0=αkq and 0=A
[12]. The requirement of spatial symmetry
[ ] ,0ˆ,ˆ =kPw NkpkkP ˆˆˆ −≡ P , (3.12)
means that a macroscopically great number of particles
can be in the state with impulse p . The symmetry of
the equilibrium state relative to rotations in
configurational and spin spaces is not violated and is
determined by the formulas
[ ] ,ˆ,ˆ 0=kw L [ ] 0=αSw ˆ,ˆ . (3.13)
Relations (3.12) allow one to find the coordinate
dependence of the order parameter in the
nonhomogeneous equilibrium state
( ) ( ) ( ) ( )xipYxwSpx ϕη 2exp,ˆˆ =∆=∆ ,
( ) ( )0ϕϕ += xpx . (3.14)
4. EQUILIBRIUM. VECTOR ORDER
PARAMETER OF FERMI SUPERFLUID
The vector order parameter is given by formulae [13]
( ) ( ) ( ) ( ) ( )xaaaxa
i
x ''ˆ'2'2ˆ
2
ˆ
µψµ µασστµψα =∆ .(4.1)
According with this definition we have
( )[ ] ( )xxaN αα ∆−=∆ ˆˆ,ˆ , 2,1=a ,
( )[ ] ( ),ˆˆ,ˆ xxSi γα β γεβα ∆−=∆
96
( )[ ] ( )xkxki αα ∆− ∇=∆ ˆˆ,ˆP , (4.2)
( )[ ] ( )xljxkjlxki αεα ∆∇−=∆ ˆˆ,ˆL .
Residual symmetry generator has the form
aNacSbiiaT ˆˆˆˆ ++≡ ααL .
Therefore we get equation
( )[ ] 021 =∆+− βα βδγα β γε ccbi .
Then we can write
( ) ( )[ ] 022
2121 =−++ bcccc .
Possible equilibrium structures of vector order
parameter and residual symmetry generators are
presented on Table 1:
Table 1
Residual symmetry
Generator
Order
Parameter
α∆
( )2
ˆ
1
ˆ
1
2ˆˆ NNcSbdiia −++ ααL αAd
( )
( )2
ˆ
1
ˆ
1
2
2
ˆ
1
ˆ
2
1ˆˆ
NNc
NNSdbiia
−+
++±+
ααL
( )αα ifeB ±
5. HOMOGENEOUS EQUILIBRIUM STATES
OF SUPERFLUID 3НE
The order parameter of a superfluid fluid with triplet
pairing contains the spin index 3,2,1=α corersponding
to the spin angular momentum 1=s , and the vectorial
index 3,2,1=k relevant, by virtue of Pauli's exclusion
principle, to the orbital momentum 1=l . As an order
parameter operator ( )xkα∆̂ it is convenient to choose
[12]
( ) ( ) ( ) ( ) ( )xxkxkxxk ψασσψψασσψα ˆ2ˆˆ2ˆˆ ∇−∇≡∆
(5.1)
Here −ασ are the Pauli matrices. According to this
definition, the following equalities are valid:
( )[ ] ( ),ˆˆ,ˆ xixiSi γα β γεβα ∆−=∆
( )[ ] ( )xixiN ββ ∆−=∆ ˆ2ˆ,ˆ , (5.2)
( )[ ] ( )xikxiki αα ∆− ∇=∆ ˆˆ,ˆP ,
( )[ ] ( ) ( )xlkilxiljxkjlxiki αεαεα ∆−∆∇−=∆ ˆˆˆ,ˆL .
The operator violating the symmetry of equilibrium
state represents a linear functional of the order
parameter operator
( ) ( )( )..ˆ3ˆ chxkfxkxdF +∆∫= αα (5.3)
The quasiaverage value of the order parameter is the
function of thermodynamic parameters and the
functional of ( )xkf α
( ) ( ) ( )( )xfYkxkwSpxk ,ˆˆ ααα ∆=∆=∆ (5.4)
By virtue of algebra (5.2) and symmetry relations (4.8)-
(4.10) we obtain the equality, defining the equilibrium
structure of the order parameter( ) 02 =∆++ jkjickjbikjia βδγ βδδα β γεαγ βδε .
The nonzero solution for the order parameter is
provided by the following condition:
kjickjbikjia δβ γδδα β γεαγ βδε 2det ++ =0.
So, we have
( )( ) ( )[ ]
( )[ ] .04
4442
22
222222
=−+×
×−−−−
cba
cbacbcaic
The results of classification by using this equation are
presented in Table 2. There are described 12 anisotropic
phases and one isotropic phase of superfluid 3Не
homogeneous states.
6. NONUNIFORM EQUILIBRIUM STATES
OF SUPERFLUID PHASES 3Нe
First we shall consider the spatial symmetry
subgroups, the generator of which consists of two
operators.
Case I: The spatial symmetry generator is
NkpkkP ˆˆˆ −≡ P . (6.1)
From this definition and taking into account algebra
(5.2)we shall obtain the following equation for the order
parameter
( ) ( )xkiipxki ββ ∆=∆∇ 2 . (6.2)
Its solution has the form
( ) ( ) ( )02
k
xiexk β
ϕ
β ∆=∆
, ( ) xpx += ϕϕ , (6.3)
here ( )0kβ∆ is the homogeneous part of the order
parameter independent of the coordinate. The conditions
of nonviolated symmetry and the spatial symmetry lead
to the equation for ( )0kβ∆ and the constraint of
parameters,
02 =∆+∆+∆ iciiblkilka βγα β γεαβε ,
0=× pa
, (6.4)
where dpcc
+≡ . For the homogeneous part of the
order parameter ( )0kβ∆ (6.3), the procedure of
classification considered above is valid.
Case II: The operator of spatial symmetry looks like
αα SkqkkP ˆˆˆ −≡ P . (6.5)
The requirement of spatial symmetry results in the
equation for the order parameter
( ) ( )xkiqxki γα β γεαβ ∆=∆∇ . (6.6)
The Jacobi identity for the operators kPiPw ˆ,ˆ,ˆ allows
one to establish the structure of parameter αiq :
αα niqiq = . (6.7)
.
97
Table 2. Classification of possible equilibrium states with tensor order parameters
Residual symmetry generator sm lm Order parameter Phase
αα SRi
ˆˆ +iL - -
iRα∆ B
N
m
l l ˆ
2
ˆ −L
NmSd s ˆ
2
ˆ −
1±
0
1±
0
0
1±
1±
0
( )kk inmd α∆
( ) klife αα ∆
( )( )kk inmife αα∆
kldα∆
А
β
A1
Polar
NmlmmSd ssl
ˆ
2
1ˆ2ˆ −− L
0 , 1±
1±
0
0
1±
1±
( )kClkBnkAmd ++α
( )( ) ( )kinkmBdifekinkmA ααα +
( ) ( )kClkBnkAmife ++αα
-
A1+A
-
NmSdmml lls
ˆ
2
1ˆ2ˆ −−
L
0
1±
1±
1,0 ±
1±
0
( ) klCdBfAe ααα ++
( )( ) ( )αααα ifekBlifekinkmA +
( ) ( )kk inmCdBfAe ααα ++
-
A1+ β
A2
NmmSdl sl ˆ
2
ˆˆ +
−+
L
1,0 ±
0
1±
± 1
1,0
1±
0
± 1
( ) ( ) klCdkAnkBmfkBnkAme ααα ++−++
( ) ( )kinkmBdifekAl ααα +
( ) ( )kinkmife αα∆
ς
ε
1A
Note. A,B,C are arbitrary complex numbers
Here kq is the magnetic spiral vector, αn is the axis of
anisotropy in the spin space. The equilibrium value of
the order parameter with this spatial symmetry has the
form
( ) ( )( ) ( )0kxnaxk γθβ γβ ∆=∆
,
( ) xqx += θθ , (6.8)
where β γa is the orthogonal matrix of spin rotation.
The requirement of residual symmetry (4.9) with due
regard for spatial symmetry (6.5) allows us to obtain the
equation for the homogeneous part of the order
parameter
02 =∆+∆+∆ iiciblkilka βγα β γεαβε ,
ααα nqdbb +≡
and the constraint of the symmetry parameters
0=× nb
, 0=nqjmnja ε .
Сase III: The spatial symmetry is defined by the
equality
jkjtkkP LP ˆˆˆ −≡ . (6.9)
The requirements of spatial and residual symmetries
(6.9) (3.5), the Jacobi identities with due regard for the
algebra (4.2) lead to the admissible structure of the
matrix ijt :
klitlikt = . 6.10)
The nonhomogeneous part of order parameter is written as
98
( ) ( )( ) ( )0kxlikaxi γψγ ∆=∆
. (6.11)
Here ( )( )xlika ψ
is the orthogonal matrix of rotation
around the axis l
in configurational space by an angle
( ) xltx += ψψ . This solution describes the helicoidal
structure. The parameter 12 −tπ determines a pitch of a
helicoid, whose direction is given by the unit vector l
.
The condition of residual symmetry with account of
(6.9),(6.11) leads to the equation for the homogeneous
part of order parameter
02 =∆+∆+∆ iiciblkilka βγα β γεαβε ,
dlitliaia
+≡
and the constraint of the parameters of symmetry
0=× la
.
Case IV: The spatial symmetry operator looks like
jkjkkkk ltlSnqNpP LP ˆˆˆˆˆ −−−≡ αα . (6.12)
The structure of the order parameter has the form
( ) ( ) ( )( ) ( )( ) ( )02
kxlikaxnaxiexi γψθβ γ
ϕ
β ∆=∆
The equation for the homogeneous part of the order
parameter ( )0kγ∆ is as follows
( ) ( ) ( ) 00200 =∆+∆+∆ kcilbliklia βγα β γεαβε ,
dlitliaia
+≡ , ααα nqdbb +≡ ,
dpcc
+≡ .
The restrictions on the parameters idcbia ,,, α of the
generator T̂ and the parameters kltnkqkp ,,,, α of the
spatial symmetry operator kP̂ result in the collinearity
of vectors alqp ,,, , and also of nb
, .
CONCLUSION
It has been demonstrated that nonhomogeneous
structures of order parameter can be presented as a
product of the nonhomogeneous part of the order
parameter dependent on spatial coordinates by the
homogeneous part. In the general case, the
nonhomogeneous part is the product of orthogonal
matrices of rotation in spin and configuration spaces by
the oscillation phase term. For the renormalized
homogeneous part of order parameter, the traditional
procedure of classification is valid.
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99
CLASSIFICATION OF SPATIALLY-NONUNIFORM EQUILIBRIUM STATES OF SUPERFLUIDS
1. INTRODUCTION
2. NOPMAL EQUILIBRIUM STATE OF FERMI LIQUID
3. EQUILIBRIUM. SINGLET PAIRING OF FERMI SUPERFLUID
4. EQUILIBRIUM. VECTOR ORDER PARAMETER OF FERMI SUPERFLUID
Table 1
5. HOMOGENEOUS EQUILIBRIUM STATES OF SUPERFLUID 3НE
6. NONUNIFORM EQUILIBRIUM STATES OF SUPERFLUID PHASES 3Нe
CONCLUSION
|
| id | nasplib_isofts_kiev_ua-123456789-80049 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:21:02Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kovalevsky, М.Y. Chekanova, N.N. Rоzhkov, A.A. 2015-04-09T16:37:27Z 2015-04-09T16:37:27Z 2001 Classification of spatially-nonuniform equilibrium states of superfluids / М.Y. Kovalevsky, N.N. Chekanova, А.А. Rоzhkov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 351-355. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 67.57.-z, 05.30.Ch https://nasplib.isofts.kiev.ua/handle/123456789/80049 The classification of equilibrium states of superfluid with scalar, vector and tensor order parameters is carried out on the basis of the quasiaveragues concept. The generalization of a requirement of the residual symmetry for nonuniform equilibrium states is given. The admissible requirements of a spatial symmetry in the terms of integrals of motion are found. The connection of these requirements with helicoidal structure of vectors of a spin and spatial anisotropy is established. At some restrictions is shown, that the equilibrium structure of an order parameter can be represented as product of a nonuniform part, depending on spatial coordinates, and homogeneous part of an order parameter. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum fluids Classification of spatially-nonuniform equilibrium states of superfluids Классификация пространственно-неоднородных состояний равновесия сверхтекучих жидкостей Article published earlier |
| spellingShingle | Classification of spatially-nonuniform equilibrium states of superfluids Kovalevsky, М.Y. Chekanova, N.N. Rоzhkov, A.A. Quantum fluids |
| title | Classification of spatially-nonuniform equilibrium states of superfluids |
| title_alt | Классификация пространственно-неоднородных состояний равновесия сверхтекучих жидкостей |
| title_full | Classification of spatially-nonuniform equilibrium states of superfluids |
| title_fullStr | Classification of spatially-nonuniform equilibrium states of superfluids |
| title_full_unstemmed | Classification of spatially-nonuniform equilibrium states of superfluids |
| title_short | Classification of spatially-nonuniform equilibrium states of superfluids |
| title_sort | classification of spatially-nonuniform equilibrium states of superfluids |
| topic | Quantum fluids |
| topic_facet | Quantum fluids |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80049 |
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