On equilibrium states of superfluid with d-wave pairing
The equilibrium states of quantum fluid with d-pairing classification is carried out on the basis of the quasiaverages concept. It is shown, that the set of such an equilibrium states can be classified in the terms of the orbital momentum quantum number, relevant to the projection of the orbital mom...
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irk-123456789-801112015-04-12T03:02:08Z On equilibrium states of superfluid with d-wave pairing Ivashin, A.P. Kovalevsky, M.Y. Chekanova, N.N. Nuclear reactions The equilibrium states of quantum fluid with d-pairing classification is carried out on the basis of the quasiaverages concept. It is shown, that the set of such an equilibrium states can be classified in the terms of the orbital momentum quantum number, relevant to the projection of the orbital momentum of the superconducting electron pair on an anisotropy direction. The explicit expression of three admissible unbroken symmetry generators and relevant order parameter equilibrium values are found. 2002 Article On equilibrium states of superfluid with d-wave pairing / A.P. Ivashin, M.Y. Kovalevsky, N.N. Chekanova // Вопросы атомной науки и техники. — 2002. — № 2. — С. 40-42. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 67.57.-z, 05.30. Ch http://dspace.nbuv.gov.ua/handle/123456789/80111 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Nuclear reactions Nuclear reactions Ivashin, A.P. Kovalevsky, M.Y. Chekanova, N.N. On equilibrium states of superfluid with d-wave pairing Вопросы атомной науки и техники |
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The equilibrium states of quantum fluid with d-pairing classification is carried out on the basis of the quasiaverages concept. It is shown, that the set of such an equilibrium states can be classified in the terms of the orbital momentum quantum number, relevant to the projection of the orbital momentum of the superconducting electron pair on an anisotropy direction. The explicit expression of three admissible unbroken symmetry generators and relevant order parameter equilibrium values are found. |
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Article |
| author |
Ivashin, A.P. Kovalevsky, M.Y. Chekanova, N.N. |
| author_facet |
Ivashin, A.P. Kovalevsky, M.Y. Chekanova, N.N. |
| author_sort |
Ivashin, A.P. |
| title |
On equilibrium states of superfluid with d-wave pairing |
| title_short |
On equilibrium states of superfluid with d-wave pairing |
| title_full |
On equilibrium states of superfluid with d-wave pairing |
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On equilibrium states of superfluid with d-wave pairing |
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On equilibrium states of superfluid with d-wave pairing |
| title_sort |
on equilibrium states of superfluid with d-wave pairing |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2002 |
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Nuclear reactions |
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http://dspace.nbuv.gov.ua/handle/123456789/80111 |
| citation_txt |
On equilibrium states of superfluid with d-wave pairing / A.P. Ivashin, M.Y. Kovalevsky, N.N. Chekanova // Вопросы атомной науки и техники. — 2002. — № 2. — С. 40-42. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT ivashinap onequilibriumstatesofsuperfluidwithdwavepairing AT kovalevskymy onequilibriumstatesofsuperfluidwithdwavepairing AT chekanovann onequilibriumstatesofsuperfluidwithdwavepairing |
| first_indexed |
2025-07-06T04:02:05Z |
| last_indexed |
2025-07-06T04:02:05Z |
| _version_ |
1836868725071937536 |
| fulltext |
ON EQUILIBRIUM STATES
OF SUPERFLUID WITH d-WAVE PAIRING
A.P. Ivashin, M.Y. Kovalevsky, N.N. Chekanova
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: ivashin@kipt.kharkov.ua
The equilibrium states of quantum fluid with d-pairing classification is carried out on the basis of the
quasiaverages concept. It is shown, that the set of such an equilibrium states can be classified in the terms of the
orbital momentum quantum number, relevant to the projection of the orbital momentum of the superconducting
electron pair on an anisotropy direction. The explicit expression of three admissible unbroken symmetry generators
and relevant order parameter equilibrium values are found.
PACS: 67.57.-z, 05.30. Ch
INTRODUCTION
Classification of condensed media equilibrium states
which, using the phenomenological Ginzburg-Landau
approach, requires knowledge of a free energy as
functions of the order parameter and essentially depends
on an considered model aspect. Other group-theoretic
approach is based on the unbroken symmetry
representation of an equilibrium degenerated state as on
the subgroup of a normal phase symmetry.
Corresponding transformational properties of an order
parameter at the transformations of Hamiltonian
symmetry are essential in this approach. This review is
free from any model suppositions about a free energy
aspect. The classification of homogeneous states within
the framework of both indicated approaches was carried
out for superfluid 3Не [1-3], which is featured by a
tensor order parameter. The problem of classification of
possible nonuniform superfluid equilibrium states of this
quantum fluid is researched in [4]. Other important
example of the degenerated condensed matter is the
superfluid in a state of d-pairing, which also is featured
by the tensor order parameter [5]. The possible states
classification for this pairing is carried out on the basis
of the phenomenological approach in [5-6]. The
microscopic approach for homogeneous states
classification of an equilibrium which is based on the
quasiaverages concept [7-8,4] was proposed in this
work. The admissible symmetry properties of a quantum
fluid equilibrium state and relevant order parameter
structures are at nonzero values of an order parameter
from the unbroken symmetry requirements.
THE BASIC EQUATIONS
The statistical physics of condensed matter with the
spontaneously broken symmetry theoretical base is the
concept of quasiaverages by N.N. Bogoliubov [7],
which extends Gibbs statistical operator on degenerated
condensed states. According to [7], the quasiaverages of
a physical value in a state of a statistical equilibrium
with the broken symmetry is determined by the formula
( ) ),(ˆˆlim
0
limˆ xawSp
V
xa νν ∞→→
≡ (1)
)ˆˆexp(ˆ FaaYw νγνν −−Ω≡ . (2)
Here −aγˆ additive integrals of motion ( H ˆ −
Hamiltonian, kPˆ − impulse operator, N̂ − particle
number operator, αŜ − spin operator),
),4,,0(
α
YYkYYaY ≡ - thermodynamic forces, relevant to
them. For convenience of the further account we
suppose, that in a laboratory co-ordinates in an
equilibrium state there are no macroscopic fluxes, i.e.
0=kY and the external magnetic field is equal to zero
0=αY . The thermodynamic potential νΩ is defined
from a normalization conditions 1ˆ =wSp . The operator
F̂ has the considered phase symmetry of a condensed
media and represents as linear functional of the order
parameter operator
( )..)(ˆ)(3ˆ chxikxikfxdF +∆∫= . (3)
Here )(xikf - some function of coordinates, conjugate
order parameter operator, which sets its equilibrium
values in sense quasiaverages )(ˆ)( xikxik ∆=∆ . The
functions structure )(xikf is determined by properties
of the quantum fluid researched states symmetry. It
allows within the microscopic theory framework to
define the additional thermodynamic parameters into
Gibbs statistical operator. We define the order
40 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2002, № 2.
Series: Nuclear Physics Investigations (40), p. 40-42.
parameter operator of d-pairing ( )xik∆̂ in the terms of
the operators of creation and annihilation of Fermi -
particle in a point x :
)x(ˆj)x(ˆjik
)x(ˆi)x(ˆk)x(ˆk)x(ˆi)x(ikˆ
ψ∇σψ∇δ−
ψ∇σψ∇+ψ∇σψ∇≡∆
23
2
22
(4),
where 2σ - Pauli matrix. We define the operators of
number of particles N̂ , impulse kPˆ , spin αŜ and orbital
moment k
ˆL
( )xnxdN ˆˆ 3∫= , ( )xˆxdˆ ii π∫= 3P , ( )xŝxdˆ
i
3S ∫=i ,
( )xl̂xdˆ
ii
3∫=L , (5)
where the relevant densities of motion integrals ( )xn̂ ,
( )xiπ̂ , ( )xsiˆ , ( )xlî , in the terms of the operators of
creation and annihilation ( ) ( )xˆ,xˆ ψψ + ,are
( ) ( ) ( )xˆxˆxn̂ σ
+
σ ψψ= ,
( ) ( ) ( ) ( ) ( ){ }xˆxˆxˆxˆixˆ iii σ
+
σσ
+
σ ψψ∇−ψ∇ψ−=π
2
, (6)
( ) ( ) ( ) ( )xˆsxˆxŝ '' σσ σα
+
σα ψψ= , ( ) ( )xxxl lkikli πε ˆˆ = .
Using definitions (4) - (6) we obtain operator algebra:
[ ] ),(ˆ2)(ˆ,ˆ xikxikN ∆−=∆
[ ] ,0)(ˆ,ˆ =∆ xikS
α
[ ] ),(ˆ)(ˆ,ˆ xiklxikli ∆− ∇=∆P
[ ]
)x(ikˆjkxlkj
)x(jiˆlkj)x(jkˆlij)x(ikˆ,l̂i
∆∇ε−
∆ε−∆ε−=∆L
. (7)
The mean values of the order parameter
)(ˆˆ)ˆ,( xikSpxik ∆=∆ ρρ , ( −ρ̂ the arbitrary statistical
operator) have properties ( ) ( )ρρ ˆ,ˆ, xkixik ∆=∆ ,
( ) 0ˆ, =∆ ρxii . We have ten independent values in this
case. We choose a parameterization of ( )ρ∆ ˆ,xik in the
form
)ˆ,()ˆ,()ˆ,( ρρρ xikiQxikQxik +≡∆ ,
]
3
1
[]
3
1
[ ikkmimBikkninAikQ δδ −+−≡ , (8)
+
−+
−≡ ikkmimDikkninCikQ δδ
3
1
3
1
( ) ( ) ( )imklkmilGilknklinFinkmknimE ++++++ .
Here GBA ,..,, - the modules of the order parameter, n ,
m , l
are the axes of an anisotropy, also represent
unitary and mutually perpendicular vectors
,1222 === lmn
0=mn , 0=ml
, 0=ln
.
The condensed matter in a normal equilibrium state
is characterized by symmetry properties
[ ] 0ˆ,ˆ =kw P , [ ] 0ˆ,ˆ =kw L , [ ] 0ˆ,ˆ =αSw , [ ] 0ˆ,ˆ =Nw . (9)
These properties reflect a translational invariance,
spatial and spin isotropy, phase invariance of a normal
phase equilibrium state of condensed matter. A
requirement of a spatial isotropy in (9) and quantum
brackets algebra (7) give the equality
0)(ˆˆ =∆ xikwSp ,
in an aspect of the chosen directions lack in the equilib-
rium state.
We consider possible equilibrium order parameter
structures in the translational-invariant states of a
superfluid state equilibrium. The analysis of
translational-invariant subgroups of the equilibrium
states unbroken symmetry, according to [4] is feasible
from relations
[ ] 0ˆ,ˆ =Tw , [ ] 0ˆ,ˆ =kw P , (10)
where T̂ - the generator of the unbroken symmetry,
which represents the linear motion integrals combination
NcSbiiaT ˆˆˆ ++≡
αα
L , (11)
here cbia ,, α - the real parameters. The unitary
transformations ( )aTiU ˆexp= form continuous
subgroups of the equilibrium state unbroken symmetries
( ) ( ) ( )( )',''' aaaUaUaU = . Agrees (10), (11) we get
[ ] ( ) 0ˆˆ,ˆ =∆ x
ik
TwSp . Therefore, taking into account
algebra (7), we get the system of linear and
homogeneous equations
0=∆ jl
ik
jlF , (12)
where
( ) ik
jljiklillkjkllijl Fica ≡++ δδδεδε 2 .
Passing in the formula (12) from double summation to
unary, at which the indexes jlik; of summation
possess the values :, βα 111 ≡ , ,212 ≡ 933... ≡ we
obtain the equation
0=∆ α
β
αF . (13)
The requirements 0det =β
αF (it is the condition of
existing of a nontrivial system linear and homogeneous
equations solution (13) ( ) 0≠∆ x
ik ) are fulfilled at с=0,
±a, ±a/2. The three solutions of the equation
0det =β
αF are as follows
41
1) 0=c , 2) ac ±= , 3) ac
2
1±= . (14)
Hence, the unbroken symmetry generators of the
superfluid fluid equilibrium state with d-pairing are
[ ] 0ˆ,ˆ =αSw , 0ˆ
2
ˆ,ˆ =
− Nm
a
aw l
i
i L . (15)
Here lm - quantum number accepting the values 0, 1, 2.
Using the second relation (15) and taking into account
(8) we obtain the equations, defining the explicit form of
the order parameter in equilibrium
( ) 0=−+ uvljuivjjviuj
i QmQQ
a
a εε , (16)
( ) 0=++ uvljuivjjviuj
i QmQQ
a
a εε .
SOLUTION OF THE EQUATIONS AND
DISCUSSION OF THEM
Solving obtained set of equations and using
representation (8) we find the order parameter structure
in the equilibrium state in this case
( )
−+=∆ uvvuuv nniCA δ
3
1
, (17)
here for simplicity the vectors a and n are chosen
collinear. The equality in (17) is similar to an order
parameter structure of a liquid crystal body with
complex amplitude.
At 2±=m we obtain the solution
( ){ }vuuvuvvuvuuv mlmlimmnnG ++±=∆ δ2 .
(18)
For a determination of the solution at the quantum
number value 1±=m , the vector a on the indicated co-
ordinate is decomposable lmnaa
γβα ++=/ . Here
γβα ,, are bound by equality 1222 =++ γβα . From
(13) we get the solution
{ ±−+=∆ uvvuvuuv mmnnA δ2 ,
[ ]
+++± vuuvvuuv lnlnnmnmi
2
, (19)
thus 2/12 −±== γβ , 0=α .
To compare the obtained outcomes with results of
works [5,6] we normalize the order parameter by a
relation
1=∆∆ ∗
kiik . (20)
Agrees (17) - (19), (20) we find 2/322 =+ CA ,
2/1±=G , 2/1±=A . According to the paper [5] we
define mean value
jiji kk ∆≡∆ , (21)
where the unit vector k
is defined by equality
{ } { }xyz k,k,kncossin,msinsin,lcosk ≡ϕθϕθθ≡
. (22)
For a solution (17) according to (21), (22) we write
( ) ( ) ( )3/121 −+=∆ zkiCA .
This solution corresponds the "real" state. The solution
(18) in according to (21) (22) gives the equality
( ) ( ) 2/22
yx ikk +=∆ ,
which coincides with "axial" state of works [5,6]. At
last, in a case (19) we obtain
( ) ( ) ( )xyz kkki −++=∆ 23 γβ . (23)
Comparing this result with "cyclic" state from papers
[5,6], we can’t notice the equivalence of them. They
coincide only with α=0 and ζ==0.
REFERENCES
1. G. Barton, M. Moore. Superfluids with l≠0.
Cooper pairs: parameterization of the Landau free
energy // J. Phys. C. 1974, v. 7, p. 2989-3000.
2. H.W. Capel. Group theory and phases of
superfluid 3He. Proceedings of the 5 th
International Symposium on Selected topics in
statistical mechanics, World Scientific, 1990, p. 73-
83.
3. D. Vollhardt, P. Wolfle. The superfluid pha-
ses of helium 3. Taylor Francis, 1990, 619 p.
4. M.Y. Kovalevsky, S.V. Peletminsky,
N.N. Chekanova. Classification of the equilibrium
superfluid states with scalar and tensor order
parameters // Low Temperature Physics. 2002,
v. 28, № 4, p. 327-337.
5. D. Mermin. d-pairing near the transition
temperature // Phys. Rev. 1974, A9, p. 868-870.
6. T.L. Ho, S.S. Yip. Pairing of Fermions with
Arbitrary Spin // Phys. Rer. Lett. 1999, v. 82, № 2,
p. 247-250.
7. N.N. Bogolyubov. Superfluidity and quasime-
ans in problems of statistical mechanics //
Proceedings of the Steklov Institute of Mathematics.
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8. N.N. Bogolyubov (Jr.), М.Y. Kovalevsky,
А.М. Kurbatov, S.V. Peletminsky, А.N. Tarasov.
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fizicheskikh nauk. 1989, v. 154, p. 585-620.
42
A.P. Ivashin, M.Y. Kovalevsky, N.N. Chekanova
THE BASIC EQUATIONS
REFERENCES
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