On spiral superfluidity in the Fermi-liquid model
In this work we propose a model of description of a superfluid Fermi liquid with a spiral ordering by spins. The method of description is similar to that of description of spiral magnetics. Self-consistency equations for the order parameter are obtained. The transition temperature and order paramete...
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irk-123456789-800482015-04-10T03:02:13Z On spiral superfluidity in the Fermi-liquid model Shul’ga, S.N. Yatsenko, A.A. Quantum fluids In this work we propose a model of description of a superfluid Fermi liquid with a spiral ordering by spins. The method of description is similar to that of description of spiral magnetics. Self-consistency equations for the order parameter are obtained. The transition temperature and order parameter in case of interaction similar to the Skyrme interaction are numerically calculated. The transition temperature and the order parameter, are proved to reach their maximal value under spirality distinct from zero. 2001 Article On spiral superfluidity in the Fermi-liquid model / S.N. Shul’ga, A.A. Yatsenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 348-350. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 67.57. z, 71.10.Ay http://dspace.nbuv.gov.ua/handle/123456789/80048 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum fluids Quantum fluids Shul’ga, S.N. Yatsenko, A.A. On spiral superfluidity in the Fermi-liquid model Вопросы атомной науки и техники |
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In this work we propose a model of description of a superfluid Fermi liquid with a spiral ordering by spins. The method of description is similar to that of description of spiral magnetics. Self-consistency equations for the order parameter are obtained. The transition temperature and order parameter in case of interaction similar to the Skyrme interaction are numerically calculated. The transition temperature and the order parameter, are proved to reach their maximal value under spirality distinct from zero. |
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Article |
| author |
Shul’ga, S.N. Yatsenko, A.A. |
| author_facet |
Shul’ga, S.N. Yatsenko, A.A. |
| author_sort |
Shul’ga, S.N. |
| title |
On spiral superfluidity in the Fermi-liquid model |
| title_short |
On spiral superfluidity in the Fermi-liquid model |
| title_full |
On spiral superfluidity in the Fermi-liquid model |
| title_fullStr |
On spiral superfluidity in the Fermi-liquid model |
| title_full_unstemmed |
On spiral superfluidity in the Fermi-liquid model |
| title_sort |
on spiral superfluidity in the fermi-liquid model |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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Quantum fluids |
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http://dspace.nbuv.gov.ua/handle/123456789/80048 |
| citation_txt |
On spiral superfluidity in the Fermi-liquid model / S.N. Shul’ga, A.A. Yatsenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 348-350. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT shulgasn onspiralsuperfluidityinthefermiliquidmodel AT yatsenkoaa onspiralsuperfluidityinthefermiliquidmodel |
| first_indexed |
2025-07-06T03:59:07Z |
| last_indexed |
2025-07-06T03:59:07Z |
| _version_ |
1836868538781925376 |
| fulltext |
ON SPIRAL SUPERFLUIDITY IN THE FERMI-LIQUID MODEL
S.N. Shul’ga, A.A. Yatsenko1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
1e-mail: yats@kipt.kharkov.ua
In this work we propose a model of description of a superfluid Fermi liquid with a spiral ordering by spins. The
method of description is similar to that of description of spiral magnetics. Self-consistency equations for the order
parameter are obtained. The transition temperature and order parameter in case of interaction similar to the Skyrme
interaction are numerically calculated. The transition temperature and the order parameter, are proved to reach their
maximal value under spirality distinct from zero.
PACS: 67.57. z, 71.10.Ay
The term “spiral ordering” means that the state of a
system remains invariant after an arbitrary spatial shift
on a vector a and a simultaneous rotation of spins on an
angle aq, with q the spirality vector, i. e. the state of the
system is invariant relatively the unitary transformation
)ˆˆ(ˆ
3exp σqpa −= iV : (1)
fVfV ˆˆˆˆ =+ , (2)
where
−= + f1g
gf
f ~ˆ
,
(3)
3σˆ – Pauli matrix and p̂ – operator of momentum,
generalized on a superfluid Fermi liquid:
−
=
3
3
3 0
0
σ
σ
σˆ
−
=
p
p
p
0
0ˆ
.
(4)
Here f and g stand for normal and anomal
distribution functions of quasi-particles in the superfluid
Fermi liquid.
Such states of the superfluid Fermi liquid are
analogous to the spiral ordering of magnetics [1] and
can be presumably realized in liquid 3He or in neutron
stars.
Generally, such a state must be spatially unhomo-
geneous. It is a superposition of singlet and triplet spin
states of the superfluid Fermi liquid.
In construction of the theory of spiral ordering for
the superfluid Fermi liquid we followed the works [2,3],
where the distribution function of the normal and
superfluid components is presented in the form of
defined above supermatrix f̂ ; the quasi-particle energy
ε̂ and the operators of physical values a read as
−∆
∆
= + ε
ε
ε ~ˆ
,
−
=
a
a
a ~ˆ
0
0 . (5)
The unitary transformation
+= UfUf ˆˆˆˆ
q , (6)
where ( )3exp σxq ˆˆ iU −= , preserves the structure of the
supermatrix qf̂ analogical to the matrix f̂ :
−= +
q
qq
q
q
f1g
gf
f ~ˆ (7)
and transfers the system into a homogeneous state:
0][ =p,q ˆf̂ . (8)
Normal and anomalous distribution functions fq
and gq that define the supermatrix qf̂ in (7) are
expressed through f and g in the following way:
33 σσ qxqx
q
ii feef −= , (9)
33 σσ xqqx
q
~ii geeg −−= . (10)
The general structure of the “homogeneous”
anomalous distribution function gq and the order
parameters Δq for case of one kind of fermions is:
( ) 20 )()()( σσ pppq ggg += , (11)
( ) 20 )(Δ)(Δ)(Δ σσ
pppq += . (12)
This describes a superposition of singlet and triplet
states of the system. On the other hand, the quantity
gq(p) can be expanded into components g|| and g⊥ that is
convenient for further calculations:
⊥+= ggg ||q , (13)
where
( ) 2330 σσggg +=|| , (14)
and
( ) 22211 σσσ ggg +=⊥ . (15)
The order parameter Δq(p) can be expanded in the same
way:
⊥∆+∆=∆ ||q , (16)
where
92 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 348-350.
( ) 2330 σσ∆+∆=∆ || , (17)
and
( ) 22211 σσσ ∆+∆=∆ ⊥ . (18)
By means of these matrices, it will be easy to
reconstruct the unhomogeneous distribution functions f
and g, the order parameter Δ and to calculate the
spatially unhomogeneous characteristics of the system.
In fact, the matrix elements of the distribution
functions f, g, fq and gq between the impulse states p
and p′ have form:
( )qpp
qpqpp
)(
2
1)(
2
1 3
1
1
3
κηδ
κ σηση
κ
η
−−′−×
×+−+= ∑
±=
±=
′ ff
, (19)
( )qpp
qpqpp
)(
2
1)(
2
1 3
1
1
3
κηδ
κ σηση
κ
η
+−′+×
×+−+= ∑
±=
±=
′ gg
(20)
The matrix elements of the order parameter Δ12 and
the energy ε12 are of analogous form. They are defined
through “homogeneous” order parameters Δq(p)12 and
εq(p)12, which are diagonal matriсes in the impulse
space.
The energy of the system can be represented in the
following form:
( ) ( ) ( )gEfEfE int0 +=ˆ . (21)
In this work we do not take into account influence of
the normal amplitudes of interaction on the superfluid
properties.
The energy E0( f) of the free Fermi liquid quasi-
particles possesses the following form:
( ) ( )∑=
p
ppp ftrfE ε0 , (22)
where ( ) m22pp =ε . Note, that the energy εpp' (p) and
εq(p) have diagonal form in the impulse space:
( ) ( ) ( )pppppp ′−=′ δεε . (23)
Then
( ) ( ) ( ) ( )pp
qp
pq 330
2
3
2
εσε
σ
ε +=
+
=
m
. (24)
The term ( )p33εσ in the energy εq (p) is not equal to
zero and substantially influences the process of
diagonalization of the distribution function ( )pqf̂ .
We choose the term of (21) that is responsible for
the superfluidity in the form which is quadratic in the
distribution function:
( ) ( )∑ +=
1234
3421int 1234
2
1 ggvgE . (25)
We choose the interaction between the Fermi liquid
quasi-particles being translationally invariant and
invariant under rotations in the spin space:
( ) ( )
( ) =′+
+′=
24131
2413012344321
σσ
δδ
kk
kkpppp
,
,
v
vv
(26)
1234
P)(u)(u 124130 σδδ kkkk ′+′= ,, ,
where
( ) 2241324131234 σσδδσ
+=P (27)
and
.
,,
4321
4321
22
pppp
ppkppk
+=+
−=′−=
(28)
We suppose that the interaction between quasi-
particles does not depend on the total momentum of the
interacting particles. As a model of interaction we
consider the effective Skyrme force that depends on
angles between momenta of the interacting quasi-
particles and quadratically – on absolute values of their
momenta. Under such a choice of interaction one can
say formally that one examines the proton-neutron
pairing in a symmetric nuclear matter, unless the
interaction is considered as an example of enough
simple force between quasi-particles.
As is known from the works [2,3], the order
parameter Δ12 and anomalous distribution function are
connected by the following equation:
( )∑=∆
43
3412 1234
,
gv . (29)
By expressing g through Δ, one can obtain a self-
consistency equation for the order parameter.
In the case of spiral ordering the order parameter
Δpp' and the anomalous distribution function gpp' express
through Δq (p) and gq (p) accordingly, and the
“homogeneous” normal distribution function gq (p) was
obtained in the work [4] as a function of the singlet and
triplet order parameters 0∆ and ∆
.
For the order parameter (Δ0± Δ3) we have the
following equation:
( ))()(
)(1)()(
2320
211310
2
pp
qpqppp
p
gg
V
±
×±±Φ=∆±∆ ∑ ,
, (30)
where
( ) ( ) ( ) ( )pppppppp ′−′−′=′Φ ,-2,,, 110 vvv , (31)
( ) ( ) ( ) ( )
T
E
th
E
gg
22
30
30
±
±
∆±∆−=± ||
||
pppp , (32)
93
( ) ( ) ( ) 2
30
22
2
ppqp ∆±∆+
−+=± µ
m
E|| . (33)
For the order parameter Δ2 ± Δ1 we have the
equation:
( ) ,
,
)()(
)(1)()(
2122
211112
2
pp
pppp
p
igg
V
i
±×
×Ψ=∆±∆ ∑
(34)
where
( ) ( ) ( )pppppp ′+′=′Ψ ,,, 10 vv , (35)
( ) ( ) ( ) ( )
T
Eth
E
iigg
22
12
12
±
⊥
±
⊥
∆±∆−=± pppp , (36)
( ) ( ) 2
12
222
2
ppqp ∆±∆+
−+=±
⊥ i
m
E µ . (37)
Equations (30, 34) describe two particular cases. For
description of general spiral ordering of superfluid
Fermi liquid we have to obtain a system of 4 nonlinear
integral self-consistency equations for the singlet (Δ0)
and triplet ( ∆
) order parameters. Nevertheless, the true
spirality is possible only in the case when all 4
components Δj, where j=0,….3 do not vanish. Then the
systems of equations are closely coupled between them-
selves. However, in the present work we do not investi-
gate this case.
It is easy to see that the spirality parameter q defines
directly the equations for Δ1 and Δ2 only, so we will
examine this unhomogeneous case. Note that equation
(34) does not depend on the sign “±”, hence Δ1=0.
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
0,000
0,005
0,010
0,015
0,020
Tc(q)
δ (q)
q
Fig. 1
The equation (30) for Δ2 has been numerically
solved for the Skyrme SKP force. For this force we
have
( ) ( )
( ) ( ) .,
,
22
2211001
22
2100
u
,u
pppppp
pppppp
′++′+=′
′++′+=′
axaxax
aaa
(38)
The constants ai and xi (i=0,….,3) are defined by
parametrisation of the potential.
Under such a choice of potential we should seek for
the order parameter in the form Δ2(p)=δ2 p z, where z is
the cosine of angle between the vectors p and q.
The order parameter Δ2 has been calculated at T = 0
as a function of the spirality vector q and is presented on
the Fig.1. At the same figure we present the transition
temperature versus q. At q ~ 0,75 the superfluidity
disappears (δ2=0, Tc=0).
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14
0,00828
0,00829
0,00830
0,00831
0,00832
q
Tc
Fig. 2
The transition temperature (and δ2 as well) reaches
its maximal value at q~0,1. It means that the most stable
state should be under q distinct from zero, unless the
maximum of Tc is very tiny. Similarly behaves the order
parameter δ2 at T = 0. The area of extremum of the
transition temperature is shown on the Fig.2. Whether
the appearance of the extermum is model-dependent,
remains an open question.
REFERENCES
1. M.Yu. Kovalevsky, S.V. Peletminsky, A.A.
Rozhkov. Green functions of spiral magnetics in the
hydrodynamical and quasi-particle approximations //
TMF. 1988, v. 75, №1, p. 86-100 (in Russian).
2. V.V. Krasilnikov, S.V. Peletminskij, A.A. Yatsenko.
On the theory of a superfluid Fermi liquid // Physica
A. 1990, v. 162, p. 513-541.
3. A.I. Akhiezer, V.V. Krasilnikov, S.V. Peletminskii,
A.A. Yatsenko. Research on superfluidity and
superconductivity on the basis of the Fermi liquid
concept // Physics Reports. 1994, v. 245, №1,2, p. 1-
110.
4. A.I. Akhiezer, A.A. Isaev, S.V. Peletminsky,
A.A. Yatsenko. To the theory of the singlet-triplet
pairing of fermions // TMF. 1998, v. 115, №3,
p. 459-476 (in Russian).
94
S.N. Shul’ga, A.A. Yatsenko1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
1e-mail: yats@kipt.kharkov.ua
REFERENcES
1.M.Yu. Kovalevsky, S.V. Peletminsky, A.A. Rozhkov. Green functions of spiral magnetics in the hydrodynamical and quasi-particle approximations // TMF. 1988, v. 75, №1, p. 86-100 (in Russian).
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