Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He
In frame of selfconsistent microscopical model of the superfluid non-relativistic Bose liquid ⁴He with suppressed due to many-particle effects single-particle condensate and intensive pair coherent condensate the ab initio calculation of the spectrum E(p), based on the realistic models for the pair...
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nasplib_isofts_kiev_ua-123456789-1118682025-02-23T17:10:13Z Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He Мiкроскопiчна теорiя, структура надплинної компоненти та спектру елементарних збуджень в бозе-рiдинi Нe-ІІ Микроскопическая теория, структура сверхтекучей компоненты и спектра элементарных возбуждений в бозе-жидкости Нe-ІІ Pashitskij, E.A. Chumachenko, A.V. Vilchynskyi, S.I. Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина In frame of selfconsistent microscopical model of the superfluid non-relativistic Bose liquid ⁴He with suppressed due to many-particle effects single-particle condensate and intensive pair coherent condensate the ab initio calculation of the spectrum E(p), based on the realistic models for the pair interaction potentials V (r), which has a finite oscillating Fourier-component, was presented. It was shown that oscillating character of the renormalized Fourier component of the fitting "hard sphere" and "semitransparent sphere" pair interacting potentials leads to non-monotonic behavior for the momentum dependence of the renormalized mass operators and, as a consequence, to emerging of the roton minimum in the quasiparticles spectrum E(p), which exactly connected with the first largest negative minimum of the renormalized interaction Fourier-component of a pair of bosons. В рамках самоузгодженої мiкpоскопiчної моделi надплинної нерелятивiстської бозе-piдини ⁴He з подавленим за pахунок взаємодiї одночастинковим i iнтенсивним парним конденсатами обчислено з пеpших пpинципiв спектp елементаpних збуджень E(p), виходячи з pеалiстичних моделей для потенцiалiв паpної взаємодiї V(r), якi мають скiнчену осцилюючу фуp’є-компоненту. Показано, що в pамках моделей "твердих" та "напiвпpозоpих сфер" осцилюючий хаpактеp пеpеноpмованої фуp’є-компоненти потенцiалу парної взаємодiї бозонiв пpиводить до немонотонної поведiнки iмпульсних залежностей перенормованих масових опеpатоpiвi, як наслiдок, до появи в спектpi квазiчастинок E(p) pотонного мiнiмуму, який однозначно пов’язаний з найбiльш глибоким пеpшим вiд’ємним мiнiмумом фур’є-компоненти перенормованої парної взаємодiї бозонiв. В рамках самосогласованной микроскопической модели сверхтекучей нерелятивистской бозе-жидкости ⁴He с подавленным за счет взаимодействия одночастичным и интенсивным парным конденсатами произведен расчет с первых принципов спектра элементарных возбуждений E(p) исходя из реалистичных моделей для потенциалов парного взаимодействия V(r), которые имеют конечную осциллирующую фурье-компоненту. Показано,что в рамках моделей "твердых" и "полупрозрачных сфер" осциллирующий характер перенормированой фурье-компоненты потенциала парного взаимодействия бозонов приводит к немонотонному поведению импульсных зависимостей перенормированных массовых операторов и, как следствие, к появлению в спектре квазичастиц E(p) ротонного минимума, который однозначно связан с наиболее глубоким первым отрицательным минимумом фурье-компоненты перенормированного парного взаимодействия бозонов. 2013 Article Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He / E.A. Pashitskij, A.V. Chumachenko, S.I. Vilchynskyi // Вопросы атомной науки и техники. — 2013. — № 3. — С. 35-41. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS: 68.03.-g https://nasplib.isofts.kiev.ua/handle/123456789/111868 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
| spellingShingle |
Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Pashitskij, E.A. Chumachenko, A.V. Vilchynskyi, S.I. Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He Вопросы атомной науки и техники |
| description |
In frame of selfconsistent microscopical model of the superfluid non-relativistic Bose liquid ⁴He with suppressed due to many-particle effects single-particle condensate and intensive pair coherent condensate the ab initio calculation of the spectrum E(p), based on the realistic models for the pair interaction potentials V (r), which has a finite oscillating Fourier-component, was presented. It was shown that oscillating character of the renormalized Fourier component of the fitting "hard sphere" and "semitransparent sphere" pair interacting potentials leads to non-monotonic behavior for the momentum dependence of the renormalized mass operators and, as a consequence, to emerging of the roton minimum in the quasiparticles spectrum E(p), which exactly connected with the first largest negative minimum of the renormalized interaction Fourier-component of a pair of bosons. |
| format |
Article |
| author |
Pashitskij, E.A. Chumachenko, A.V. Vilchynskyi, S.I. |
| author_facet |
Pashitskij, E.A. Chumachenko, A.V. Vilchynskyi, S.I. |
| author_sort |
Pashitskij, E.A. |
| title |
Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He |
| title_short |
Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He |
| title_full |
Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He |
| title_fullStr |
Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He |
| title_full_unstemmed |
Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He |
| title_sort |
microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic bose-liquid ⁴he |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2013 |
| topic_facet |
Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111868 |
| citation_txt |
Microscopical theory, structure of superfluid component and elementary excitation spectrum of the non-relativistic Bose-liquid ⁴He / E.A. Pashitskij, A.V. Chumachenko, S.I. Vilchynskyi // Вопросы атомной науки и техники. — 2013. — № 3. — С. 35-41. — Бібліогр.: 21 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| fulltext |
MICROSCOPICAL THEORY,
STRUCTURE OF SUPERFLUID COMPONENT AND
ELEMENTARY EXCITATION SPECTRUM
OF THE NON-RELATIVISTIC BOSE-LIQUID 4He
E.A.Pashitskij1, A.V.Chumachenko2, S.I.Vilchynskyi2∗
1Ukrainian National Academy of Science, Institute of Physics, Kiev, Ukraine
2Kiev National Taras Shevchenko University, 03022, Kiev, Ukraine
(Received January 22, 2013)
In frame of selfconsistent microscopical model of the superfluid non-relativistic Bose liquid 4He with suppressed due
to many-particle effects single-particle condensate and intensive pair coherent condensate the ab initio calculation of
the spectrum E(p), based on the realistic models for the pair interaction potentials V (r), which has a finite oscillating
Fourier-component, was presented. It was shown that oscillating character of the renormalized Fourier component of
the fitting ”hard sphere” and ”semitransparent sphere” pair interacting potentials leads to non-monotonic behavior
for the momentum dependence of the renormalized mass operators and, as a consequence, to emerging of the roton
minimum in the quasiparticles spectrum E(p), which exactly connected with the first largest negative minimum of
the renormalized interaction Fourier-component of a pair of bosons.
PACS: 68.03.-g
Investigation of a core insight of physical phe-
nomena, namely the demonstration of the quantum
laws on the macroscopical level, always was one of
the primary interests for P.I. Fomin scientific activ-
ities. Strong scientific intuition, unique approach to
analysis of experimental data, which were inherent
to P.I. Fomin, allow him successfully use ideas and
principles of quantum field theory for the creation
of a relativistic theory of the jet quasar activity, ap-
plicable for quasars and radio galaxies; theoretical es-
timations for some parameters of the jets, which were
done using this theory, gives a good correspondence
with experimental data.
Superfluidity phenomenon in the Bose-liquid 4He
(discovered by P.L. Kapitsa in 1938) and Fermi-liquid
3He (discovered by D. Osheroff, R. C. Richardson
and D. M. Lee in 1972), together with superconduc-
tivity of electron Fermi-liquid in metals, also were a
in the list of interest for P.I. Fomin. Along with the
construction of the relativistic theory of superfluidity
P.I. Fomin were extremely interested in the investi-
gation of the structure of the Bose liquid 4He su-
perfluid component, which was reflected in our com-
mon with P.I. Fomin paper dedicated to this prob-
lem (Fomin P.I., Pashitskij E.A, Vilchynskyy S.I.,
On pair-wise character of superfluid condensate in
helium-II. (1997) Low Temperature Physics, v.23,
N12, p.12671271). Here, starting from the empiri-
cal data for the dynamics of 3He atoms impurity in
superfluid 4He (what suggests an abnormal high ef-
fective mass of 4He) was proposed that helium atoms
form a pairs below the -point and the role of a paired
condensate in 4He were discussed. Further, for inves-
tigation of the superfluidity P.I. Fomin has been using
methods and ideas of solid state physics and physics
of crystals and E.P. together with S.V. developed the
self-consistent model of the superfluid (SF) state of
a Bose liquid with strong interaction between bosons
and a weak single-particle Bose-Einstein condensate
(BEC).
In series of papers renormalized perturbation the-
ory was used, it was built on combined hydrodynamic
variables with analytic normal and anomalous self-
energy functions and a nonzero SF order parameter,
proportional to the density of the SF component. On
the base of this theory a closed system of nonlin-
ear integral equations for the normal and anomalous
self-energy parts were obtained. Unlike in the Bo-
goliubov theory of a quasi-ideal Bose gas, were small
parameter is the ratio of the number of supracon-
densate excitations to the number of particles in an
intensive BEC, the ratio of the BEC density to the
total particle density of the Bose liquid was used as
a small parameter of the model. Quasiparticle spec-
trum, obtained within this approach, is in a good
agreement with experimental spectrum of elementary
excitations in superfluid 4He. And, as it was shown,
the roton minimum in the spectrum is associated with
negative minimum of the Fourier component of the
pair interaction potential.
This article holds a brief discussion on current sta-
tus in the microscopic theory of superfluidity of the
∗Corresponding author E-mail address: sivil@univ.kiev.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85).
Series: Nuclear Physics Investigations (60), p.35-41.
35
non-relativistic Bose liquid and review of the results
obtained by the authors during the last decade.
In the basis of the theory, which describes super-
fluidity and superconductivity phenomena, lays an
appearance, as additional complex macroscopic order
parameter, of the wave function of bosons or fermi-
onic Cooper pairs, which stays in the same quantum
state, in other words - appearance of the coherent
condensate is the foundation of the superfluid and
superconductivity phenomena. In Bose-systems such
condensate appears because of direct accumulation of
bosons in the ground state, but in the Fermi-systems
due to the formation of the Cooper pairs of fermions.
Great amount of experimental and theoretical in-
vestigations performed over last 80 years allowed to
achieve a high level of understanding of the proper-
ties of non-relativistic superfluid state in a Bose liq-
uid 4He. Most important achievements in the theory
of superfluidity reached at a phenomenological level,
in particular, for the description of the properties of
superfluid 4He (so-called Ne-II) on the basis of two-
fluid Landau hydrodynamics, according to which the
superfluid helium can be divided into two components
- the superfluid component with density ρs and ve-
locity vs that describes the non-dissipative motion of
quantum fluid and normal component with density
ρn and velocity vn which describes dissipative flow of
the gas of excitations (phonons and rotons). Numeri-
cal calculations of thermal conductivity and viscosity
of superfluid helium, obtained according to this the-
ory, are in good agreement with experiment. One
of the greatest achievements of the phenomenologi-
cal approach to the phenomenon of superfluidity is
that Landau, basing on the temperature dependence
of heat capacity of superfluid 4He on the basis of
his superfluidity criteria for the quantum liquids pre-
dicted the form of the superfluid helium elementary
excitation spectrum with a linear (phonon) disper-
sion law E(p) ≈ pc for the small momentum p → 0
and so called ”roton” minimum for p 6= 0 Later this
form of the spectrum was brilliantly confirmed in the
experiments for the scattering of the slow neutrons in
the liquid helium.
But in the frame of phenomenological theory of
superfluidity exact calculation (from first principles)
of the elementary excitations spectrum in superfluid
Bose liquid with strong interaction between particles
may not be possible. This is the task for the micro-
scopic theory of superfluidity of Bose liquids. The-
oretical study of the properties of superfluid Bose
liquid 4He at the microscopic level hampered by a
number of fundamental problems associated with the
strong interaction between bosons and complex quan-
tum structure of effective coherent condensate, which
is the main part of the superfluid component, unlike
of almost ideal Bose gas, where this role is played by
single-particle Bose Einstein condensate (BEC).
The first microscopic theory of superfluidity,
based on a model of weakly non-ideal Bose gas, was
proposed by N.N. Bogoliubov [1] more than 50 years
ago. The main advantage of this theory is rejecting of
the standard perturbation methods, which are based
on series expansions on the weak interaction coupling
constant. The insight of the theory is the idea of ex-
istence in the superfluid system the intensive BEC
- namely that the number of particles in the single
BEC is macroscopically large and close to the total
number of particles (per unit volume). Consequently
we can neglect the fact that operators of creation and
annihilation of particles with zero momenta are not
commute, diagonalize the initial Hamiltonian of the
system and find an expression for the renormalized
quasiparticle spectrum.
Bogoliubov theory has been improved further and
in the most of these improved models of superfluid-
ity by the selection of variational parameters one can
achieve a good agreement between theoretical and
experimental spectrum of elementary excitations in
superfluid 4He for a certain range of the momen-
tum. But such agreement is more coincidental and
is not true, because, as it was shown in the later ex-
perimental studies, a part of single-particle BEC in
the superfluid 4He is small and ranges from 2 to 10
percent, which is the opposite to a condition of weak
non-ideality of the Bose-gas in the Bogoliubov theory.
Therefore in order to give an adequate microscopic
description of the Bose-liquid superfluid properties
the most promising is the Greens function method,
which for the first time was used in the papers of
S.T. Beliaev and widely developed in the future.
Analysis of the modern experimental and theoret-
ical works testifies to the fact that the study of unique
phenomena of superfluidity of helium 4He is far from
the end. There are number of contradictions between
theory and experiment, which are related to both the
hydrodynamics of superfluid Bose liquid 4He, and
the form of quasi-particle spectra, which have not yet
satisfactory explanation in frame of the microscopic
theory. We shortly focus on the most significant con-
tradictions between the fundamental principles and
conclusions of a microscopic theory of superfluidity
and experimental data (for more details see [2]).
Firstly, experiments on inelastic scattering of slow
neutrons in liquid helium, which confirmed the form
of proposed by Landau curve for the spectrum of el-
ementary excitations, indicated a weak temperature
dependence of the spectrum, as the consequence the
size of the ”roton” gap ∆r = 8.65 K at minimum
of E(p), that determines the critical velocity of nor-
mal fluid, changes only to the value of ∆r = 5.24 K
near the temperature of the phase transition from
superfluid to normal state, i.e. the so-called λ-point
Tλ = 2.17 K . At the same time, for the spectrum of
quasiparticles in the electronic Fermi liquid in super-
conductors superfluidity criterion holds for the phase
below critical temperature Tc, when there is a finite
energy gap in the spectrum, and fail in the normal
state at T > Tc when gap in the spectrum becomes
zero.
Moreover, obtaining of the quasiparticle spectrum
from the ”first principles” also fails. Using the latest
calculation methods such as Monte-Carlo it is possi-
36
ble to obtain a good agreement between calculated
and experimental data almost for all range of mo-
mentum, but remains unclear the physical reason for
the roton minimum in the quasiparticle spectrum of
the Bose-liquid.
Secondly, in the superfluid Bose-liquid, unlike the
Bose gas, single-particle Bose-condensate (SPC) due
to strong interaction between bosons must be signifi-
cantly impoverished by the particles of zero energy
and momentum (i.e. ”depleted” condensate) even
at temperature. Analysis of the experimental data
from neutron scattering [3] and quantum evapora-
tion of the helium atoms [4] shows that superfluid
4He at low temperature in the Bose-condensate state
contain less than 10% of the full density of liquid
4He, whereas density of the superfluid component
at the temperatures according to classical measure-
ments of the viscosity of superfluid helium [5], al-
most equal to the liquid helium density. This means
that against widespread conviction superfluidity of
the Bose-liquid 4He cannot be connected only with
Bose-condensation phenomena. Microscopical struc-
ture of the superfluid component, as it was for the
first time shown in [6], should have more complicated
quantum nature in the form of effective coherent con-
densate.
These contradictions are somehow reduced to the
question of the quantum structure of the superfluid
component in liquid 4He below λ-point. This is cen-
tral question for the construction of the consistent
microscopical theory of superfluidity of Bose-liquids.
Let us shortly focus on the results of investiga-
tion of the superfluid component microscopical quan-
tum structure and the elementary excitation spec-
trum in the superfluid 4He, which based on the con-
clusion that superfluid component at T → 0 is the
coherent superposition of weak BEC and intensive
pair coherent condensate (PCC). This assumption is
based on the fact that, as was already indicated, that
according to numerous experimental data on inelas-
tic scattering of neutrons and quantum evaporation
of 4He atoms the maximal density of single-particle
BEC in the Bose liquid 4He, even at low tempera-
tures T ¿ Tλ does not exceed 10% of total density of
liquid 4He, although density of the superfluid com-
ponent ρs → ρ when T → 0 . Such a low density
of the BEC, appeared due to the strong interaction
between atoms of 4He, shows that quantum struc-
ture of the superfluid condensate in He-II with ”ex-
cess” density (ρs − ρ0 À ρ0) requires a more detailed
and deep investigation. Our approach, based on the
microscopic model of superfluidity of the Bose-liquid
with depleted BEC and intensive PCC, proposed by
Pashitskij E.A. and Nepomnyashij Yu.A. [6]. Such
PCC may occur at sufficiently high ”effective attrac-
tion” between bosons in some ranges of momentum
due to the effects of the quantum diffraction of Bosons
in the process of their interaction, and is similar to
attraction between fermions near the Fermi surface.
The ratio of BEC density to the full density of the liq-
uid was taken as a small parameter (i.e. ρ0/ρ ¿ 1 ),
in contrast to the Bogoliubov [1] theory were small
parameter is taken as ratio of the number of supra-
condensate excitations to the number of particles in
the intensive BEC. Due to a small BEC density it
is possible to derive a closed self-consistent system
of integral equations for the normal and anomalous
self-energy parts by cutting an infinite series of per-
turbation theory and keeping only a first order terms
by small parameter ρ0/ρ:
∑
11
(ε, p) = n0Λ (ε, p)V (ε, p) + n1V (0) + ψ11 (ε, p) ,
∑
12
(ε, p) = n0Λ (ε, p) V (ε, p) + ψ12 (ε, p) ,
here V (p) – the Fourier component of the inocu-
lating two-particle interaction potential of bosons;
V (ε, p) = V (p) [1− V (p)Π (ε, p)]−1 – the renor-
malized (”shielded”) due to many-particle collective
effects Fourier component of non-local interaction;
Π (ε, p) – bosonic polarization operator; n1 = n− n0
– the number of ”supra-condensate” particles. Func-
tions ψ11 (ε, p) and ψ12 (ε, p) , taking into account
poles of single-particle Green functions, are defined
by the following expressions:
ψ11 (ε, p) = −1/2
∫
d3k (2π)−3 Γ (ε, p, k, E(k))×
×V (p− k, ε− E(k))
[
A (k,E(k)) E−1(k)− 1
]
,
ψ12 (ε, p) = −1/2
∫
d3k (2π)−3×
×Γ (ε, p, k, E(k)) E−1(k)V (p− k, ε− E(k))×
× [n0Λ (k, E(k)) V (k,E(k)) + ψΛ (k, E(k))] ,
here Γ (ε, p, k, E(k)) and Λ (ε, p) = Γ (ε, p, 0, 0) – the
vortex functions, which describes many-particle cor-
relations; function A (p,E(p)) defined as
A (p,E(p)) = n0Λ (p,E(p)) V (p,E(p))+
p2/2m + ψ11 (p,E(p))− ψ12(0, 0) + ψ0, 0,
and E(p) – quasiparticle spectrum, which in our ap-
proximation have the following form
E(p) =
(
A2 (p,E(p)) −
− [n0Λ (p,E(p)) V (p,E(p)) + ψ12 (p, E(p))]2
)−1/2
+
+1/2 (ψ11 (p,E(p))− ψ12 (p,E(p))) .
From the last expression, because of analiticity of
the ψ (ε, p) functions follows the fact that quasipar-
ticle spectrum is acoustic at p → 0 and its struc-
ture at p 6= 0 is strongly dependent from the proper-
ties of the renormalised two-particle interaction be-
tween bosons. In case when BEC is absent (n0 = 0)
integral equation for the ψ12 (ε, p) function became
homogeneous and degenerate over the phase of the
function. Thus it is similar to the momentum space
Bethe-Goldstone integral equation for a pair of parti-
cles with zero binding energy, which has a nontrivial
37
solution ψ 6= 0 only in case of attraction V (p) < 0 in
a sufficiently wide region of values of the transmitted
momentum p. Follow this analogy, function ψ12 (ε, p)
can be considered as order parameter for bosonic
PCC, which describe condensation of bosonic pairs
in the momentum space (identically to the Cooper
condensate of the fermion pairs).
Different fenomenological and semi-impirical po-
tentials are used to describe interaction between
helum atoms in a real space. Thay show strong re-
pulsion at small distances and a weak Van der Vaals
attraction at small distances. Howewer, there are a
few reasons why these potentials are inappropriate in
case of microscopical field theory of superfluidity of
the Bose-liquids, which formulated in terms of mo-
mentum space Green functions:
1. These are the fitting potentials and intended to
agree experimental data with theoretical pre-
dictions using a large number of arbitrary pa-
rameters.
2. These potentials are divergent or characterized
by the non-analytical exponential behavior of
the radial dependence at r → 0 , which can be
obtained on the base of the microscopical quan-
tum theory of the atomic interactions. This
non-analyticity can lead to uncontrolled non-
physical features in the behavior of the Fourier
components of the potential.
3. Potentials, which describe interaction of two he-
lium atoms, generally speaking, are not suitable
for describing a pair interaction of the helium
atoms in the quantum Bose-liquid, where the
average interparticle distance is of the order or
less then its De Broglie wave length at the tem-
peratures T < 1K. In this case quantum dif-
fraction and microscopic quantum coherence, as
it is known, plays the main role. Therefore, ef-
fective pair interaction potential in the Bose-
liquid (more precisely, pseudo-potential) may
be very different from the interaction potential
in the vacuum.
In this connection, pair interaction between
bosons was chosen in the form of regularized repulsion
potentials in models of ”hard” and ”semitransparent
spheres”, which Fourier components, due to diffrac-
tion of particles of one another, are oscillating and
alternating functions of transmitted momentum and
can be determined by the spherical Bessel functions
of zero and first orders.
Many-particle collective effects in Bose liquid lead
to a significant renormalization of the pair interac-
tion, which determines the normal and anomalous
self-energy parts (Fig. 1). An important feature of
the renormalized interaction is that, as was shown
in [7], in those areas of the phase volume (ε, p) ,
were the real part of the bosonic polarization opera-
tor ReΠ(p, ω) is negative, repulsion become weaker
(when V (p) > 0) and attraction effectively increased
(when V (p) < 0) (see Fig. 1). The key point in
the behavior of the renormalized (screened) potential
V (p,E(p)), as it was shown by the numerical calcula-
tions in [8], is that for all values of momentum p > 0
the real part of the bosonic polarization operator is
negative (ReΠ(p, ω) < 0, upper corner of Fig. 1) if
quasiparticle spectrum is stable with respect to decay
on a pairs of quasiparticles.
Fig.1. Fourier components of the non- renormalized
(curve 1) and renormalized (curve 2) interactions in
the model of “semitransparent spheres”. In the up-
per corner depicted the momentum dependence of the
real part of the bosonic polarization operator
In the numerical calculations [8], based on the
Fourier-component of the ”hard spheres” potential
V (p) = V0 sin(pa)/pa, were used a simplified model
of the renormalized potential in the form
V (p, ω) = V0 sin(pa)/ (pa + α sin(pa)) ,
here α = −V0Π = const, and Π – the value of the
polarization operator on the mass surface ω = E(p)
averaged over the momentum p which was taken as
negative constant value in case if quasiparticle spec-
trum not decay. Resulting spectrum qualitatively
agrees with experimental spectrum Eexp in super-
fluid 4He (Fig. 2), but numerical correspondence of
the minimal and maximal values of quasiparticle en-
ergies are not satisfactory. In addition, calculated
value of the speed of the first (hydrodynamical) sound
c = 2.08×104 cm/s appeared to be lower than exper-
imental value c = 2.36× 104 cm/s, and total concen-
tration of particles is higher - n = 2.57× 1022 cm−3;
the value of the BEC particle concentration is lower
n0 = 3% then obtained from experiment.
Further calculations, carried in papers [9, 10],
show that oscillating pseudo-potential in the model
of ”semitransparent spheres”
V (p) = V0 (sin(pa)− pa cos(pa)) /(pa)−3
is more appropriate than ”hard sphere” po-
tential both for the stability of the spec-
trum and its correspondence to the empiri-
cal spectrum in 4He. In the ”semitransparent
spheres” model the explicit momentum depen-
dence of the polarization operator Π (p, ω) is used.
38
Fig.2. Solid curve – spectrum obtained in the model
of “hard spheres”, dot curve – experimental data
[11]-[14]
Iterative numerical calculations of the self-energy
and bosonic polarization operator, the two-particle
order parameter and the quasiparticle spectrum at
T = 0 allowed to find a conditions when theoretical
spectrum E(p) is in a good agreement with experi-
mental spectrum of elementary excitations in 4He.
The roton minimum of the quasiparticle spectrum
E(p) in the Bose liquid, as it was shown in [10],
clearly associated with the first negative minimum of
the Fourier-component renormalized potential. The
only fitting parameter in these calculations was the
amplitude of the starting pseudo-potential with the
parameter a = 2.44 A−1, which is equal to the twice
of a quantum radius of the 4He atom. For the cal-
culations we take an experimental value of the BEC
density n0 = 9% n = 1.95× 1022 cm−3 [3, 4].
As a result, after numerical calculations it was
possible to get a quite satisfactory agreement of the
theoretical E(p) and experimental Eexp(p) spectrum
under condition p < 2.5 A−1 (Fig. 3, curve 1).
In calculations of E(p) the only fitting parameter
was taken in order to satisfy a condition that qua-
siparticle phase velocity E(p)/p at p → 0 coincide
with hydrodynamical speed of sound c1
∼= 236 m/sec
in the liquid 4He, which corresponds to the value
U0 = V0/4πa3 = 1552 K for the amplitude of the
”semitransparent spheres” potential at a = 2.44 A−1.
In the range p < 2.5 A−1 theoretical spectrum E(p)
lies slightly above Eexp(p). This is, most likely, due
to the fact that vertex function Γ(k, p) which decay
with increasing of the momentum p, was taken as
constant Γ = 1.5. This value was obtained from
the exact asymptotic of the polarization operator
Π(0, 0) = n/mc2. In this connection vertex function
on the interval 2.1 A−1 < p < 3.8 A−1 was approx-
imated by the linear function, which slowly changed
from Γ = 1.5 to Γ = 1.1. Spectrum, obtained within
this approximation (see Fig. 3, curve 2), is in good
agreement with experimental curve for all range of
the momentum.
Self-consistency of the given model is confirmed
by the value of the total particle density nteor =
2.12 × 1022 cm−3, which is close to the experi-
mental value of the particle density n = 2.12 ×
1022 cm−3 of the liquid 4He (when n0 = 9%n).
On the other hand, independent calculation of the
over-condensate density of particles n1 for the given
parameters gives a value of about 93%n, which
also agrees with experiment under condition that
BEC density is determined with accuracy of ±1%.
Fig.3. Curve 1 – theoretical quasiparticle spectrum
E(p) , obtained with a model of the “semitransparent
spheres” for the constant value of the vertex func-
tion Γ = 1.5. Curve 2 – theoretical quasiparticle
spectrum E(p), obtained with a model of the “semi-
transparent spheres” for the weakly decaying (from
Γ = 1.5 to Γ = 1.1 ) vertex function, on the inter-
val 2.1 A−1 < p < 3.8 A−1. In both cases the value
of the fitting parameter is U0 = V0/4πa3 = 1552K.
Circles show experimental spectrum, obtained by in-
elastic neutron scattering in the liquid 4He [11]-[14],
stars – show the results of the experiment [15] be-
yond the roton minimum 2 A−1 < p < 3.6 A−1.
Inserted box shows weak oscillations of the spectrum
with maximum Emax = 17.2 K at p = 2.99 A−1 and
minimum Emax = 16.7 K at p = 3.39 A−1
Thus, proposed in [6] model of superfluid Bose liq-
uid with a suppressed BEC and intensive PCC, which
is based on renormalized perturbation field theory
with combined variables [16], allow to cut an infi-
nite series by the low BEC density and obtain ”trun-
cated”, closed system of nonlinear integral equations
for the self-energy parts
∑
ij (ε, p) , (j = 1, 2). In this
way it is possible to build a self-consistent micro-
scopic theory of superfluid Bose liquid and calculate
”from first principles” the spectrum of elementary
excitations E(p), based on realistic models for the
pair interaction potential V (r), with oscillating finite
Fourier component. For a wide class of repulsion po-
tentials, which characterized by the infinite (”hard
sphere” model) or finite (”semitransparent spheres”
model) jump or inflection point (jump of the first
derivative) of the V (p) Fourier components, and are
the alternating functions of transmitted momentum p
which form an effective attraction V (p) < 0 at p 6= 0
at certain areas of the momentum space, which is
not associated with the presence of Van der Waals
forces and have a quantum mechanical diffraction na-
ture. This attraction significantly enhanced by the
collective effects of renormalization (”screening”) of
the initial interaction, which are described by the
bosonic polarization operator. Enhancement of the
”attraction” occurs because of the negative value of
the bosonic polarization operator on the ”mass shell”
ω = E(p) in the all range of the momentum where the
39
quasiparticle spectrum E(p) is stable against decay-
ing processes. Rather strong ”attraction” between
a pairs of bosons that arise in the momentum space
create an intensive PCC, which together with a weak
BEC form an entire coherent condensate. Such con-
densate is the microscopical basis for the superfluid
component of Bose liquid.
On the other hand, the oscillating character
of the renormalized Fourier component of the po-
tential leads to a non-monotonic behavior of the
momentum dependencies of the mass operators∑
ij (ε, p) , (j = 1, 2) and, as a result, to the ap-
pearance of the roton minimum in the spectrum of
quasiparticles E(p), which definitely connected with
first deepest negative minimum of the renormalized
Fourier component of the potential. Thus, for suffi-
ciently large values of the amplitude of the initial po-
tential V0 quasiparticles excitation spectrum become
unstable in some domain of momenta p 6= 0 where
E2(p) < 0.
For the conclusion it is necessary to emphasize
that mentioned property of the polarization opera-
tor Π (p,E(p)) < 0 is typical only for Bose-systems,
where single-particle and many-particle spectrum are
coincide and have a common zero energy reference
point, unlike Fermi-systems, where single-particle
excitation spectrum due to the Pauli principle is
counted from the Fermi energy. Therefore, corre-
sponding effective increasing of the negative values
of the polarization operator cannot take place for the
Fermi-liquid 3He, so formation of the Cooper pairs
is possible only for non-zero values of the orbital mo-
mentum and a real Van der Waals attraction between
fermions. Probably, this is the reason for the three
order difference between critical temperatures of su-
perfluid transition in 4He and 3He.
Finally, we briefly focus on the calculation of
the temperature dependence of superfluid component
density based on the model of the coherent structure
of the condensate of the Bose liquid 4He. Tempera-
ture dependence of the superfluid component density
in this model is given by the following expression [10]
ρs(T )
ρ
=
ψ0(T )
Ṽ (0)n
[
1− ψs(T )
Ṽ (0)n
]−1
, ρ = mn,
here
ψ0(T ) = −1
2
∫
dq
(2π)3
Ṽ (q)×
[
A12 −D12
c1q
coth
(c1q
2T
)
+
B12
c2q
coth
(c2q
2T
)]
,
ψs(T ) = −1
2
∫
dq
(2π)3
Ṽ (q)×
[
D12
c1q
coth
(c1q
2T
)
+
B12
c2q
coth
(c2q
2T
)]
.
In the last expression Ṽ (q) - the Fourier compo-
nent of the renormalized due to many-particle effects
a pair interaction potential. For the calculation of
the temperature dependences we use model potentials
(”hard sphere” and ”semitransparent sphere” poten-
tial), Aziz potential and expressions for the velocity
of the first and second sounds in liquid 4He and they
temperature corrections, obtained in [18] from the so-
lutions of a kinetic type equations:
E2
1(q) = c2
1q
2 = c2
0
(
1 +
29
16
ρn
ρ
)
q2,
E2
2(q) = c2
2q
2 =
c2
0
3
(
1− 33
8
ρn
ρ
)
q2.
Temperature dependence of the superfluid com-
ponent, calculated in [10] and [19], is given on
Fig. 3. For the comparison result obtained by
GinzburgV.L. [20], presented on the same figure.
So, field theoretical methods, which have been
used in order to explain the maxon-roton spectru
leads to a good agreement between theory and an
experiment only for the case T = 0, but these at-
tempts do not seem to be fully complete for case of
nonzero temperature. This case requires futher in-
vestigations.
Fig.4. Temperature dependencies of the Bose- liq-
uid 4He superfluid density obtained in [10] (curve
1); and in [19] (curve 2); experimental data obtained
by Andronikashvilli E.L. [21] (curve 3); result of
Ginzburg V.L. [20] (curve 4)
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МИКРОСКОПИЧЕСКАЯ ТЕОРИЯ, СТРУКТУРА СВЕРХТЕКУЧЕЙ
КОМПОНЕНТЫ И СПЕКТРА ЭЛЕМЕНТАРНЫХ ВОЗБУЖДЕНИЙ
В БОЗЕ-ЖИДКОСТИ He-II
Е.А.Пашицкий, А.В.Чумаченко, С.И.Вильчинский
В рамках самосогласованной микроскопической модели сверхтекучей нерелятивистской бозе-жидкости
4He с подавленным за счет взаимодействия одночастичным и интенсивным парным конденсатами про-
изведен расчет с первых принципов спектра элементарных возбуждений E(p) исходя из реалистичных
моделей для потенциалов парного взаимодействия V (r), которые имеют конечную осциллирующую
фурье-компоненту. Показано, что в рамках моделей "твердых"и "полупрозрачных сфер"осциллиру-
ющий характер перенормированой фурье-компоненты потенциала парного взаимодействия бозонов
приводит к немонотонному поведению импульсных зависимостей перенормированных массовых опе-
раторов и, как следствие, к появлению в спектре квазичастиц E(p) ротонного минимума, который
однозначно связан с наиболее глубоким первым отрицательным минимумом фурье-компоненты пере-
нормированного парного взаимодействия бозонов.
МIКРОСКОПIЧНА ТЕОРIЯ, СТРУКТУРА НАДПЛИННОЇ КОМПОНЕНТИ
ТА СПЕКТРУ ЕЛЕМЕНТАРНИХ ЗБУДЖЕНЬ В БОЗЕ-РIДИНI He-II
Є.А.Пашицький, А.В.Чумаченко, С.I.Вiльчинський
В рамках самоузгодженої мiкpоскопiчної моделi надплинної нерелятивiстської бозе-piдини 4He з подав-
леним за pахунок взаємодiї одночастинковим i iнтенсивним парним конденсатами обчислено з пеpших
пpинципiв спектp елементаpних збуджень E(p), виходячи з pеалiстичних моделей для потенцiалiв паp-
ної взаємодiї V (r), якi мають скiнчену осцилюючу фуp’є-компоненту. Показано, що в pамках моделей
"твердих"та "напiвпpозоpих сфер"осцилюючий хаpактеp пеpеноpмованої фуp’є-компоненти потенцiа-
лу парної взаємодiї бозонiв пpиводить до немонотонної поведiнки iмпульсних залежностей перенормо-
ваних масових опеpатоpiв i, як наслiдок, до появи в спектpi квазiчастинок E(p) pотонного мiнiмуму,
який однозначно пов’язаний з найбiльш глибоким пеpшим вiд’ємним мiнiмумом фур’є-компоненти пе-
ренормованої парної взаємодiї бозонiв.
41
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