Advanced Bogolyubov model of imperfect bose gas and superfluidity

The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estim...

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Date:	2012
Main Authors:	Bogolyubov, N.N. Jr., Sankovich, D.P.
Format:	Article
Language:	English
Published:	V.A. Steklov Mathematical Institute of RAS 2012
Series:	Вопросы атомной науки и техники
Subjects:	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/107119
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Cite this:	Advanced Bogolyubov model of imperfect bose gas and superfluidity / N.N. Bogolyubov, Jr., D.P. Sankovich // Вопросы атомной науки и техники. — 2012. — № 1. — С. 245-247. — Бібліогр.: 7 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1071192025-02-09T21:23:20Z Advanced Bogolyubov model of imperfect bose gas and superfluidity Расширенная модель Боголюбова неидеального бозе-газа и сверхтекучесть Розширена модель Боголюбова неідеального бозе-газу і надплинність Bogolyubov, N.N. Jr. Sankovich, D.P. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estimate for temperature and density in the λ-point. С микроскопической точки зрения изучены равновесные свойства системы взаимодействующих бозонов. Мы вычисляем плотность сверхтекучей компоненты в модели Боголюбова неидеального бозе-газа. Рассмотрен модельный суперстабильный гамильтониан. Мы изучаем случай некоторого парного потенциала и получаем оценку для температуры и плотности в λ-точке. З мікроскопічної точки зору вивчені рівноважні властивості системи взаємодіючих бозонів. Ми обчислюємо густину надплинної компоненти в моделі Боголюбова неідеального бозе-газу. Розглянуто модельний суперстабільний гамільтоніан. Ми вивчаємо випадок деякого парного потенціалу і одержуємо оцінку для температури і густини в λ-точці. 2012 Article Advanced Bogolyubov model of imperfect bose gas and superfluidity / N.N. Bogolyubov, Jr., D.P. Sankovich // Вопросы атомной науки и техники. — 2012. — № 1. — С. 245-247. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 05.30.Jp, 03.75.Fi, 67.40.-w https://nasplib.isofts.kiev.ua/handle/123456789/107119 en Вопросы атомной науки и техники application/pdf V.A. Steklov Mathematical Institute of RAS
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
spellingShingle	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Bogolyubov, N.N. Jr. Sankovich, D.P. Advanced Bogolyubov model of imperfect bose gas and superfluidity Вопросы атомной науки и техники
description	The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estimate for temperature and density in the λ-point.
format	Article
author	Bogolyubov, N.N. Jr. Sankovich, D.P.
author_facet	Bogolyubov, N.N. Jr. Sankovich, D.P.
author_sort	Bogolyubov, N.N. Jr.
title	Advanced Bogolyubov model of imperfect bose gas and superfluidity
title_short	Advanced Bogolyubov model of imperfect bose gas and superfluidity
title_full	Advanced Bogolyubov model of imperfect bose gas and superfluidity
title_fullStr	Advanced Bogolyubov model of imperfect bose gas and superfluidity
title_full_unstemmed	Advanced Bogolyubov model of imperfect bose gas and superfluidity
title_sort	advanced bogolyubov model of imperfect bose gas and superfluidity
publisher	V.A. Steklov Mathematical Institute of RAS
publishDate	2012
topic_facet	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
url	https://nasplib.isofts.kiev.ua/handle/123456789/107119
citation_txt	Advanced Bogolyubov model of imperfect bose gas and superfluidity / N.N. Bogolyubov, Jr., D.P. Sankovich // Вопросы атомной науки и техники. — 2012. — № 1. — С. 245-247. — Бібліогр.: 7 назв. — англ.
series	Вопросы атомной науки и техники
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first_indexed	2025-11-30T23:03:19Z
last_indexed	2025-11-30T23:03:19Z
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fulltext	ADVANCED BOGOLYUBOV MODEL OF IMPERFECT BOSE GAS AND SUPERFLUIDITY N.N. Bogolyubov, Jr. and D.P. Sankovich ∗ V.A. Steklov Mathematical Institute of RAS, Moscow, Russia (Received October 17, 2011) The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estimate for temperature and density in the λ-point. PACS: 05.30.Jp, 03.75.Fi, 67.40.-w 1. INTRODUCTION It took 30 years after the liquefaction of 4He in 1908 to make the discovery that liquid helium was not just a “cold” liquid. Below 2.18K, it is a “quantum” liq- uid which exhibits spectacular macroscopic quantum behavior that can be seen with our eyes. The first ev- idence for superfluidity was, in fact, a seminal experi- ment in 1935 by Misener. He found that the viscosity decreased sharply as one went just below 2.18K, al- though it was still finite. This was the first evidence that He II was a new kind of liquid (this work was published under the name of E.F. Barton, the head of the Toronto lab, causing some later confusion [1]). The superfluid behavior is commonly associated with the phenomenon of a Bose-Einstein condensa- tion. However, the detailed theoretical investigation of superfluidity and condensation in liquid helium is a great challenge, as it is a strongly interacting quan- tum system and cannot be effectively described with a mean-field or perturbative approaches. Experimental achievement of Bose-Einstein con- densation in dilute trapped atomic gases gives a unique possibility to study their superfluid properties under condition of weak interactions. It has opened an opportunity for revisiting the concept of superflu- idity. In this context the issue of the atomic Bose- Einstein condensation under rotation has attracted great interest. On the other hand, a little is known about superfluidity in a translational motion of the atomic condensate. There are still no experiments that would correspond to the demonstration of fric- tionless non-rotary flow, which is the most intuitive manifestation of this phenomenon. In Section 2 we analyze the superfluid properties of Bogolyubov quasiparticles. We derive analytically superfluid density of the system. In Section 3 we present some generalization of a standard Bogolyubov model. This model is ex- actly solvable in thermodynamic limit. The estimates for temperature and total density in the λ-point are found. 2. BOGOLYUBOV MODEL. SUPERFLUIDITY Let us consider a system of N spinless identical non- relativistic bosons of mass m enclosed in a centered cubic box Λ ⊂ R 3 of volume V = \|Λ\| = L3 with periodic boundary conditions for the wave functions. The Hamiltonian of the system can be written in the second quantized form as: ĤΛ(μ) ≡ ĤΛ − μN̂Λ = ∑ k∈Λ∗(εk − μ)â†kâk + 1 2V ∑ p,q,k∈Λ∗ ν(k)â†pâ † qâp+kâq−k. (1) Here â# p = {â†p or âp} are the usual boson creation (annihilation) operators for the one-particle state ψp(x) = V −1/2 exp(ipx), p ∈ Λ∗, x ∈ Λ, acting on the Fock space FΛ = ⊕∞ n=0H(n) B , where H(n) B ≡ [L2(Λn)]symm is the symmetrized n-particle Hilbert space appropriate for bosons, and H(0) B = C. The sums in (1) run over the dual set: Λ∗ = {p ∈ R 3 : pα = 2π L nα, nα = 0,±1,±2, . . . , α = 1, 2, 3}, εp = \|p\|2/(2m) is the one-particle energy spectrum of free bosons in the modes p ∈ Λ∗ (we propose � = 1), N̂Λ = ∑ k∈Λ∗ â † kâk is the total particle-number oper- ator, μ is the chemical potential, ν(k) is the Fourier transform of the interaction pair potential Φ(x). We suppose that Φ(x) = Φ(\|x\|) ∈ L1(R3) and ν(k) is a real function with a compact support such that 0 ≤ ν(k) = ν(−k) ≤ ν(0) for all k ∈ R3. Under these conditions the Hamiltonian (1) is superstable. ∗Corresponding author E-mail address: sankovch@mi.ras.ru PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 245-247. 245 So long as the rigorous analysis of the Hamil- tonian (1) is very knotty problem, Bogolyubov in- troduced the model Hamiltonian of the superfluidity theory [2, 3]. He proposed to disregard the terms of the third and fourth order in operators â# p , p �= 0 in the Hamiltonian (1): ĤB Λ (μ) = ∑ k∈Λ∗ (εk − μ)â†kâk + 1 V â†0â0 ∑ k �=0 ν(k)â†kâk + 1 2V ∑ k �=0 ν(k)(â†k â † −kâ0â0 + â†0â † 0â−kâk) + ν(0) V â†0â0 ∑ k �=0 â†kâk + ν(0) 2V â†0â † 0â0â0. (2) Then Bogolyubov takes advantage of the macro- scopic occupation of the zero momentum one-particle state to replace the corresponding creation and anni- hilation operators â# 0 by c-numbers: â†0√ V → c̄, â0√ V → c, (3) where c ∈ C and the bar means complex conjugation. The validity of substituting a c-number for the k = 0 mode operators a# 0 was established rigorously for the Bogolyubov Hamiltonian (2) in papers [4, 5]. So one can prove that the model Hamiltonian ĤB Λ (μ) is thermodynamically equivalent to the approximate Hamiltonian: ĤB Λ (μ, c) = ∑ k �=0 [εk − μ+ \|c\|2(ν(0) + ν(k))]â†k âk + 1 2 ∑ k �=0 ν(k)(c2â†kâ † −k + c̄2âkâ−k) + 1 2 ν(0)\|c\|4V − μ\|c\|2V. (4) The self-consistency parameter c in the method is determined by the condition that the approximate pressure p[ĤB Λ (μ, c)] be maximal. In the same time, the stability condition μ ≤ ν(0)\|c\|2 must be fulfilled. A necessary condition for p[ĤB Λ (μ, c)] to be maxi- mum (self-consistency equation) in the case of the Bogolyubov model is〈 ∂ĤB Λ (μ, c) ∂c 〉 ĤB Λ (μ,c) = 0. (5) This equation always has the trivial solution c = 0 (no Bose condensation). By explicit calculations we get the following equation to obtain a nontrivial so- lution: μ− xν(0) = 1 2V ∑ k �=0 [ −ν(k) hk Ek coth βEk 2 +(ν(0) + ν(k)) ( fk Ek coth βEk 2 − 1 )] , (6) where uk = √ 1 2 ( fk Ek + 1 ) , vk = − √ 1 2 ( fk Ek − 1 ) , fk = εk − μ+ x(ν(0) + ν(k)), hk = xν(k), Ek = √ f2 k − h2 k and we denote x ≡ \|c\|2. We shall use the function ν(k) = { ν(0) for \|k\| ≤ k0, 0 for \|k\| > k0 (7) as the Fourier transform of the pair potential. Here ν(0) = 4πr0/m, k0r0 = 1, r0 = 2.56 Å, m = mHe4 . In this case it was shown [4–6], that if the poten- tial ν(k) in the Bogolyubov model of superfluidity (2) satisfies the condition ν(0) ≥ 1 2V ∑ k �=0 ν2(k) εk , (8) then there exists the domain of stability on the phase diagram {0 < μ ≤ μ∗, 0 ≤ θ ≤ θ0(μ)}, where the non- trivial solution of the self-consistency equation takes place. In this domain there is the non-zero Bose con- densate. At the boundary θ = θ0(μ) of this domain the Bose condensate density equals ρ0 = μ/ν(0). In this case the quasi-particles spectrum of the Bo- golyubov Hamiltonian (2) Ek = √ εk(εk + 2ρ0ν(k)) has a gapless type and the famous Landau’s criterion of superfluidity min k Ek \|k\| > 0 holds. We shall now study the fluid motion of our sys- tem at a uniform, constant velocity. This will provide a means for defining normal and superfluid compo- nents. The normal density may be defined by the effective mass for drift, as was originally asserted by Landau [7]. The mass density of the normal fluid is found from ρn = − 1 3V ∑ k k2 ∂nk ∂εk = β 3V ∑ k k2nk(1 + nk). (9) The total mass density ρ = mn = ρn + ρs, where ρs is the density of superfluid. Of course, ρs is not to be interpreted as the density of the particles in the zero-momentum state ρ0 = mn0. On evaluation of equation (9), we find (ρ−ρs)/ρ ≈ 0.1, (ρ − ρ0)/ρ ≈ 0.1, ρs > ρ0 for μ = μ∗ and θ = θ0(μ∗). For the model pair potential (7) one can find ρ ≈ 0.02 g/cm3 and θ0 ≈ 0.18 K. (For real He4 θ0 ≈ 2.18 K and ρ ≈ 0.13 g/cm3.The Bogolyubov HamiltonianHBog does not take into account interac- tions between the excitations. So the physical results based on the Hamiltonian HBog are far from “real superfluidity”. 3. BOGOLYUBOV MODEL. λ-POINT We attempt to take into account the terms of fourth order in operators a# k , k �= 0 in the full Hamiltonian (1). Consider the model superstable Hamiltonian Ĥadv Λ (μ) = ĤB Λ (μ) + ν(0) 2V ˆ̃ N 2 , (10) 246 where ˆ̃ N = ∑ k �=0 â † kâk. It is possible to prove that this Hamiltonian is equivalent to the approximate Hamiltonian Ĥadv Λ (μ, x) = ĤB Λ (μ) + ν(0)x ˆ̃ N + ν(0) 2 x2V, (11) where the self-consistency equations are x = 1 2V ∑ k �=0 ( fk Ek coth βEk 2 − 1 ) , (12) μ− ν(0)(n0 + x) = 1 2V ∑ k �=0 ν(k) ( fk − hk Ek coth βEk 2 − 1 ) . (13) Here fk = εk − μ+ ν(0)(n0 + x) + ν(k)n0, (14) hk = ν(k)n0, E 2 k = f2 k − h2 k. (15) When n0 = 0 the equation (12) reads x = 1 V ∑ k �=0 1 exp [β(εk − μ+ ν(0)x)] − 1 . (16) This equation has the unique positive solution for μ ≤ μc = ν(0)ρ(0) c , where ρ(0) c = 1 V ∑ k �=0 1 exp (βεk) − 1 . (17) Then, the condensate density n0 �= 0 for μ > μc. So, for n0 �= 0 we must solve the pair of self-consistency equations (12)-(13). For μ = ν(0)n the Landau’s cri- terion of superfluidity holds. In this case the equation (9) and the condition ρ = ρn give an estimate for the λ-point θs ≈ 1.9 K for ρ ≈ 0.08 g/cm3. This estimate is in sufficiently good-enough agreement with exper- imental data. As is easy to see this method does not describe a case of small density. Yet it permits to investigate the neighborhood of the λ-point. 4. CONCLUSIONS We have investigated the superfluid properties of a Bogolyubov’s weakly interacting Bose gas at thermal equilibrium. Using the conventional definition of the normal fraction, we find that the gas has a signifi- cant superfluid fraction only in the Bose condensed regime. However, it is impossible to describe the λ- point region by a standard Bogolyubov model. To investigate more carefully the λ-point regime where the superfluid density tends to zero, we examined the advanced Bogolyubov model. In this model we take into account the terms of fourth order in operators â# k , k �= 0 in the full Hamiltonian of a non-ideal Bose- gas. Quantitatively, we have found that the super- fluid phase transition temperature is in good agree- ment with experimental data. References 1. A. Griffin. New light on the intriguing history of superfluidity in liquid 4He // J. Phys.: Condens. Matter. 2009, v. 21, 164220, 9 p. 2. N.N. Bogolyubov. On the theory of superfluidity // J. Phys. (USSR). 1947, v. 11, p. 23-32. 3. N.N. Bogolyubov. Energy levels of the imperfect Bose-Einstein gas // Bull. Moscow State Univ. 1947, v. 7, p. 43-56. 4. N.N. Bogolyubov, Jr. and D.P. Sankovich. As- ymptotic exactness of c-number substitution in Bogolyubov’s theory of superfluidity // Cond. Matter Phys. 2010, v. 13, 23002, 6 p. 5. D.P. Sankovich. Bogolyubov’s theory of super- fluidity, revisited // Int. J. Mod. Phys. 2010, v. B24, p. 5327-5336. 6. D.P. Sankovich. Bogolyubov’s theory of super- fluidity // Phys. of Part. and Nucl. 2010, v. 41, p. 1068-1070. 7. L.D. Landau. The theory of superfluidity of He- lium II // J. Phys. (USSR). 1941, v. 5, p. 71-93. �� !�� ! �� λ�� " �#��# ��$ �� # �#��%�# ��# �� &��#� �� #�� &�� $ �� # �� #� �� #�� #��#�� &�� !#�� # �� %�&�� !#�� # �� λ�� !#� '()

Advanced Bogolyubov model of imperfect bose gas and superfluidity

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