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A study of the temperature dependence of single-particle and pair coherent condensate densities for the Bose liquid with the “depleted” single-particle Bose–Einstein condensate
Journal of Molecular Liquids 124 (2006) 72 – 77 www.elsevier.com/locate/molliq
A study of the temperature dependence of single-particle and pair coherent condensate densities for the Bose liquid with the ‘‘depleted’’ single-particle Bose–Einstein condensate A. Chumachenko *, S. Vilchynskyy Department of Physics, Taras Shevchenko Kiev National University, Kiev 03022, Ukraine Received 11 March 2005; accepted 20 October 2005
Abstract In this paper we present the studies of the temperature dependence of single-particle and pair coherent condensate densities for the Bose liquid with the ‘‘depleted’’ single-particle Bose – Einstein condensate (BEC) at T m 0. Our investigations are based on the field-theoretical Green functions approach. Superfluid state is described at T m 0 on the basis of the use of empirical data for the first and second sound speeds. The structure of the superfluid state is studied taking into account the appearance of the normal component within non-zero temperatures and the branch of the second sound, where speed approaches zero at T Y T k. The obtained temperature dependences of the single-particle and pair coherent condensate qualitatively agreed with the experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Condensed matter; Bose liquid; Superfluidity
1. Introduction Despite the big progress achieved in the theory of superfluidity since the pioneer works by Landau [1], Bogolyubov [2], Feynman [3] and others [4 –22], the task of constructing a microscopic theory of a superfluid (SF) state of the 4He Bose liquid cannot be considered complete. There are some unsolved contradictions between the theory and the experiment. One of them is the problem with the structure of quasi-particle spectrum in the case of superfluid hydrodynamics. These contradictions could be considered as the question of quantum-mechanical structure of superfluid 4He component below the k point. This question is crucial for the creation of the consistent microscopic superfluid theory of Bose liquid for the cases of both zero and non-zero temperatures. In the works [23,24] was introduced the microscopic model of the SF state of a Bose liquid at T = 0 with a single-particle Bose –Einstein condensate (BEC) suppressed by interaction, based upon a renormalized field perturbation theory with * Corresponding author. E-mail addresses: [email protected] (A. Chumachenko), [email protected] (S. Vilchynskyy). 0167-7322/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2005.10.003
combined variables [11 –14]. That allows one to obtain a selfconsistent ‘‘short’’ system of nonlinear integral equations for self-energy parts R˜ij ( p,() by means of truncating the infinite series at the small density of the BEC (n 0/n b 1). By the same token, one can work out a self-consistent microscopic theory of the superfluid Bose liquid and perform an ab initio calculations of the spectrum of elementary excitations E( p), starting from the realistic models of the pair interaction potential U(r). In this work we study a superfluid state structure of the 4He Bose liquid under a non-zero temperature (0 < T < T k ). The similar problem was investigated in work [25] on the basis of Chester [26] IBG wave function W IBG by Jastrov factors F = Pf ij . We use the microscopic model proposed in [27] for the superfluid Bose liquid with a ‘‘depleted’’ BEC. The small parameter in this model is the ratio of the BEC density to the total density of the Bose liquid q 0/q s b 1 in contrast to the Bogolyubov theory [2] for the almost ideal Bose gas, where the small parameter is the ratio of the number of the overcondensate excitations to the number of particles in the intense BEC (n n 0) / n 0 b 1. Such an approach is based on the analysis of the numerous precise experimental data on the neutron inelastic scattering [22,28 – 30] and on the results of the quantum evaporation of
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77 4
He atoms [31]. According to this experiments the maximal density q 0 of the single-particle BEC in the 4He Bose liquid does not exceed 10% of the total density q of 4He liquid even at very low temperatures T b T k, whereas the density of the SF component q s Y q at T Y 0 [10]. Such a low density of the BEC is implied by the strong interaction between the 4He atoms and is an indication of the fact that the quantum structure of the part of the SF condensate in He II carrying the ‘‘excess’’ density (q s q 0) b q 0 calls for a more thorough investigation. So, in this case the density of the superfluid component q s is determined by the quantity of the renormalized anomalous selfenergy parts R 12(0,0), which are a superposition of the ‘‘depleted’’ single-particle BEC and the intense ‘‘Cooper’’ pair coherent condensate (PCC), with the coincident phases (sings) of the corresponding order parameters. Such a PCC emerges due to the effective attraction between bosons in some regions of the momentum space, which results from the oscillating sign-changing momentum dependence of the Fourier component V( p) of the interaction potentials U(r) with the inflection points in the radial dependence. The system of Dyson-Belyaev equations [5], that describes the superfluid (SF) state of the Bose liquid with strong interaction between bosons and a weak single-particle Bose–Einstein ˜ 11 and condensate (BEC), allows one to express the normal G ˜ anomalous G12 renormalized single-particle boson Green functions in terms of the respective self-energy parts R˜11 and R˜12. These equations can be presented in the next analytic form of [27]: R˜ 11 ðpY ; eÞ ¼ n0 KðpY ; eÞV˜ ðY p ; eÞ þ n1 V ð0Þ þ W˜ ðY p ; eÞ; ð1Þ 11
R˜ 12 ðpY ; eÞ ¼ n0 KðpY ; eÞV˜ ðY p ; eÞ þ where
Z
Y
Y W12 ðY p ; eÞ;
Y
d3 k
Y where C Y p ; e; k ; xÞ is the vertex part (three-pole) describing many-particle correlations of the local field’s type; KðpY ; eÞ ¼ C p ; e; 0; 0Þ ¼ Cð0; 0; Y ðY p ; eÞ is the vertex part with zero values of the input momentum and energy. Taking into account the residues at the poles of single-particle Green functions G˜ij ðpY ; eÞ and neglecting the contributions of eventual poles of the functions Y C Y p ; e; k ; xÞ and V˜ ðY p ; eÞ, that do not coincide with the poles of G˜ ij ðY p ; eÞ, the functions W˜ ij ðY p ; eÞ on a ‘‘mass shell’’ (=E( p) can be presented in the following form (at T =0): 1 p ; E ð pÞ Þ ¼ W˜ 11 ðY 2
Z
Y
d3 k
ð2pÞ
3
Y C pY ; E ð pÞ; k ; E ðk Þ
3 2 Y A k ; E ðk Þ Y 15; V˜ pY k ; E ð pÞE ðk Þ 4 E ðk Þ ð7Þ Z 3Y 1 d k Y ˜ W12 ðp ; Eð pÞÞ ¼ 2 ð2pÞ3 Y Y V˜ Y p k ; E ð pÞE ðk Þ C pY ; Eð pÞ; k ; E ðk Þ Y Y Y n0 K k ; Eðk Þ V˜ k ; E ðk Þ þW˜ 12 k ; Eðk Þ ; E ðk Þ ð8Þ where AðY p Þ ¼ n0 KðY p ; E ðY p ÞÞV˜ðpY ; EðY p ÞÞ þ n1 V ð0Þ þ W˜ 11 ðY pÞ Y
Z
dx Y ˜ Y Y Gij k V p k ; e x W ij ðp ; eÞ ¼ i 3 2p ð2pÞ Y C Y p ; e; k ; x ˜
ð2Þ
73
þ
ð3Þ
V˜ ðY p ; eÞ ¼ V ðpY Þ½1 V ðpY ÞPðY p ; eÞ1 ; ð4Þ where V( p) is the Fourier component of the bare potential of the pair interaction of the bosons; V˜ ðY p ; eÞ is the renormalized (screened) pair interaction between bosons due to the manyparticles collective effects; n0 is the number of particles in BEC; n1 =n n0 is the number of overcondensate particles (n1 H n0), which is determined by the condition of the total particle number conservation: Z 3Y Z Y d k dx n ¼ n0 þ n1 ¼ n0 þ i G ð5Þ 11 k ; x : 2p ð2pÞ3
p2 l; 2m
ð9Þ
where l represents the chemical potential of the quasi-particles, and satisfies the Hugengoltz–Pines relation [6] l ¼ R˜ 11 ð0; 0Þ R˜ 12 ð0; 0Þ:
ð10Þ
The quasi-particle spectrum corresponding to the poles of Green’s function is determined in general, with allowance for relations (1) an (2), by the following expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E ðY p Þ ¼ A2 ðY p Þ n0 K ð Y p ; E ðY p ÞÞV˜ ðY p ; EðpY ÞÞ þ W˜ 12 ðY pÞ : ð11Þ 2. The structure of the superfluid Bose liquid state at T =/ 0
Y
The boson’s polarization operator Pðp ; eÞ describes manyparticles correlation effects and can be presented as Z 3Y Z dx Y Y n Y dk Y C p ; e; ; k ; x G11 k ; x P ðp ; eÞ ¼ i 3 2p ð2p Þ Y Y G11 k þ pY ; e þ x þ G12 k ; x Y o G12 k þ pY ; e þ x ð6Þ
=The main purpose of our research is to investigate the superfluid helium condensate structure by means of finding temperature dependence of the single-particle and pair coherent condensate densities under the assumption that the superfluid component is a superposition of a weak single-particle BEC and an intensive PCC. Let us consider the superfluid Bose liquid state at T m 0, when both superfluid component density q s(T) and normal
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component density q n(T) exist simultaneously. As it is shown in [13,14], the expression for the renormalized Green function G˜ij ð pÞ is constructed with the help of the combined variables: W˜ ð xÞ ¼ W˜ L ð xÞ þ W˜ sh ð xÞ:
ð12Þ Y
In the long-wavelength region jk j < k0 (where k 0 is some characteristic momentum) this variables are just the hydrody˜ L(x) close to those described in the Landau namics variables W quantumY hydrodynamics [1], and in the short-wavelength region jk j > k0 they coincide with the usual field operators ˜ sh(x): W qffiffiffiffiffiffiffiffiffi n˜ L b˜nL ˜ ˜ ˜ þ i/L ; W˜ sh ¼ wsh ei/L ; WL ð xÞ ¼ b˜nL 1 þ 2b˜nL 1 wsh ¼ w wL ; wL ðrÞ ¼ pffiffiffiffi ~ ak eikr ¼ V jkjk0
qffiffiffiffiffiffiffiffiffi ˜ b˜nL ei/L :
ð13Þ
Such an approach means that the separation of the Bose system into a macroscopic coherent condensate and a gas of supracondensate excitations is made not on the statistical level, like in the case of weakly non-ideal Bose gas [2], but on the level of ab initio field operators, which are used to construct a microscopic theory of the Bose liquid. In the region of small p m 0 at T Y 0 Green’s functions have the form G˜ 11 ð pÞ ¼ ns guu ð pÞ igup ð pÞ
G˜ 12 ð pÞ ¼ ns guu ð pÞ where Uuu ð pÞ ¼
Z jqjq0
1 ns gpp ð pÞ Uuu ð pÞ N 4qs 2
ð14Þ ð15Þ
d4 q guu ðqÞguu ð p qÞ; p 2p4
Y ¼ k ; e ; q ¼ ðqY ; xÞ
ð16Þ
with g lm ( p) being the ‘‘hydrodynamics’’ Green functions, which are associated with the long-wavelength fluctuations of the phase and density of the condensate (lm = u p ). The expressions for g uu ( p), g up ( p), g pp ( p) are calculated in [12] for T > 0. They contain sums of two pole terms, corresponding to the first and second sounds with the velocities c 1 and c 2 in the Bose liquid with the normal and superfluid components: alm dlm qn =q blm q =q glm ðk; eÞ þ 2 n2 2 e2 c21 k 2 e c2 k l; m u; p
We henceforth assume expression (18) to be valid in the entire temperature interval T < T k. It follows from (18) that for T Y 0, where q n Y 0 the main contribution to the integral over energy ( in (3) comes from the first-sound pole ( = c 1k of the Green functions. However, at higher temperatures T > 1 K, where q n ¨ q s the main contribution because of the strong inequality c 1 H c 2 is given by the low-energy pole ( = c 2k, which corresponds to the second sound. Taking into account the contributions of the first and second poles of the Green functions (18) for finite temperatures (T m 0) we obtain the following approximation for the self-energy parts: Z 3Y Y 1 d q ˜ Y Y qn ðT Þ 1 R˜ ij k ; T ¼ V k q D A ij ij 2 q c1 q ð2pÞ3 c q q ðT Þ 1 c1 q 1 coth þ Bij n : coth 2T q c2 q 2T ð19Þ
1 gpp ð pÞ 4qs
ns uuu ð pÞ N 2
between the microscopic field theory of superfluidity [5,7] and the macroscopic two-fluid hydrodynamics [8,10]. It follows from (14), (15) and (17) that the pole parts of the renormalized Green’s functions G˜ij could be presented in the form Y A D q =q Bij q =q ij ij n ˜ Gij k ; e ¼ þ 2 n2 2 i; j ¼ 1; 2 ð18Þ e c2 k e2 c21 k 2
ð17Þ
where q = q n + q s is the total density of the liquid, and the coefficients a lm , d lm and b lm are independent from T at low temperatures. This result establishes a unique correspondence
It should be emphasized that the long-wavelength approximation for the Green functions (18) in this case is valid because of the divergence of the temperature factor coth(c 2q/2T) at q Y 0 and the rather rapid decay of the interaction kernel ( q Y V). Moreover, the system of equations (19) does not need to be matched with the expression for the renormalized quasi-particle spectrum E(k), as it is usually done in the microscopic field theory for T Y 0. It is true because all the necessary renormalizations would be automatically taken into account after the substitution of the empirical spectra of the first and second sound (with the experimental values of the velocities c 1 ˜ ij ( p). and c 2) into the expressions for the Green functions G Using (19), one can determine the superfluid order parameter for T m 0: q ðT Þ ; R˜ 12 ð0; T Þ ¼ W0 ðT Þ þ Ws ðT Þ s q
ð20Þ
where W0 ðT Þ¼
1 2
Z
1 W s ðT Þ ¼ 2
Z
c q B c q d3 Y q ˜ A12 D12 1 12 2 coth coth V ð q Þ þ ; c1 q c2 q 2T 2T ð2pÞ3 ð21Þ c q B c q d3 Y q ˜ D12 1 12 2 V ðq Þ coth coth : c1 q c2 q 2T 2T ð2pÞ3 ð22Þ
Since at T = 0 the density of the superfluid component q s coincides with the total mass density of the Bose liquid
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
q = mn and is proportional to R 12(0,0) (which plays a part of superfluid order parameter) we can obtain the following expression qs ¼ q0 þ q˜ s ¼ bm
R12 ð0Þ ¼ bm½n0 ð1 cÞ þ h; Kð0ÞV˜ ð0Þ
mined by (4), so we are calculating the boson polarization operator (6). The pair interaction between bosons can be chosen in the form of the Aziz regularized repulsive potential [33,34]:
ð23Þ UA ðrÞ ¼
where cu
8 h i 2 2 > 2k6 > ; < Aexpð ar br2 Þ exp ðr0 =r 1Þ ~ c2kþ6 r
rr0
k¼0
2 > > : Aexpð ar br2 Þ ~ c2kþ6 r2k6 ;
r r0
k¼0
ZV
1 ð2pÞ2 Kð0ÞV˜ ð0Þ
h¼
75
2 k 2 dk Kðk ÞV˜ ðk Þ ; E ðk Þ
ð24Þ
0
ZV
1 2
ð29Þ
˜
ð2pÞ Kð0ÞVð0Þ
k 2 dk Kðk ÞV˜ ðk ÞWðk Þ; E ðk Þ
ð25Þ
0
with b as a constant to be identified. Starting from the definition of the single-particle BEC density q 0 as q 0 = mn 0, we obtain b = (1 c) 1. Thus the ‘‘Cooper’’ PCC density takes the form: q˜ s ¼ mn1 ¼
mh ; 1c
ð26Þ
where q˜ s is the density of ‘‘Cooper’’ PCC. On the other hand, the temperature dependence of BEC density, q 0(T) = mn 0(T), due to the entire particle number conservation condition (6) can be determined by the following expression Z q0 ðT Þ 1 d3 q q n ðT Þ 1 ¼1 A11 D11 q 2 q c1 q ð2pÞ3 c q c q q ðT Þ 1 1 2 coth þ B11 n : coth 2T q c2 q 2T ð27Þ The density of the pair coherent condensate is
qs ðT Þ W 0 ðT Þ W s ðT Þ ¼ 1 q V˜ ð0Þn V˜ ð0Þn
˚ 1, b = where A = 1.8443101 105 K, a = 10.43329537 A 2 6 ˚ ˚ 2.27965105 A , c 6 = 1.36745214 K A , c 8 = 0.42123807 K ˚ 8, and c 10 = 0.17473318 K A ˚ 10. Such potential remains finite A at r = 0 due to the nonanalytic exponential dependence on r, which suppresses any power divergence at r Y 0. The Fourier component V( p) of this potential is an oscillatory and signvarying function of the momentum p which is the result of the ‘‘excluded volume’’ effect and the quantum diffraction of the particles on one another. The Asis potential (29) is convenient for calculation in the real space using Jastrov-like wave functions. However, employing its Fourier component in solving of the nonlinear integral equation is technically difficult. But in an essentially many-body problem like the one described here, the quantum effects are certain to play a very important role and therefore all subtleties of the form of the two-body potential are not very important to take into account. So to be able to go forward while retaining the crucial features of the interaction, one should employ a model potential with a simpler analytic expression. As it was shown in [23,24], such model potential can be taken in the form of a Fermi type function in the real space 2 1 r a2 UF ðrÞ ¼ V0 exp : ð30Þ þ 1 b2 The Fourier component of this potential is expressed in terms of the first order spherical Bessel function V ð pÞ ¼ V0
1 :
ð28Þ
Thus the ‘‘Cooper’’ PCC density at T = 0 in our model can be found in two independent ways. We can find it directly with help of expression (26) or we can subtract (26) from (28) and then find a result at T = 0. So we can verify the self-consistency of the proposed model via comparison of the results of this two independent calculations. 3. Results and the iterative scheme of the calculation In order to calculate the temperature dependence of the single-particle and pair coherent condensate densities within the model of a Bose liquid with the suppressed BEC we have to find the Fourier component of the bare potential of the pair interaction of bosons and after that to find the renormalized (screened) pair interaction between bosons, which is deter-
j1 ð paÞ sinx xcosx ; j ð xÞ ¼ : pa x2
ð31Þ
Then, in order to make a numerical calculation for the polarization operator, the functions U1 ð pÞuW˜ 11 ðY p ; E0 ðY p ÞÞ and ˜ Y Y W1 ð pÞuW12 ðp ; E0 ðp ÞÞ were used as the first approximation of Eqs. (7) and (8). For the zero order approximation, the ‘‘screened’’ potential (4) is taken at some constant negative value of P 0: V˜ 0 ð pÞ ¼
V0 j1 ð paÞ : pa V0 P0 j1 ð paÞ
ð32Þ
Then, using the functions u 1( p) and W 1( p), and (14)(16), (18) at C = 1, the first approximation for the polarization operator P 1( p) was calculated. The limiting value P 1(0) was compared with the exact thermodynamic value of the 4He Bose liquid polarization operator P(0,0) = n/mc 12 which determines the compressibility of the Bose system [13] at p = 0 and x = 0. The absolute value |P(0,0)| turned out to be almost 1.5 times greater than the calculated value |P 1(0)|. In the first
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A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
approximation this gives [23,24] as an estimate value C 1 K K 1 *1.5 for the vertex part C at p = 0. The second approximation u 2( p) and W 2( p) was obtained from (7) and (8) with the constant value C 1 K K1 and the first approximation for the renormalized pseudopotential (4): V˜1 ð pÞ ¼
V0 j1 ð paÞ : pa V0 P1 ð pÞj1 ð paÞ
ð33Þ
Such an iterative procedure was repeated from four to six times and used to improve precision in the calculation of the polarization operator. At each stage, Eqs. (9) and (11) were used to reproduce the quasi-particle spectrum E( p). The rate of convergence of the iterations was tracked, as well as the degree of proximity of E( p) to the empirical spectrum E exp( p). The amplitude V 0 of the potential was the fitting parameter in these ˚ , which is twice calculations. The a value was taken a = 2.44 A the quantum radius of the 4He atom. The BEC density was fixed at n 0 = 9% n = 1.95 I 1021 cm 3 in accordance with the experimental data [28 – 31]. For the numerical computation of the single-particle and pair coherent condensate density temperature dependence we have to take into account that the first (hydrodynamics) sound velocity is practically independent on T and in the given approximation can be determined as c 1 = [V˜ (0)n/m*]1/2 (here m* is the effective mass of quasi-particles). Whereas the velocity of second sound pffiffiffi c 2 is substantially T-dependent, varying from c2 ð0Þ ¼ c1 = 3 at T = 0 to the value c 2(T) * 20 m/ s in the region T > 1 K [8] and at T Y T k the velocity c 2 Y 0. Thus approaching the k point due to the strong inequality c 1 O c 2, the main contribution is given by the last terms in the integrands in Eqs. (21) and (27), which are proportional to B 12 and B 11 and contain the temperature factor 1 c2 ðT Þq 2T coth ; c2 q < T f ðq; T Þ ¼ ¼ 2 c2 ðT Þq 2kB T c2 ðT Þq2 ð34Þ which diverges according to the square-law as q Y 0. At the same time the width of the singular peak increases rapidly with increasing T and decreasing c 2. (The temperature factor momentum dependence obtained for the different temperatures is depicted in Fig. 1). As a result, with increasing T the contribution to the integral (21) from the repulsive part of the potential V˜ ( q) > 0 in the
Fig. 2. The temperature dependence of the density of BEC (lower curve) and total density of superfluid component (upper curve), obtained according (27) and (28) for next parameters A 11 = 6.14 K, D 11 = 2.03 K, B 11 = 0.00018 K, A 12 = 6.21 K, D 12 = 3.12 K, B 12 = 0.00225 K. The empirical temperature dependence of superfluid component [32] is shown in circles.
long-wavelength region q < p/a is increasing and the function W 0 (T) is decreasing. The function W 0(T) can be treated as the superfluid order parameter and it is positive at low T < c 2q because of the strong attraction V˜ ( q) < 0 in the region p/ a < q < 2p/a. At a certain critical temperature T = Tc the function P 0(T) approaches zero and then becomes negative (for T > Tc) that corresponds to the destruction of the superfluid state (q s = 0), i.e. Tc coincides with the k point. In a similar way with increasing T, there is an increase in the negative contribution to the integral (27) and decrease of q 0(T) until the density of the BEC vanishes at a certain point T = T 0 and formally becomes negative for T > T 0. The results of the numerical calculations for the single part and the paired parts of the superfluid density temperature dependence within the temperature range from zero to the ktransition point are depicted in Fig. 2. The received results agreed qualitatively with the experimental data and the regular deviation of the received curve from the experimental is caused first of all by using the model potential in our calculations and also that pulse dependence of a vertex part has not been taken into account. Nevertheless, the received results as represented are not trivial as within the framework of the suggested self-consistent model of the superfluid state of the Bose liquid; it is possible to explain the known contradiction according to which the maximal density q 0 of the single-particle Bose –Einstein condensate in the 4He Bose liquid even at very low temperatures T b T k does not exceed 10% of the total density q of liquid 4He, whereas the density of the SF component q s ¨ q at T < T k [8]. 4. Conclusions
Fig. 1. The momentum dependence of the temperature factor (34), obtained for different temperatures.
Thus, the description of the superfluid state on the basis of the microscopic superfluid state model of Bose liquid with the depleted single-particle BEC within the temperature range from T = 0 up to the k-transition point area was presented in this work. In the frame of this model the superfluid state structure was studied taking into account the emergence of the normal component and the branch of the second sound, where speed approaches zero at T Y T k. We obtained the analytic expressions (27), (28) for the computation of q 0 and q s densities and realized the numerical scheme for the single-
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
particle and pair coherent condensate density temperature dependence computation. It should be mentioned that the self-consistency of this model is corroborated by the fact that the theoretical value of total particle density calculated from (5), n th = 2.1 I1022 cm 3, is quite close to the experimental value of the particle density in the 4He liquid, n = 2.17 I 1022 cm 3 (at n 0 = 9% n). On the other hand at T = 0, the density n 1 of the supracondensate particles, calculated from (26), is about 0.93n which is also in a good agreement with experiment, taking into account that the BEC density is determined up to T0.01n. It goes without saying, that the given results, which correspond to the approximation of self-consistent field, cannot be used directly near the k point, where thermodynamic fluctuations play a very important role. But the calculations, depicted in Fig. 2, give the qualitatively correct description of the superfluid component density temperature dependence. References [1] L.D. Landau, Z. Eksp. Teor. Fiz. 11 (1941) 592; L.D. Landau, Z. Eksp. Teor. Fiz. 17 (1947) 91. [2] N.N. Bogolubov, Izv. Acad. Nauk USSR. Ser. Fiz. 11 (1947) 77; N.N. Bogolubov, Physyca 9 (1947) 23. [3] R.P. Feynmann, Phys. Rev. 94 (1954) 262. [4] J. Gavoret, P. Noziers, Ann. Phys. (N.Y.) 28 (1964) 349; P. Nozie`res, D. Pines, Theory of Quantum Liquids, Academic, New York, 1969. [5] S.T. Belyaev, JETP, 34 (1958) 417, 433. [6] N. Hugengoltz, D. Pines, Phys. Rev. 116 (1959) 489. [7] A.A. Abrikosov, L.P. Gor’kov, I.E. Dzyaloshinskij, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, 1963. [8] Khalatnikov, I.M., ‘‘Theory of Superfluidity’’, ‘‘An Introduction to the Theory of Superfluidity’’, reissue in Perseus Books (2000). [9] L. Reatto, C.V. Chester, Phys. Rev. 155 (1967) 88. [10] S.J. Pautterman, Superfluid Hydrodynamics, Mir, Moscow, 1968. [11] Yu.A. Nepomnyashchii, A.A. Nepomnyashchii, Z. Eksp. Teor. Fiz. 75 (1978) 976.
77
[12] Yu.A. Nepomnyashchii, Z. Eksp. Teor. Fiz. 85 (1983) 1244; Yu.A. Nepomnyashchii, Z. Eksp. Teor. Fiz. 89 (1985) 511. [13] V.N. Popov, Continuous Integrals in Quantum Field Theory in Statistical Physics, Nauka, Moscov, 1973. [14] V.N. Popov, A.V. Serednyakov, Z. Eksp. Teor. Fiz. 77 (1979) 377. [15] R.J. Donnelly, J.A. Donnelly, R.H. Hills, J. Low Temp. Phys. 44 (1981) 471. [16] H.R. Glyde, E.C. Swensson, in: D.L. Price, K. Skold (Eds.), Neutron Scattering, Methods of Experimental Physics, vol. 23, p. B, Academic Press, New York, 1987, p. 303. [17] H.R. Glyde, A. Griffin, Phys. Rev. Lett. 65 (1990) 1454. [18] A. Griffin, Excitations in a Bose-Condensed Liquid, Cambridge University Press, Cambridge, 1993. [19] E. Krotscheck, M.D. Miller, R. Zillich, Physica B 280 (2000) 59. [20] B. Krishnamachari, G.V. Chester, Phys. Rev., B 61 (14) (2000) 9677. [21] A.F.G. Wyatt, M.A.H. Tucker, I.N. Adamenko, K.E. Nemchenko, A.V. Zhukov, Phys. Rev., B 62 (14) (2000) 9402. [22] H.R. Glyde, R.T. Azuah, W.G. Stirling, Phys. Rev., B 62 (21) (2000) 14337. [23] E.A. Pashitskij, S.V. Mashkevich, S.I. Vilchynskyy, Phys. Rev. Lett. 89 (7) (2002) 075301. [24] E.A. Pashitskij, S.V. Mashkevich, S.I. Vilchynskyy J. Low Temp. Phys. 134 (314) (2004) 851. [25] Torsten Fliessbach, Phys. Rev., B 59 (6) (1999) 4334. [26] G.V. Chester, Phys. Rev. 100 (1955) 455. [27] Yu.A. Nepomnyashchii, E.A. Pashitskii, Z. Eksp. Teor. Fiz. 98 (1990) 178. [28] T.R. Sosnic, W.M. Snow, P.E. Sokol, Phys. Rev., B 41 (1990) 11185. [29] B. Fa˚k, J. Bossy, J. Low Temp. Phys. 113 (1998) 531. [30] R.T. Azuah, W.G. Stirling, H.R. Glyde, P.E. Sokol, S.M. Bennington, Phys. Rev., B 51 (1995) 605. [31] A.F.G. Wyatt, Nature 391 (6662) (1998) 56. [32] I.M. Khalatnikov, Theory of Superfluidity, ‘‘An Introduction to the Theory of Superfluidity’’, Perseus Books, 2000. [33] R.A. Aziz, V.P.S. Nain, J.S. Earley, W.L. Taylor, G.T. McConville, J. Chem. Phys. 70 (1979) 4330; M.H. Kalos, P.A. Whitlock, G.V. Chester, Phys. Rev., B 38 (1988) 4218; R.A. Aziz, M.J. Slaman, J. Chem. Phys. 94 (1991) 8047. [34] R.A. Aziz, F.R.W. McCourt, C.C.K. Wong, Mol. Phys. 61 (1987) 1487; A.R. Janzen, R.A. Aziz, J. Chem. Phys. 103 (1995) 9626.
A study of the temperature dependence of single-particle and pair coherent condensate densities for the Bose liquid with the ‘‘depleted’’ single-particle Bose–Einstein condensate A. Chumachenko *, S. Vilchynskyy Department of Physics, Taras Shevchenko Kiev National University, Kiev 03022, Ukraine Received 11 March 2005; accepted 20 October 2005
Abstract In this paper we present the studies of the temperature dependence of single-particle and pair coherent condensate densities for the Bose liquid with the ‘‘depleted’’ single-particle Bose – Einstein condensate (BEC) at T m 0. Our investigations are based on the field-theoretical Green functions approach. Superfluid state is described at T m 0 on the basis of the use of empirical data for the first and second sound speeds. The structure of the superfluid state is studied taking into account the appearance of the normal component within non-zero temperatures and the branch of the second sound, where speed approaches zero at T Y T k. The obtained temperature dependences of the single-particle and pair coherent condensate qualitatively agreed with the experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Condensed matter; Bose liquid; Superfluidity
1. Introduction Despite the big progress achieved in the theory of superfluidity since the pioneer works by Landau [1], Bogolyubov [2], Feynman [3] and others [4 –22], the task of constructing a microscopic theory of a superfluid (SF) state of the 4He Bose liquid cannot be considered complete. There are some unsolved contradictions between the theory and the experiment. One of them is the problem with the structure of quasi-particle spectrum in the case of superfluid hydrodynamics. These contradictions could be considered as the question of quantum-mechanical structure of superfluid 4He component below the k point. This question is crucial for the creation of the consistent microscopic superfluid theory of Bose liquid for the cases of both zero and non-zero temperatures. In the works [23,24] was introduced the microscopic model of the SF state of a Bose liquid at T = 0 with a single-particle Bose –Einstein condensate (BEC) suppressed by interaction, based upon a renormalized field perturbation theory with * Corresponding author. E-mail addresses: [email protected] (A. Chumachenko), [email protected] (S. Vilchynskyy). 0167-7322/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2005.10.003
combined variables [11 –14]. That allows one to obtain a selfconsistent ‘‘short’’ system of nonlinear integral equations for self-energy parts R˜ij ( p,() by means of truncating the infinite series at the small density of the BEC (n 0/n b 1). By the same token, one can work out a self-consistent microscopic theory of the superfluid Bose liquid and perform an ab initio calculations of the spectrum of elementary excitations E( p), starting from the realistic models of the pair interaction potential U(r). In this work we study a superfluid state structure of the 4He Bose liquid under a non-zero temperature (0 < T < T k ). The similar problem was investigated in work [25] on the basis of Chester [26] IBG wave function W IBG by Jastrov factors F = Pf ij . We use the microscopic model proposed in [27] for the superfluid Bose liquid with a ‘‘depleted’’ BEC. The small parameter in this model is the ratio of the BEC density to the total density of the Bose liquid q 0/q s b 1 in contrast to the Bogolyubov theory [2] for the almost ideal Bose gas, where the small parameter is the ratio of the number of the overcondensate excitations to the number of particles in the intense BEC (n n 0) / n 0 b 1. Such an approach is based on the analysis of the numerous precise experimental data on the neutron inelastic scattering [22,28 – 30] and on the results of the quantum evaporation of
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77 4
He atoms [31]. According to this experiments the maximal density q 0 of the single-particle BEC in the 4He Bose liquid does not exceed 10% of the total density q of 4He liquid even at very low temperatures T b T k, whereas the density of the SF component q s Y q at T Y 0 [10]. Such a low density of the BEC is implied by the strong interaction between the 4He atoms and is an indication of the fact that the quantum structure of the part of the SF condensate in He II carrying the ‘‘excess’’ density (q s q 0) b q 0 calls for a more thorough investigation. So, in this case the density of the superfluid component q s is determined by the quantity of the renormalized anomalous selfenergy parts R 12(0,0), which are a superposition of the ‘‘depleted’’ single-particle BEC and the intense ‘‘Cooper’’ pair coherent condensate (PCC), with the coincident phases (sings) of the corresponding order parameters. Such a PCC emerges due to the effective attraction between bosons in some regions of the momentum space, which results from the oscillating sign-changing momentum dependence of the Fourier component V( p) of the interaction potentials U(r) with the inflection points in the radial dependence. The system of Dyson-Belyaev equations [5], that describes the superfluid (SF) state of the Bose liquid with strong interaction between bosons and a weak single-particle Bose–Einstein ˜ 11 and condensate (BEC), allows one to express the normal G ˜ anomalous G12 renormalized single-particle boson Green functions in terms of the respective self-energy parts R˜11 and R˜12. These equations can be presented in the next analytic form of [27]: R˜ 11 ðpY ; eÞ ¼ n0 KðpY ; eÞV˜ ðY p ; eÞ þ n1 V ð0Þ þ W˜ ðY p ; eÞ; ð1Þ 11
R˜ 12 ðpY ; eÞ ¼ n0 KðpY ; eÞV˜ ðY p ; eÞ þ where
Z
Y
Y W12 ðY p ; eÞ;
Y
d3 k
Y where C Y p ; e; k ; xÞ is the vertex part (three-pole) describing many-particle correlations of the local field’s type; KðpY ; eÞ ¼ C p ; e; 0; 0Þ ¼ Cð0; 0; Y ðY p ; eÞ is the vertex part with zero values of the input momentum and energy. Taking into account the residues at the poles of single-particle Green functions G˜ij ðpY ; eÞ and neglecting the contributions of eventual poles of the functions Y C Y p ; e; k ; xÞ and V˜ ðY p ; eÞ, that do not coincide with the poles of G˜ ij ðY p ; eÞ, the functions W˜ ij ðY p ; eÞ on a ‘‘mass shell’’ (=E( p) can be presented in the following form (at T =0): 1 p ; E ð pÞ Þ ¼ W˜ 11 ðY 2
Z
Y
d3 k
ð2pÞ
3
Y C pY ; E ð pÞ; k ; E ðk Þ
3 2 Y A k ; E ðk Þ Y 15; V˜ pY k ; E ð pÞE ðk Þ 4 E ðk Þ ð7Þ Z 3Y 1 d k Y ˜ W12 ðp ; Eð pÞÞ ¼ 2 ð2pÞ3 Y Y V˜ Y p k ; E ð pÞE ðk Þ C pY ; Eð pÞ; k ; E ðk Þ Y Y Y n0 K k ; Eðk Þ V˜ k ; E ðk Þ þW˜ 12 k ; Eðk Þ ; E ðk Þ ð8Þ where AðY p Þ ¼ n0 KðY p ; E ðY p ÞÞV˜ðpY ; EðY p ÞÞ þ n1 V ð0Þ þ W˜ 11 ðY pÞ Y
Z
dx Y ˜ Y Y Gij k V p k ; e x W ij ðp ; eÞ ¼ i 3 2p ð2pÞ Y C Y p ; e; k ; x ˜
ð2Þ
73
þ
ð3Þ
V˜ ðY p ; eÞ ¼ V ðpY Þ½1 V ðpY ÞPðY p ; eÞ1 ; ð4Þ where V( p) is the Fourier component of the bare potential of the pair interaction of the bosons; V˜ ðY p ; eÞ is the renormalized (screened) pair interaction between bosons due to the manyparticles collective effects; n0 is the number of particles in BEC; n1 =n n0 is the number of overcondensate particles (n1 H n0), which is determined by the condition of the total particle number conservation: Z 3Y Z Y d k dx n ¼ n0 þ n1 ¼ n0 þ i G ð5Þ 11 k ; x : 2p ð2pÞ3
p2 l; 2m
ð9Þ
where l represents the chemical potential of the quasi-particles, and satisfies the Hugengoltz–Pines relation [6] l ¼ R˜ 11 ð0; 0Þ R˜ 12 ð0; 0Þ:
ð10Þ
The quasi-particle spectrum corresponding to the poles of Green’s function is determined in general, with allowance for relations (1) an (2), by the following expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E ðY p Þ ¼ A2 ðY p Þ n0 K ð Y p ; E ðY p ÞÞV˜ ðY p ; EðpY ÞÞ þ W˜ 12 ðY pÞ : ð11Þ 2. The structure of the superfluid Bose liquid state at T =/ 0
Y
The boson’s polarization operator Pðp ; eÞ describes manyparticles correlation effects and can be presented as Z 3Y Z dx Y Y n Y dk Y C p ; e; ; k ; x G11 k ; x P ðp ; eÞ ¼ i 3 2p ð2p Þ Y Y G11 k þ pY ; e þ x þ G12 k ; x Y o G12 k þ pY ; e þ x ð6Þ
=The main purpose of our research is to investigate the superfluid helium condensate structure by means of finding temperature dependence of the single-particle and pair coherent condensate densities under the assumption that the superfluid component is a superposition of a weak single-particle BEC and an intensive PCC. Let us consider the superfluid Bose liquid state at T m 0, when both superfluid component density q s(T) and normal
74
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
component density q n(T) exist simultaneously. As it is shown in [13,14], the expression for the renormalized Green function G˜ij ð pÞ is constructed with the help of the combined variables: W˜ ð xÞ ¼ W˜ L ð xÞ þ W˜ sh ð xÞ:
ð12Þ Y
In the long-wavelength region jk j < k0 (where k 0 is some characteristic momentum) this variables are just the hydrody˜ L(x) close to those described in the Landau namics variables W quantumY hydrodynamics [1], and in the short-wavelength region jk j > k0 they coincide with the usual field operators ˜ sh(x): W qffiffiffiffiffiffiffiffiffi n˜ L b˜nL ˜ ˜ ˜ þ i/L ; W˜ sh ¼ wsh ei/L ; WL ð xÞ ¼ b˜nL 1 þ 2b˜nL 1 wsh ¼ w wL ; wL ðrÞ ¼ pffiffiffiffi ~ ak eikr ¼ V jkjk0
qffiffiffiffiffiffiffiffiffi ˜ b˜nL ei/L :
ð13Þ
Such an approach means that the separation of the Bose system into a macroscopic coherent condensate and a gas of supracondensate excitations is made not on the statistical level, like in the case of weakly non-ideal Bose gas [2], but on the level of ab initio field operators, which are used to construct a microscopic theory of the Bose liquid. In the region of small p m 0 at T Y 0 Green’s functions have the form G˜ 11 ð pÞ ¼ ns guu ð pÞ igup ð pÞ
G˜ 12 ð pÞ ¼ ns guu ð pÞ where Uuu ð pÞ ¼
Z jqjq0
1 ns gpp ð pÞ Uuu ð pÞ N 4qs 2
ð14Þ ð15Þ
d4 q guu ðqÞguu ð p qÞ; p 2p4
Y ¼ k ; e ; q ¼ ðqY ; xÞ
ð16Þ
with g lm ( p) being the ‘‘hydrodynamics’’ Green functions, which are associated with the long-wavelength fluctuations of the phase and density of the condensate (lm = u p ). The expressions for g uu ( p), g up ( p), g pp ( p) are calculated in [12] for T > 0. They contain sums of two pole terms, corresponding to the first and second sounds with the velocities c 1 and c 2 in the Bose liquid with the normal and superfluid components: alm dlm qn =q blm q =q glm ðk; eÞ þ 2 n2 2 e2 c21 k 2 e c2 k l; m u; p
We henceforth assume expression (18) to be valid in the entire temperature interval T < T k. It follows from (18) that for T Y 0, where q n Y 0 the main contribution to the integral over energy ( in (3) comes from the first-sound pole ( = c 1k of the Green functions. However, at higher temperatures T > 1 K, where q n ¨ q s the main contribution because of the strong inequality c 1 H c 2 is given by the low-energy pole ( = c 2k, which corresponds to the second sound. Taking into account the contributions of the first and second poles of the Green functions (18) for finite temperatures (T m 0) we obtain the following approximation for the self-energy parts: Z 3Y Y 1 d q ˜ Y Y qn ðT Þ 1 R˜ ij k ; T ¼ V k q D A ij ij 2 q c1 q ð2pÞ3 c q q ðT Þ 1 c1 q 1 coth þ Bij n : coth 2T q c2 q 2T ð19Þ
1 gpp ð pÞ 4qs
ns uuu ð pÞ N 2
between the microscopic field theory of superfluidity [5,7] and the macroscopic two-fluid hydrodynamics [8,10]. It follows from (14), (15) and (17) that the pole parts of the renormalized Green’s functions G˜ij could be presented in the form Y A D q =q Bij q =q ij ij n ˜ Gij k ; e ¼ þ 2 n2 2 i; j ¼ 1; 2 ð18Þ e c2 k e2 c21 k 2
ð17Þ
where q = q n + q s is the total density of the liquid, and the coefficients a lm , d lm and b lm are independent from T at low temperatures. This result establishes a unique correspondence
It should be emphasized that the long-wavelength approximation for the Green functions (18) in this case is valid because of the divergence of the temperature factor coth(c 2q/2T) at q Y 0 and the rather rapid decay of the interaction kernel ( q Y V). Moreover, the system of equations (19) does not need to be matched with the expression for the renormalized quasi-particle spectrum E(k), as it is usually done in the microscopic field theory for T Y 0. It is true because all the necessary renormalizations would be automatically taken into account after the substitution of the empirical spectra of the first and second sound (with the experimental values of the velocities c 1 ˜ ij ( p). and c 2) into the expressions for the Green functions G Using (19), one can determine the superfluid order parameter for T m 0: q ðT Þ ; R˜ 12 ð0; T Þ ¼ W0 ðT Þ þ Ws ðT Þ s q
ð20Þ
where W0 ðT Þ¼
1 2
Z
1 W s ðT Þ ¼ 2
Z
c q B c q d3 Y q ˜ A12 D12 1 12 2 coth coth V ð q Þ þ ; c1 q c2 q 2T 2T ð2pÞ3 ð21Þ c q B c q d3 Y q ˜ D12 1 12 2 V ðq Þ coth coth : c1 q c2 q 2T 2T ð2pÞ3 ð22Þ
Since at T = 0 the density of the superfluid component q s coincides with the total mass density of the Bose liquid
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
q = mn and is proportional to R 12(0,0) (which plays a part of superfluid order parameter) we can obtain the following expression qs ¼ q0 þ q˜ s ¼ bm
R12 ð0Þ ¼ bm½n0 ð1 cÞ þ h; Kð0ÞV˜ ð0Þ
mined by (4), so we are calculating the boson polarization operator (6). The pair interaction between bosons can be chosen in the form of the Aziz regularized repulsive potential [33,34]:
ð23Þ UA ðrÞ ¼
where cu
8 h i 2 2 > 2k6 > ; < Aexpð ar br2 Þ exp ðr0 =r 1Þ ~ c2kþ6 r
rr0
k¼0
2 > > : Aexpð ar br2 Þ ~ c2kþ6 r2k6 ;
r r0
k¼0
ZV
1 ð2pÞ2 Kð0ÞV˜ ð0Þ
h¼
75
2 k 2 dk Kðk ÞV˜ ðk Þ ; E ðk Þ
ð24Þ
0
ZV
1 2
ð29Þ
˜
ð2pÞ Kð0ÞVð0Þ
k 2 dk Kðk ÞV˜ ðk ÞWðk Þ; E ðk Þ
ð25Þ
0
with b as a constant to be identified. Starting from the definition of the single-particle BEC density q 0 as q 0 = mn 0, we obtain b = (1 c) 1. Thus the ‘‘Cooper’’ PCC density takes the form: q˜ s ¼ mn1 ¼
mh ; 1c
ð26Þ
where q˜ s is the density of ‘‘Cooper’’ PCC. On the other hand, the temperature dependence of BEC density, q 0(T) = mn 0(T), due to the entire particle number conservation condition (6) can be determined by the following expression Z q0 ðT Þ 1 d3 q q n ðT Þ 1 ¼1 A11 D11 q 2 q c1 q ð2pÞ3 c q c q q ðT Þ 1 1 2 coth þ B11 n : coth 2T q c2 q 2T ð27Þ The density of the pair coherent condensate is
qs ðT Þ W 0 ðT Þ W s ðT Þ ¼ 1 q V˜ ð0Þn V˜ ð0Þn
˚ 1, b = where A = 1.8443101 105 K, a = 10.43329537 A 2 6 ˚ ˚ 2.27965105 A , c 6 = 1.36745214 K A , c 8 = 0.42123807 K ˚ 8, and c 10 = 0.17473318 K A ˚ 10. Such potential remains finite A at r = 0 due to the nonanalytic exponential dependence on r, which suppresses any power divergence at r Y 0. The Fourier component V( p) of this potential is an oscillatory and signvarying function of the momentum p which is the result of the ‘‘excluded volume’’ effect and the quantum diffraction of the particles on one another. The Asis potential (29) is convenient for calculation in the real space using Jastrov-like wave functions. However, employing its Fourier component in solving of the nonlinear integral equation is technically difficult. But in an essentially many-body problem like the one described here, the quantum effects are certain to play a very important role and therefore all subtleties of the form of the two-body potential are not very important to take into account. So to be able to go forward while retaining the crucial features of the interaction, one should employ a model potential with a simpler analytic expression. As it was shown in [23,24], such model potential can be taken in the form of a Fermi type function in the real space 2 1 r a2 UF ðrÞ ¼ V0 exp : ð30Þ þ 1 b2 The Fourier component of this potential is expressed in terms of the first order spherical Bessel function V ð pÞ ¼ V0
1 :
ð28Þ
Thus the ‘‘Cooper’’ PCC density at T = 0 in our model can be found in two independent ways. We can find it directly with help of expression (26) or we can subtract (26) from (28) and then find a result at T = 0. So we can verify the self-consistency of the proposed model via comparison of the results of this two independent calculations. 3. Results and the iterative scheme of the calculation In order to calculate the temperature dependence of the single-particle and pair coherent condensate densities within the model of a Bose liquid with the suppressed BEC we have to find the Fourier component of the bare potential of the pair interaction of bosons and after that to find the renormalized (screened) pair interaction between bosons, which is deter-
j1 ð paÞ sinx xcosx ; j ð xÞ ¼ : pa x2
ð31Þ
Then, in order to make a numerical calculation for the polarization operator, the functions U1 ð pÞuW˜ 11 ðY p ; E0 ðY p ÞÞ and ˜ Y Y W1 ð pÞuW12 ðp ; E0 ðp ÞÞ were used as the first approximation of Eqs. (7) and (8). For the zero order approximation, the ‘‘screened’’ potential (4) is taken at some constant negative value of P 0: V˜ 0 ð pÞ ¼
V0 j1 ð paÞ : pa V0 P0 j1 ð paÞ
ð32Þ
Then, using the functions u 1( p) and W 1( p), and (14)(16), (18) at C = 1, the first approximation for the polarization operator P 1( p) was calculated. The limiting value P 1(0) was compared with the exact thermodynamic value of the 4He Bose liquid polarization operator P(0,0) = n/mc 12 which determines the compressibility of the Bose system [13] at p = 0 and x = 0. The absolute value |P(0,0)| turned out to be almost 1.5 times greater than the calculated value |P 1(0)|. In the first
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A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
approximation this gives [23,24] as an estimate value C 1 K K 1 *1.5 for the vertex part C at p = 0. The second approximation u 2( p) and W 2( p) was obtained from (7) and (8) with the constant value C 1 K K1 and the first approximation for the renormalized pseudopotential (4): V˜1 ð pÞ ¼
V0 j1 ð paÞ : pa V0 P1 ð pÞj1 ð paÞ
ð33Þ
Such an iterative procedure was repeated from four to six times and used to improve precision in the calculation of the polarization operator. At each stage, Eqs. (9) and (11) were used to reproduce the quasi-particle spectrum E( p). The rate of convergence of the iterations was tracked, as well as the degree of proximity of E( p) to the empirical spectrum E exp( p). The amplitude V 0 of the potential was the fitting parameter in these ˚ , which is twice calculations. The a value was taken a = 2.44 A the quantum radius of the 4He atom. The BEC density was fixed at n 0 = 9% n = 1.95 I 1021 cm 3 in accordance with the experimental data [28 – 31]. For the numerical computation of the single-particle and pair coherent condensate density temperature dependence we have to take into account that the first (hydrodynamics) sound velocity is practically independent on T and in the given approximation can be determined as c 1 = [V˜ (0)n/m*]1/2 (here m* is the effective mass of quasi-particles). Whereas the velocity of second sound pffiffiffi c 2 is substantially T-dependent, varying from c2 ð0Þ ¼ c1 = 3 at T = 0 to the value c 2(T) * 20 m/ s in the region T > 1 K [8] and at T Y T k the velocity c 2 Y 0. Thus approaching the k point due to the strong inequality c 1 O c 2, the main contribution is given by the last terms in the integrands in Eqs. (21) and (27), which are proportional to B 12 and B 11 and contain the temperature factor 1 c2 ðT Þq 2T coth ; c2 q < T f ðq; T Þ ¼ ¼ 2 c2 ðT Þq 2kB T c2 ðT Þq2 ð34Þ which diverges according to the square-law as q Y 0. At the same time the width of the singular peak increases rapidly with increasing T and decreasing c 2. (The temperature factor momentum dependence obtained for the different temperatures is depicted in Fig. 1). As a result, with increasing T the contribution to the integral (21) from the repulsive part of the potential V˜ ( q) > 0 in the
Fig. 2. The temperature dependence of the density of BEC (lower curve) and total density of superfluid component (upper curve), obtained according (27) and (28) for next parameters A 11 = 6.14 K, D 11 = 2.03 K, B 11 = 0.00018 K, A 12 = 6.21 K, D 12 = 3.12 K, B 12 = 0.00225 K. The empirical temperature dependence of superfluid component [32] is shown in circles.
long-wavelength region q < p/a is increasing and the function W 0 (T) is decreasing. The function W 0(T) can be treated as the superfluid order parameter and it is positive at low T < c 2q because of the strong attraction V˜ ( q) < 0 in the region p/ a < q < 2p/a. At a certain critical temperature T = Tc the function P 0(T) approaches zero and then becomes negative (for T > Tc) that corresponds to the destruction of the superfluid state (q s = 0), i.e. Tc coincides with the k point. In a similar way with increasing T, there is an increase in the negative contribution to the integral (27) and decrease of q 0(T) until the density of the BEC vanishes at a certain point T = T 0 and formally becomes negative for T > T 0. The results of the numerical calculations for the single part and the paired parts of the superfluid density temperature dependence within the temperature range from zero to the ktransition point are depicted in Fig. 2. The received results agreed qualitatively with the experimental data and the regular deviation of the received curve from the experimental is caused first of all by using the model potential in our calculations and also that pulse dependence of a vertex part has not been taken into account. Nevertheless, the received results as represented are not trivial as within the framework of the suggested self-consistent model of the superfluid state of the Bose liquid; it is possible to explain the known contradiction according to which the maximal density q 0 of the single-particle Bose –Einstein condensate in the 4He Bose liquid even at very low temperatures T b T k does not exceed 10% of the total density q of liquid 4He, whereas the density of the SF component q s ¨ q at T < T k [8]. 4. Conclusions
Fig. 1. The momentum dependence of the temperature factor (34), obtained for different temperatures.
Thus, the description of the superfluid state on the basis of the microscopic superfluid state model of Bose liquid with the depleted single-particle BEC within the temperature range from T = 0 up to the k-transition point area was presented in this work. In the frame of this model the superfluid state structure was studied taking into account the emergence of the normal component and the branch of the second sound, where speed approaches zero at T Y T k. We obtained the analytic expressions (27), (28) for the computation of q 0 and q s densities and realized the numerical scheme for the single-
A. Chumachenko, S. Vilchynskyy / Journal of Molecular Liquids 124 (2006) 72 – 77
particle and pair coherent condensate density temperature dependence computation. It should be mentioned that the self-consistency of this model is corroborated by the fact that the theoretical value of total particle density calculated from (5), n th = 2.1 I1022 cm 3, is quite close to the experimental value of the particle density in the 4He liquid, n = 2.17 I 1022 cm 3 (at n 0 = 9% n). On the other hand at T = 0, the density n 1 of the supracondensate particles, calculated from (26), is about 0.93n which is also in a good agreement with experiment, taking into account that the BEC density is determined up to T0.01n. It goes without saying, that the given results, which correspond to the approximation of self-consistent field, cannot be used directly near the k point, where thermodynamic fluctuations play a very important role. But the calculations, depicted in Fig. 2, give the qualitatively correct description of the superfluid component density temperature dependence. References [1] L.D. Landau, Z. Eksp. Teor. Fiz. 11 (1941) 592; L.D. Landau, Z. Eksp. Teor. Fiz. 17 (1947) 91. [2] N.N. Bogolubov, Izv. Acad. Nauk USSR. Ser. Fiz. 11 (1947) 77; N.N. Bogolubov, Physyca 9 (1947) 23. [3] R.P. Feynmann, Phys. Rev. 94 (1954) 262. [4] J. Gavoret, P. Noziers, Ann. Phys. (N.Y.) 28 (1964) 349; P. Nozie`res, D. Pines, Theory of Quantum Liquids, Academic, New York, 1969. [5] S.T. Belyaev, JETP, 34 (1958) 417, 433. [6] N. Hugengoltz, D. Pines, Phys. Rev. 116 (1959) 489. [7] A.A. Abrikosov, L.P. Gor’kov, I.E. Dzyaloshinskij, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, 1963. [8] Khalatnikov, I.M., ‘‘Theory of Superfluidity’’, ‘‘An Introduction to the Theory of Superfluidity’’, reissue in Perseus Books (2000). [9] L. Reatto, C.V. Chester, Phys. Rev. 155 (1967) 88. [10] S.J. Pautterman, Superfluid Hydrodynamics, Mir, Moscow, 1968. [11] Yu.A. Nepomnyashchii, A.A. Nepomnyashchii, Z. Eksp. Teor. Fiz. 75 (1978) 976.
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