Microscopic theory of heat capacity of liquid helium-4 for temperatures above the critical point
In this paper, with the corresponding formula for internal energy obtained in Ref. , combined with a simple calculation of the effective mass of interacting Bose particles, the behavior of the heat capacity of liquid He is analyzed numerically for the entire temperature range. The results agree quite well with experimental data.
Key words: liquid He, heat capacity, effective mass
Notwithstanding a great number of papers (starting from Refs. [1, 2]) concerned with the microscopic study of Bose system’s properties a good description of the heat capacity of liquid helium-4 in the whole temperature range has not yet been created. The first attempts were made by Brout  where it was shown in the first order of perturbation theory for the free energy of non-ideal Bose system that the presence of interaction does change the order of the phase transition. In [4, 5] the thermodynamic functions of liquid helium-4 at all temperatures were obtained using the two-time temperature Green’s function formalism. A good agreement of the specific heat at low temperatures with experimental data was obtained and the temperature of phase transition was calculated as . The success of such an approach lies in the application of the experimentally measured structure factor of liquid helium-4 extrapolated to zero temperature  instead of the interparticle interaction potential. A good agreement of the heat capacity curve with relevant experimental data for the temperatures below the temperature of the -transition was obtained in Ref.  where the calculations were made using the quantum-statistical approach based on the density matrix of Bose liquid. At a higher temperatures the specific heat curve was shifted upward almost in a parallel way.
In Ref.  the density matrix formalism with the functional optimization of the Jastrow wave-function parameters was used to describe the properties of liquid helium. The results for internal energy agree well with experimental data for the temperatures below the critical one. It was shown that by taking into account the dynamic two-particle correlations only one can obtain the value of the critical temperature 3.4 K. Thus for good agreement with experiments one needs to take into account higher-order approximations, which are specifically related to the concept of effective mass of the helium atom in a liquid.
In recent years much attention has been paid to the study of the atom’s effective mass in liquid helium because in this way part of the interaction could be taken into account accordingly to Feynman’s idea . However, there is no satisfactory formula for the effective mass of the helium atom in the liquid at arbitrary temperatures. Various scholars were mostly concerned with the value of the effective mass at . Isihara and Samulski  have used the value of to agree the theoretically calculated sound branch of the excitation spectrum with the corresponding experimental data. In Ref.  the effective mass was obtained on the basis of the liquid helium-4 structure factor measurements. In Ref.  the interatomic potential was preserved as the input information, but in part the contribution of higher correlations was “transferred” to the kinetic energy term. In this way the mass of particles was renormalized that is somehow in correlation with the approach of Ref. . As a result of such a renormalization the value of was obtained using Green’s function method. It was shown in Refs.  that the above-mentioned mass renormalization leads to the expressions obtained for the effective mass of the He impurity atom in liquid He but with the replacement of the “pure” He atom mass by the He atom mass.
The aim of this paper is to calculate the heat capacity of liquid helium above the temperature of phase transition. The formula for the internal energy of Bose liquid, obtained in Ref.  with the help of the method proposed in Ref.  (where the effective mass is a free parameter of the theory), forms the basis of these calculations. Thus, further we calculate step by step with the help of thermodynamic perturbation theory the quasi-particle spectrum of the Bose system at the temperatures higher than the critical one, then we obtain the effective mass and numerically analyze the behavior of the heat capacity.
2 Perturbation theory for the grand canonical potential at
Consider a collection of spinless particles embedded into volume . The Hamiltonian of the system which takes into account only pair interaction between particles may be written using the secondary quantization language
The creation and destruction operators of the particle with the momentum satisfy the usual bosonic commutation relations. The notations stands for the Fourier transform of the potential and for the free-particle spectrum are introduced. It is more convenient to work in the grand canonical ensemble. That is why we introduced the chemical potential and fugacity (, where is the temperature) of the system that can be found with the help of the following equation
Using the secondary quantization formalism it is easy to write down the Fourier transform of the particle density fluctuation operator
Further, our task is to calculate the partition function of a many-boson system above the temperature of phase transition. Of cause, the most interesting features of these calculations occur at the region in a close vicinity of the temperature of Bose condensation. In the statistical operator, let us pass to the interaction representation and write down the partition function in the following way:
where the quantity
and the braces stand for statistical averaging with the Hamiltonian . The first multiplier is the partition function of the ideal Bose gas. The second one and the third one take into account the inter-particle interaction completely.
Next, we rewrite, with the help of the Hubbard-Stratonovich transformation, the -exponent in terms of the functional integral and we also average it by the states of the ideal Bose gas
where we use the notations and is the Matsubara frequency and
Here the symbol denotes the integration over real and imaginary parts of the variables from the half space of all possible values of due to the symmetry . The polarization operator
Here and thereafter is the filling factor of the ideal Bose gas. We also introduced notations for the symmetrical functions
The one-particle Green’s function of noninteracting bosons is
The representation Eq. (5) when the partition function is written in terms of functional integrals was used successfully in the theory of Fermi systems with Coulombic interaction in Ref. . The fact that in the Gaussian approximation of the calculation of the functional integral we recover Random Phase Approximation (RPA) correctly is a great advantage of our method. This approximation in the case of interacting fermions generalizes at finite temperatures the well-known result of Gell-Mann–Brueckner for the high-density electron gas. The non-Gaussian part can be taken into account approximately by means of perturbation theory.
The thermodynamic potential up to the first order of the perturbation theory (in two-sum approximation over the “4-vector”) is
where the ideal gas contribution
and the RPA-part
As we work in the grand canonical ensemble we take the average number of particles of the function of the chemical potential .
The one-loop contribution to the thermodynamic potential is
The function , and the correlator
were obtained in the Random Phase Approximation. The structure of expressions (10), (11) clearly shows that they are “in correlation” with the formulae obtained in Ref.  where an entirely different method of calculations was used.
3 Renormalization of the one-particle spectrum
It is clear that the basis of further calculations is fully determined by the renormalization of the quasi-particle spectrum. For our analysis we use RPA. Notwithstanding the simplicity of this approximation it “catches” certain important features of the behavior of the system. It is not surprising because RPA effectively sums up an infinite set of terms of the perturbation theory divergent near phase transition point.
At first let us use the thermodynamic equality to find the average number of particles in the system. The explicit calculation of the corresponding derivative with the first two terms of Eq. (8) gives
Let us construct the Bose filling factor with a new spectrum using the expression in braces. Making use of equality we finally obtained the formula for the renormalized one-particle spectrum
where the correction to the quasi-particle spectrum is
and the correction to the chemical potential is
It is easy to obtain the above-mentioned expression for the spectrum in a different way. To do this one has to recall that variational derivative of the -potential with respect to equals the renormalized one-particle filling factor . After simple calculations we obtained the following formula:
It is easy to argue by making summation over the wave-vector of the left-hand and right-hand sides of the previous equality that the second term in braces vanishes. So, after the summation of expression (16) we arrive at equality (12) and thus we get the same expression (14) for the correction to a one-particle spectrum again. It is interesting to note that the calculation of the variation derivative in RPA gives the same result. Finally, Eq. (14) coincides with the result derived in Ref.  where calculations were made in terms of temperature Green’s function technique.
Let us analyze expression (14). First, the potential problems with the integration over the wave-vector may occur only in the critical region and at the zero frequency . That is why we write down this term apart and immediately set apart the Hartree-Fock-like term
where , and is equilibrium density of the system. Secondly, to go further we have to investigate the properties of polarization operator (6)
here for convenience the following notations are used: , . We are interested in a long wave-length behavior of the polarization operator. To find the leading order asymptote of at zero frequency in the critical point it is sufficient to replace the Bose filing factor in integral (18) by . Then after a simple integral calculation we get . For higher temperatures ()
The dots stand for higher than quadratic terms in the expansion over the wave-vector. It is easy to see from definition that for non-zero frequencies .
Now, let us consider the contribution to the one-particle spectrum (14) from zero frequency. Using designation for this term of spectrum it is easy to write down
For self-consistency of our calculations, especially near the critical point, the chemical potential should be changed by in the right-hand side of Eq. (20) (the critical point is determined by the equation , respectively). Admittedly, the ideal gas dispersion relation should be replaced by the exact one-particle spectrum, but further analysis will not be influenced by this replacement qualitatively.
Let us consider the value of the integral in Eq. (20) at a small and assume for definiteness that the temperature is higher than the critical one. Then substituting , and we obtain
It turns out that this integral equals zero identically. Moreover even after the substitution of (3), the integral in Eq. (20) equals zero too. Thus it is shown, with a realistic restriction on the Fourier transform of the potential energy, i.e. the absence of linear and quadratic terms in the expansion of at a small , that .
The situation is quite different in the critical region. Here the leading order asymptote is (it is not hard to ascertain taking into account the properties of in this region), which obviously is a hint at the following behavior of the one-particle spectrum () at the critical temperature. Clearly one cannot obtain this result using a simple perturbative approach.
Hence, having separated the non-analytic problematic part of the spectrum (the second term in Eq. (17)) we can consider the “non-universal” one, i.e. the remainder of Eq. (17). Precisely this expression will determine the observed non-universal properties of the Bose liquid. The latter calculations are linked to the summation over the Matsubara frequency in the last term of Eq. (17) and coincide with those in Ref. . That is why we do not dwell on the details of these calculations. Now the leading-order non-vanishing term of the quasi-particle dispersion relation is quadratic over the wave-vector. We recall that and hence its contribution is not significant. So, for reasons of simplification we assume the spectrum to be a quadratic free-particle one, but with the normalized mass.
As we single out the “problematic” contribution to the quasi-particle spectrum expanding second and third sums of Eq. (17) into a series in we obtain for the one-particle spectrum
The effective mass is
where the quantity
Here we use the following notations: is Bogoliubov spectrum, and . The effective mass in the low-temperature region is always larger than its “bare” one which means that the renormalized temperature of the Bose condensation of interacting particles is always lower than the critical temperature of the ideal gas. This is the most important result of Eqs. (22) and (23). At low temperatures carefully calculating the limit of it is easy to ascertain that the effective mass tends to the mass of particles. It is important that the temperature-independent part of formula (23) coincides with the effective mass derived in Ref.  where a different method was used for the calculations.
At the end of this section one more remark on the applicability of the formula for has to be made. The approximation of the exact spectrum of collective modes by the Bogoliubov spectrum on the one hand made it possible to obtain the analytical expression (23), on the other hand it brought us beyond the limits of RPA. It is hard to assess the accuracy of such a trick, but it can be justified only considering that the Fourier transform of a two-particle potential is a rapidly decreasing function. Then the main contribution to the integral over the wave-vector comes from a lower limit of integration where the Bogoliubov spectrum completely coincides with the exact one.
4 Internal energy and heat capacity
The expression for the grand canonical potential derived in the second section of this paper is applicable only for the temperatures higher than the critical one. To describe the -transition phenomenon, in particular thermodynamic functions, let us use an approach based on the density matrix of the Bose liquid . Using this approach the dependence of an internal energy of the Bose liquid on the effective mass of the helium-4 atom was found in the approximation of pair-particle correlations in Ref. . In the case of in Ref.  a good agreement of the heat capacity with experimental data in the region below the critical point was obtained, but above the temperature of phase transition the heat capacity curve was shifted upward. Hence, the calculation of heat capacity for the case of is an interesting problem.
We take the expression for the internal energy in the approximation of pair-particle correlations from Ref. :
Here the following notations are introduced:
is the structure factor of the ideal Bose gas with a renormalized mass.
The quantity is an average number of particles of the ideal Bose gas with zero momentum and is its fugacity.
The next quantity
is the pair structure factor of the Bose liquid, and
Obviously, if we turn off interparticle interaction , and take into account that sound velocity in the ideal Bose gas at equals zero (there are no zero density fluctuations in the Bose system) and (27) we obtain a well-known formula for the energy of the ideal Bose gas
Let us analyze the total energy Eq.(24) in the low temperature region where it coincides with the formulas obtained in Ref. [13, 16]. At low temperatures , when only small values of the wave-vector are important in the expression for the spectrum we obtain:
where the ground state energy is
and the heat capacity, respectively,
Exactly the same temperature dependence of the heat capacity of liquid He at is observed. Since we obtained the correct behavior of the heat capacity at low temperatures using the energy of the Bose liquid Eq. (24) we expect to derive the correct behavior of the heat capacity for the entire temperature range.
5 Numerical results
Our numerical calculations are carried out at the equilibrium density of liquid helium Å, mass of particles u, sound velocity m/s in the limit of , and at the critical temperature of the ideal gas K. We use the liquid structure factor extrapolated to  as the output information, instead of the interparticle potential, i.e.
where is the experimentally measured structure factor at .
It is logical to start calculations with the formula for the renormalized one-particle spectrum and thus with the formula for the effective mass of particles. Despite the complexity of the last sum over the wave-vector in Eq. (23) the main contribution to the effective mass arises from the temperature-independent part. In Fig. 1 the dependence of a dimensionless value of the effective mass as a function of temperature is presented. Formally we extrapolated a curve of the effective mass in the condensate region where obviously it becomes a parameter of the theory. It is important that at zero temperature coincides with the effective mass of the impurity atom in the Bose liquid .
The first three terms of a low-temperature expansion are (, )
where -coefficients are
Then the effective mass equals approximately
This formula reproduces the curve in Fig. 1 quite well up to the critical temperature.
Now we are in position to calculate the renormalized temperature of the Bose condensation. We can find using condition at the zero value of the renormalized chemical potential . A simple calculation gives that agrees quite well with experimental measurements of the temperature of the -transition despite the simplicity of the approximations.
Let us pass on to the heat capacity calculation:
We calculate heat capacity using the difference method and build the plot of its temperature dependence .
A comparison of different heat capacity curves is depicted in Fig. 2. As is seen from the comparison of the calculated curve 1 with the experimental one the agreement is quite good at low temperatures . At the temperatures the inconsistency occurs: the behavior of the calculated heat capacity is very similar to the behavior of the experimental curve, but shifted upward almost in a parallel way. This inconsistency is related to the fact that three- and four-particle correlations should be taken into account for the quantitative description. The contribution of three- and four-particle correlations, as is shown in Refs. [16, 18], improves significantly the ground-state results and gives a fairly good agreement at .
Further, let us calculate the heat capacity of liquid helium-4 with taking into account the effective mass of the helium atom in the liquid (curve 2). At low temperatures the heat capacity with taking into account the effective mass of the Bose particles practically coincides with curve 1, which agrees well with experimental data. This shows a weak dependence of the heat capacity on the effective mass below the temperature of phase transition. As is seen from Fig. 2, the calculated curve 2 (unlike curve 1) agrees quite well with the experimental one. It is related to the fact that by using the effective mass we partially take into account a contribution from three- and four-particle correlations. It is not surprising that in close vicinity of the Bose condensation point the theoretically calculated heat capacity deviates most significantly from the experimental curve. It is solely related to the inconsistency of our description near the critical point because the non-analytical part (20) of the one-particle spectrum, which makes a significant contribution in the thermodynamic functions at , is disregarded in our approach. One has to use renormalization group methods [23, 24, 25] for the correct description of the heat capacity in this temperature region.
In this paper we succeed in deriving quite well an agreement of the heat capacity curve of liquid helium with experimental data practically for all temperatures. The application of the thermodynamic functions of the Bose liquid obtained with the help of the Hamiltonian averaging combined with a one-particle spectrum derivation were the key moments of the calculations. Notwithstanding the simplicity of the spectrum calculation we obtained quite interesting results. In particular, we can decompose the part of the quasiparticle spectrum that is responsible for the non-analyticity in the Bose condensation point and show that this term of spectrum has no effect on the physical observables in the undercritical temperature region. So, an attempt is made to justify microscopically the idea that the -transition in a real quantum liquid is very similar to the Bose-Einstein condensation phenomena of the ideal gas “slightly” deformed by the interaction between the particles (keeping in mind the non-universal properties of the system).
The calculation found that the long wave-length limit of the “non-universal” part of the one-particle spectrum is quadratic over the wave-vector, i.e. very similar to the dispersion relation of the ideal gas but with a new mass. In the general case it is shown that this new mass at low temperatures is always greater than the mass of particles, and thus, the presence of the interaction at least in our approximation always lowers the critical temperature.
Another feature of the developed theory, perhaps a bit unexpected, is that even this simple temperature dependence of the effective mass improves the behavior of the heat capacity curve in the undercritical region and does not affect it in the condensate phase. Hence, the quantum-statistical approach based on the density matrix is suitable for describing thermodynamic properties of such a strongly-interacting Bose liquid as the helium-4 liquid not only in the limits of low and high temperatures, but for the entire temperature range.
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