Momentum distributions in ^3He-^4He liquid mixtures

We present variational calculations of the one-body density matrices and momentum distributions for ^3He-^4He mixtures in the zero temperature limit, in the framework of the correlated basis functions theory. The ground-state wave function contains two- and three-body correlations and the matrix elements are computed by (Fermi)Hypernetted Chain techniques. The dependence on the ^3He concentration (x_3) of the ^4He condensate fraction $(n_0^{(4)})$ and of the ^3He pole strength (Z_F) is studied along the P=0 isobar. At low ^3He concentration, the computed ^4He condensate fraction is not significantly affected by the ^3He statistics. Despite of the low x_3 values, Z_F is found to be quite smaller than that of the corresponding pure ^3He because of the strong ^3He-^4He correlations and of the overall, large total density \rho. A small increase of $n_0^{(4)}$ along x_3 is found, which is mainly due to the decrease of \rho respect to the pure ^4He phase.


Abstract
We present variational calculations of the one-body density matrices and momentum distributions for 3 He- 4 He mixtures in the zero temperature limit, in the framework of the correlated basis functions theory. The ground-state wave function contains two-and three-body correlations and the matrix elements are computed by (Fermi)Hypernetted Chain techniques. The dependence on the 3 He concentration (x 3 ) of the 4 He condensate fraction (n (4) 0 ) and of the 3 He pole strength (Z F ) is studied along the P = 0 isobar. At low 3 He concentration, the computed 4 He condensate fraction is not significantly affected by the 3 He statistics. Despite of the low x 3 values, Z F is found to be quite smaller than that of the corresponding pure 3 He because of the strong 3 He- 4 He correlations and of the overall, large total density ρ. A small increase of n (4) 0 along x 3 is found, which is mainly due to the decrease of ρ respect to the pure 4 He phase.
Typeset using REVT E X The momentum distributions (MD) of atoms in quantum liquids is a challenging problem of fundamental interest. 1,2 They provide essential information on the correlations present in the system, which do not show up explicitly in other quantities. In the past years, accurate inelastic neutron scattering experiments have allowed for studying several aspects of the momentum distribution in helium liquids, 4 He, 3,4 3 He 5 and 4 He-3 He mixtures. 6,7 However, a clean extraction of information on the Helium MD's is somehow tampered by the need of a sound theoretical understanding of the final state effects in the analysis of the dynamic structure function, even at high momentum transfers.
The theoretical methods to evaluate momentum distributions of many-body interacting, dense systems at zero temperature have also made a significant progress in recent years. 1 At present, there are results for the pure Helium phases obtained within different many-body techniques, i.e., variational theory (using either integral equations 8,9 or Monte Carlo methods 10 ) and almost exact stochastic methods as Green's Function Monte Carlo (GFMC) 11,12 or Path-Integral Monte Carlo (PIMC ). 13 The MD's of liquid 4 He ( 3 He) are influenced by the Bose (Fermi) statistics of the atoms.
The macroscopic occupation of the zero momentum state, as given by the condensate fraction n (4) 0 , characterizes the momentum distribution of bosonic, liquid 4 He and it is strictly linked to its superfluid behavior. On the other hand, the discontinuity Z F at the Fermi momentum k F is a characteristic of the 3 He system when it is studied as a normal Fermi liquid.
In this paper we consider the interesting case of isotopic 3 He-4 He mixtures where, due to its fermion -boson nature, both quantities Z F and n (4) 0 are simultaneously present. Recent neutron scattering experiments on Helium mixtures at high momentum transfers 6,7 give additional motivations to undertake a microscopic, theoretical study of their momentum distributions and one-body density matrices. Special emphasis will be devoted to the dependence on the 3 He concentration, x 3 , of the single-particle kinetic energies of the isotopes and of Z F and n (4) 0 .
The investigation is carried on in the framework of the variational approach. The trial wave function for the mixture contains two-body (Jastrow) and triplet correlations. This type of correlated wave function has been useful in effectively studying the pure phases. 8,9,14,15 Two of us 16  The paper is organized as follows: in the second section we will shortly present the HNC/FHNC theory to calculate n(k) for mixtures described by correlated wave functions containing two-and three-body correlations. The treatment of the elementary diagrams in the so called scaling approximation is discussed in some details in the second part of the section. Results for n (4) (k), n (3) (k) and for the one-body density matrices are presented in Section II, together with a critical discussion of the discrepancies with the available analysis of the deep inelastic neutron scattering measurements on mixtures, which (in contrast with our results) point to a large enhancement of the 4 He condensate fraction.
In homogeneous mixtures, with constant particle densities s satisfy the normalization conditions ν α ρ (α) (0) = 1, ν α being the spin degeneracy (ν 4 = 1, ν 3 = 2). Notice that in the definition of ρ (3) (r) the spin variables have not been explicitly written. We will henceforth omit the subindex in the degeneracy factor and assume that it always refers to 3 He.
The momentum distribution of the α component, or rather the occupation probability for single-particle states with momentum k and given spin projection, can be obtained as the Fourier transform of the corresponding density matrix, where n The ground state of the mixture is well described by a generalization of the correlated wave function used in the pure phases: φ(1, . . . , N 3 ) is the Slater determinant of plane waves corresponding to the Fermi component of the mixture, and f (α,β) (i α , j β ) ( f (α,β,γ) (i α , j β , k γ )) are the 2 (3)-body correlation functions involving 2 (3) particles of types α, β (α, β, γ), respectively. Similar trial wave functions have been used in previous works to study the structure and energetic ground-state properties of 3 He-4 He mixtures. 16,19,20 A cluster analysis of ρ (α) (r) in powers of as that carried out in the pure phases, 24,25 gives the following structural decomposition for ρ (α) (r): where massive re-summations of the diagrams, as defined in Refs. 8,9,16,25, may be performed in practice by using HNC/FHNC techniques. 16,20,26 The strength factor n (α) 0 is given by and sums up all the irreducible diagrams with external points 1 α and 1 ′ α . In Eq.(6), l(x) = 3j 1 (x)/x is the Slater function and k F = (6π 2 ρ/ν) 1/3 is the 3 He Fermi momentum. and where and n c (k) = −X 2 ωcc X yc = g yc − N yc + l/ν for y = ω c , c andX xy (k) stands for the Fourier transform X xy (k) = ρ 3 dr e ik·r X xy (r) (11) The strength factor n (4) 0 is the asymptotic value of the 4 He one-body density matrix, and corresponds to the 4 He condensate fraction. The decomposition of n (3) (k) in a continuous (n c (k)) and a discontinuous (n d (k)) piece explicitly links the A. Scaling approximation for the elementary diagrams The HNC/FHNC equations can be solved once a given prescription for the contributions of the elementary diagrams has been given. However, as no exact method to compute them is presently known, at least in the frame of the integral equations, one has to resort to some approximation. Among the available schemes [27][28][29] we have chosen the scaling approximation (SA), developed for both the energy and the one-body density matrix of pure phases, 8,9,14,15 and satisfactorily reproducing VMC calculations. Although the number of elementary diagrams in the mixture is much larger, it is straightforward to generalize the pure phases scaling approximation to our case.
The SA is based on the evaluation of the 4-points elementary diagrams constructed with the combinations of the distribution functions g (α,β) xy (r) allowed by the diagrammatic rules and it has already been used in the calculation of the energy and of the static structure functions of the mixture. 20 The elementary diagrams are approximated by where E [4] g (r) and E [4] t (r) are the four-points elementary diagrams without and with explicit threebody correlations into their basic structure, respectively. These diagrams are constructed by using as internal links an averaged dressed correlationĝ(r) − 1, with  The additional elementary diagrams needed for the one-body density matrices are similarly evaluated: with ωd,g (r) + E [4] ωd,t (r), and ωω,t (r), [4] ωcωc,g + E [4] ωcωc,t (r).
The average distribution function (20) has been used to compute the above four-points elementary diagrams.
Finally, the set of single external point elementary diagrams, appearing in the strength factors n (α) 0 expressions, are approximated, as in the pure phases, 8,9 by x,g + E [4] x,t , x = ω, d .
and T JF is the ground-state expectation value of the kinetic energy operator computed by the Jackson-Feenberg identity. Moreover, the fulfillment of the normalization conditions of the momentum distributions, i.e., These conditions are used to determine the remaining scaling parameters (s (α) ωω , s ωcωc ). As a matter of fact, the use for the triplet correlated wave function of the same s (α) ωω and s ωcωc parameters, as determined in the Jastrow case, produces significant deviations of the above normalizations from their exact values. For this reason and to ensure the correct normalizations of the density matrices, we have recalculated the scaling factors s ωd , s (4) ωω , s (3) ωω and s ωcωc when the wave function contains three-body correlations, as in Ref. 9.

II. RESULTS
In this section we report results for the momentum distributions of 3 He-4 He liquid mixtures using the Aziz potential (HFDHE2) 30 for the variational determination of the groundstate correlations. This interaction effectively describes the equation of state of the pure phases. 12,31 The interatomic potential in isotopic mixtures is the same between any pair of particles. Based on this fact, we have used the average correlation approximation, ACA.
The ACA approach, which has been carefully analyzed for the impurity problem, 32 has also been used in the past to study finite concentration Helium mixtures. 20,33,34 The potential is The long range, r −2 behavior ensures the proper linear dependence of the 4 He structure function at k → 0.
The f (r) parameters at the 4 He energy variational minimum, at equilibrium density The three-body correlation function f (r ij , r ik , r jk ) has the parameterized form: 8,9,14,15 f (r ij , r ik , r jk ) = exp where The 0 can be strongly model dependent.
We start the analysis of the mixture by studying the x 3 -dependence of 4 He momentum distribution. Fig. 1 shows kn (4)   The value of n (4) 0 in the mixture is slightly larger than in the pure phase (see also Table I) due mainly to the smaller total density of the mixture. The fermionic nature of the 3 He does not affect n  Table I). This behavior is qualitatively explained by considering the change of both the total and partial 3 He densities. Fig. 4 with the free fermionic case (νρ(r)/ρ = l(k F r)) and with that of pure 3 He at the same ρ 3 . In this density region it is necessary to reach where ρ W F (r) is almost indistinguishable from the exact ρ (3) (r).
Eq. (30) explicitly decouples the statistical and dynamical correlations contributions to ρ (3) (r) and has also recently proved to describe quite accurately even the pure 3 He density matrix. 37 In this approximation, n (3) (k) is given by Therefore, the discontinuity Z F coincides with the value of the condensate fraction associated to n W F (k) can be expressed as where T B3 /N 3 is the kinetic energy associated to n  Table I.
n (4) 0 is shown in Fig. 5 as a function of the pressure, P , for pure 4 He (diamonds) and for a x 3 = 0.066 mixture (circles). The condensate fraction, in both cases, decreases with pressure as a consequence of the corresponding increase of density. The density of pure 4 He is larger than the one of the mixture at the same pressure and therefore the condensate fraction in the mixture is larger than in 4 He. However, as P increases, the differences between the densities become smaller and the condensate fractions of both systems get closer.
The low values of Z F imply a large value of the energy-dependent effective mass at the Fermi surface, where Σ(p, E) is the self-energy of the 3 He atoms in the mixture. At which is around three times larger than for pure 3 He at the saturation density. i.e., the value in the impurity case. Fig. 6 shows n (4) (k)/ρ 4 and νn (3) /ρ 3 for a 6 % mixture (solid and long-dashed lines respectively) together with n (4) (k)/ρ 4 for pure 4 He at the equilibrium density (short-dashed).
The three momentum distributions are very close above k F , as the large-k behavior is essentially dominated by the short-range dynamical correlations. As in the pure phases, the tails of the momentum distributions (k > 3.5Å −1 ) are taken to have an exponential behavior.
Their contribution at x = 6.6% to the total kinetic energy is ∼ 8%. On the other hand, the kinetic energy of the free Fermi sea (that would give an upper-bound to the contribution to T 3 /N 3 below k F ) is 0.58 K. That means that more than 97% of the 3 He kinetic energy comes from momenta above k F , clearly showing the importance of the correlations between 3 He and 4 He atoms.
It is also of interest to consider the dependence of T 3 /N 3 on the concentration. Fig. 7 gives T 3 /N 3 in function of the 3 He partial density in the mixture along the P = 0 isobar.
Obviously, the kinetic energy ends up with the kinetic energy of pure 3 He (∼ 12 K) which corresponds to a density value that lies out of the plot. Therefore the kinetic energy of the 3 He should be in average a decreasing function of the concentration except for the behavior at the origin where the term associated with the free Fermi kinetic energy dominates the overall decreasing behavior driven by the decrease of the total density. Actually, the kinetic energy in the interval considered here is well parameterized as the sum of the free Fermi gas energy plus a linear term describing the decrease of the kinetic energy with the density The numerical value of the parameter A may be estimated by calculating the x 3 dependence of the kinetic energy in the underlying boson-boson mixture and it results to be A = 27.2 Kσ 3 .

III. DISCUSSION AND CONCLUSIONS
The results obtained in this paper for the 4   The sums of the nodal diagrams contributions, N (3) ωcωc and N (α) ωω , are obtained by solving the integral equations and The notation (A(r ij ) | B(r jk )) stands for the convolution product The summations over z and y (where z, y = d, e, c) always extend to all possible connections allowed by the diagrammatic rules of the HNC/FHNC theory. 16,17 Besides the distribution functions g (α,β) de , g (3,3) ee and g (3,3) cc ), which have been defined elsewhere, 20,26 it is necessary to introduce the auxiliary distribution functions: where and L ω (r) = −l(k F r) + νB (3,3) ωcc (r) .
The nodal functions N (α,β) ωz (r) are solutions of the following integral equations: Finally, the functions C (α,β) ωx (r) give the contribution of the dressed triplet correlations, where E (α) x is the sum of the one-point elementary diagrams. 8,9,17 By setting ρ 3 = 0 (ρ 4 = 0), expression (2.15) reduces to the pure phases Γ x . 8,9 TABLES TABLE I. 4 He condensate fraction, 3 He Z F factor and partial kinetic energies in the mixtures as a function of the 3 He concentration at zero pressure. The first lines are the Jastrow values. The second lines include the effect of the triplet correlations.