Phys. Rev. B 43, 13587 (1991)

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Condensate fraction and momentum distribution in the ground state of liquid 4He

E. Manousakis
Department of Physics, Center for Materials Research and Technology Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32306
V. R. Pandharipande
Department of Physics and Materials Research Laboratory, University of Illinois at Urbana(hyChampaign, Urbana, Illinois 61802
Q. N. Usmani
Department of Physics, Jamia Millia Islamia, Jamia Nagar, New Delhi, 110025, India
(Received 26 December 1990)

In an earlier paper we calculated the condensate fraction and momentum distribution of atoms in liquid 4He, using a variational wave function which takes into account Jastrow and three-body correlations. In the present paper we point out an error in the expression used to compute the condensate fraction when three-body correlations were included. We report the corrected numerical results for both condensate fraction and momentum distributions for three densities. In addition, we calculate the condensate fraction and momentum distribution with and without the l=0 part of the three-body correlations in the wave function. Even though the l=0 part gives a small contribution to the ground-state energy, we find that the l=0 and l=1 terms have opposite and almost canceling effects on the condensate fraction and momentum distribution; each term separately alters the results obtained with pure Jastrow correlations by about 15—20 %, while when considered together they give only a small contribution.

©1991 The American Physical Society.

PACS: 67.40.-w

In our calculation of the condensate fraction and momentum distribution of liquid ^{4}He, we used^{1} a variational wave function which included optimized Jastrow (J) pair correlations and three body correlations [Jastrow plus triplet ( J + T ) ] . The wave function is given by

psi_{0}= ^ prod from { 1 <= i < j <= N } ^ f ( r_{ij}) ^ prod from { 1 <= i < j < k <= N } ^ f_{3}( r_{ij}, r_{jk}, r_{ik}) .   (1)

The Jastrow correlation factor f ( r_{ij}) _ takes into account the short- and long ranged correlations between the atoms, and it is optimized by solving the Euler Lagrange equations with the asymptotic behavior

f ( r -> inf ) = 1 - mc over { 2 pi^{2}hbar rho } ^ 1 over r^{2} ,  

which is a consequence of the long wavelength phonons. Here c is the velocity of sound and rho is the particle density. The three body correlation is written as

f_{3}( r_{ij}, r_{jk}, r_{ik})   L

= exp ( { case 1 over 2 } ^ sum from cyc sum from l=0,1,2 ^ xi_{l}( r_{ij}) xi_{l}( r_{ik}) P_{l}( r bhat_{ij}cdot r bhat_{ik}) right ) .   R

Here tsum_{roman}cyc _ represents a sum of the three terms obtained by replacing i j k with j k i and k i j . P_{l}_ represents Legendre polynomials of l th order, and r bhat_{ij}_ are unit vectors. The xi_{l}_'s are functions of the distance between the pairs of particles in the triplet and are determined variationally. The l = 1 term describes Feynman Cohen back flow correlations in the ground state and gives the dominant contribution to the energy; the l = 0 term gives a small contribution to the energy, and the l = 2 has a negligible effect in the calculation of the ground state energy.

The expression (3.7) of Ref. 1, however, which is used to calculate the condensate fraction, when the effect of three body correlations is taken into account, is incorrect. The correct expression is given by R_{x}= mark rho int d^{3}r [ g_{xd}( r ) - 1 - N_{xd}( r ) ]   (2)

lineup - cf12 rho int d^{3}r [ g_{xd}( r ) - 1 ] N_{xd}( r )  

lineup - rho int d^{3}r ^ g_{xd}( r ) [ E_{xd}( r ) + C_{xd}( r ) ] + E_{x}+ C_{x} ,  

with x = w , d , and the condensate fraction n_{0}_ is given by n_{0}= exp ( 2 R_{w}- R_{d}) _. Here the functions g_{xd}, N_{xd}, E_{xd}_, and C_{xd}_ are regular ( x = d ) and auxiliary ( x = w ) distribution functions, the sum of nodal and elementary diagrams, and the dressed triplet, respectively, which are obtained by solving the HNC/S equations with the inclusion of triplet correlations.^{1,2} E_{w} ( E_{d}) _ are elementary diagrams with one external point which is the extremity of an f_{ij}- 1 ( f_{ij^{2}-}1 ) _ correlation line. Such diagrams include both E^{g}- or E^{t}-type elementary diagrams which, respectively, do not or do involve explicit triplets joining the points of the diagram. C_{x}_ is the dressed triplet with one external point. Equation (3) can be simply obtained from Eq. (2.10) of Ref. 1 (which is the form for the pure Jastrow case) by simply substituting everywhere E by E + C with the corresponding subscript. Equation (2.14) of Ref. 1 should also be corrected to (4) E_{x}apeq ( 1 + { case 9 over 8 } S_{xd}) E_{x,4} ,   (3)

with x = w , d . However, as was noted in Ref. 1, because 2 E_{w}- E_{d}app 0 _, their contribution is negligible.

The calculation is performed with exactly the same approximations used in Ref. 1. The results with the full wave function are summarized in Tables I (replacing Table II of Ref. 1), II, and III (replacing Table III of Ref. 1), and Fig. 1 (replacing Fig. 9 of Ref. 1) and Fig. 2. The results have also been normalized according to Eq. (4.6) of Ref. 1. Note that the results for the normalized momentum distribution are not significantly affected by the error. Thus we do not redraw Fig. 8, in which we compared the results obtained with the J and J + T wave function, with the Green's function Monte Carlo (GFMC) and the experimental results, because the new and the momentum distributions of Ref. 1 are identical within the resolution of this figure. The error on the condensate fraction is larger; for example, at the equilibrium density with the J+T wave function, we find n_{0}= 0. 103 _ a value which is very close to that with pure J wave function.

In Ref. 1, n_{0}_ was calculated including both the l = 0 and 1 terms in Eq. (2). Here we have calculated the condensate fraction with and without the l = 0 term in f_{3}_. Including only the l = 1 term, we find that the condensate fraction at the equilibrium density is reduced from n_{0}= 0.098 _ obtained with the J wave function to n_{0}= 0. 082 _ with the J + T wave function. However, if we include both the l = 1 and 0 terms, the calculated condensate fraction at the equilibrium density is n_{0}= 0. 103 _, which is close to the value obtained with the J wave function.

Figure 1 replaces Fig. 9 of Ref. 1, where we compare our results obtained with the wave function which includes both the l = 0 and 1 terms with other available results. In Fig. 2 we compare the momentum distribution obtained with and without the l = 0 part in the wave function. We note that the effect of either term on n ( k ) is larger than the effect of both. In addition, the presence of the l = 0 part of the three body correlation puts back particles removed from the k = 0 state and its neighborhood by the l = 1 part of the correlations.

Because of the fact that the contribution to the ground state energy of the l = 0 is small, this part of the correlation function cannot be accurately optimized. Thus the contribution of this term to the condensate fraction and momentum distribution, which is rather significant, is crudely estimated by our approach.

We would like to thank A. Fabrocini and J. Boronat for bringing to our attention the error in the expression (3.7) of Ref. 1. This work was supported in part by the Supercomputer Computations Research Institute, which is partially supported by the U.S. Department of Energy through Contract No. DE FC05 85ER250000, and in part by the U.S. National Science Foundation through Grant No. PHY 89 21025.

FIGURE AND TABLE CAPTIONS

0 1 TABLE II. Results for the condensate fraction with optimized J and J+T wave functions.
  • TABLE III. Kinetic energy calculated from the normalized momentum distribution using J + T wave function ( l = 0 , 1 ) is compared with T_{JF}_ at three densities.



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