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- E. Manousakis and V. R. Pandharipande
*Department of Physics and Materials Research Laboratory, University of Illinois at UrbanaChampaign, 1110 W. Green Street, Urbana, Illinois 61801*

We study the momentum-space structure of the elementary excitations in liquid * ^{4}*He by calculating the change

©1985 *The American Physical Society.*

**PACS:** 67.40.-w

The aim of this paper is to study the momentum space structure of elementary excitations in liquid sup 4He. The momentum distribution n_{0}(p)_ of the atoms in the ground state of liquid sup 4He has been studied by the Green's-function Monte Carlosup 1,2 (GFMC) and variationalsup 3,4 methods. The initial variational calculationssup 3 used the Jastrow (J) wave function:

PSI_{0}= OMEGA^{-N/2}^prod from i<j ^ f( r_{ij}) , (1.1)

for the ground state. Here OMEGA is the normalization volume and N is the number of particles. The thermodynamic limit N -> inf, OMEGA -> inf at fixed density rho =N/ OMEGA is assumed. Recently wesup 4 calculated the n_{0}(p)_ with a more realistic wave function, which contains optimized Jastrow and three body [Jastrow plus triplet (J+T)] correlations:

PSI_{0}= OMEGA^{-N/2}^sum from i<j ^ f(r_{ij}) ^prod from i<j<k ^ f_{3}(r_{ij},r_{jk},r_{ki}) . (1.2)

The GFMC and variational n_{0}(p)_ are in reasonable agreement with each other and the experimental data.sup 5,6

In this work we consider the change delta n_{{k}vec} ( p vec )_ in the momentum distribution of atoms due to an elementary excitation of momentum k vec. The wave functions and spectra of elementary excitations in liquid sup 4He have been studied by many authors. The first approximation for the wave function of an excitation of momentum k vec is Feynman's:

PSI_{{k}vec} = rho_{F}( k vec ) PSI_{0} , (1.3)

rho_{F}( k vec ) = ^sum from i=1 to N ^ e^{{i}k vec cdot r vec_{i}} . (1.4)

An improved wave function,

PSI_{{k}vec} = rho_{B}( k vec ) PSI_{0} , (1.5)

rho_{B}( k vec ) = ^sum from i=1 to N ^ e^{{i}k vec cdot r vec_{i}}( 1+i ^sum from {j != i} ^ eta ( r_{ij}) k vec cdot r vec_{ij}right ) , (1.6)

was proposed by Feynman and Cohensup 7 to account for the backflow current. Wesup 8 have recently carried out detailed variational calculations of the spectrum of elementary excitations with the rho_{B}( k vec )_ and the J+T PSI_{0_.}A brief review of the earlier work on the wave functions and the spectrum of excitations in liquid sup 4He is also given in Ref. 8. The energies obtained with these wave functions are satisfactory for the phonons, but they are app20% too high for the maxons and rotons. We developed a perturbation expansion using correlated basis functions (CBF) generated by the rho_{B}( k vec )_ operators. The second order corrections to the spectrum improve the agreement with experiment very significantly.

In the next section we calculate the delta n_{{k}vec} ( p vec )_ using the Jastrow PSI_{0_}and the Feynman rho_{F}( k vec )_. The general structure of delta n_{{k}vec} ( p vec )_ is discussed in detail, and the small and large k limits are calculated analytically. In Secs. III and IV we introduce the effects of the three body and backflow correlations, respectively. At very low temperatures (T<1 K) the liquid can be described as low density gas of excitations. In this limit we can use the delta n_{{k}vec} ( p vec )_ to study the low temperature behavior of the momentum distribution of atoms in the liquid. This calculation is described in Sec. V.

The delta n_{{k}vec} ( p vec )_ is given by

delta n_{{k}vec} ( p vec ) mark = n_{{k}vec} ( p vec ) - n_{{0}vec} ( p vec ) L

lineup = ^int^ d^{3}r_{{11}prime} [ n_{{k}vec} ( r vec_{{11}prime} )-n_{0}( r vec_{{11}prime} ) ] e^{{-i}p vec cdot r vec_{{11}prime}} , (2.1)

where n_{x}( r vec_{{11}prime} )_ is the one body density matrix in the state x=0, k vec:

n_{x}( r vec_{{11}prime} ) = {N ^int^ PSI_{x^{star}(1}prime ,2^,^.^.^.^,^ N) PSI_{x}(1,2^,^.^.^.^,^ N) d^{3}r_{2}... d^{2}r_{N}}over {int ^ | PSI (1,2^,^.^.^.^,^ N)|^{2}^ d^{3}r_{1}... d^{3}r_{N}} , (2.2)

normalized so that

OMEGA over {(2 pi )^{3}}^int^ n_{x}( p vec ) d^{3}p=N .

Both n_{{k}vec} ( p vec )_ and n_{0}(p)_ are of order 1, and the difference delta n_{{k}vec} ( p vec )_ is of order 1/N.

In this section we calculate the delta n_{{k}vec} ( p vec )_ using Jastrow's approximation for the PSI_{0_}[Eq. (1.1)], and the Feynman wave function [Eq. (1.3)] for the PSI_{{k}vec}_. The cluster expansion of delta n_{{k}vec} ( p vec )_ is carried out in Sec. II^A. The small- and large-k limits are calculated in Sec. II^B, and the methods used for numerical calculations are discussed in Sec. II^C.

We use the diagrammatic method developed for nuclear mattersup 9 to calculate the cluster expansion of delta n_{{k}vec} ( p vec )_. Fantoni'ssup 3 calculation of n_{0}(p)_ is repeated by using methods of Ref. 9. The cluster expansion of n_{0}( r_{{11}prime} )_ is obtained by expanding the numerator of n_{0}(r_{{11}prime} )_ in powers of the functions:

h(r_{ij}) =f^{2}(r_{ij})-1, i,j != 1,1 prime (2.3)

zeta (r_{mj}) =f( r_{mj})-1, m=1,1 prime . (2.4)

The integrals of this expansion are represented by diagrams which contain external points 1 and 1prime and any number of internal points denoting particle coordinates to be integrated. The functions h(r_{ij})_ and zeta (r_{mj})_ are represented by lines joining the points ij and mj. We obtain

Y_{0}== N ^int mark ^ PSI_{0}( r vec_{{1}prime} , r vec_{2}^,^.^.^.^,^ r vec_{N})

lineup times PSI_{0}( r vec_{1}, r vec_{2}^,^.^.^.^,^ r vec_{N}) d^{3}r_{2}... d^{3}r_{N

}

= rho "{" [I]+[I][i]+ cf12 [I][i][j]+ ... "}" , L (2.5)

where [I] denotes a connected diagram having the points 1 and 1 prime, and [i], [j], etc., denote connected diagrams having only h-lines. The products [I][I], [i][i][j], etc., represent disconnected diagrams. A sum over all diagrams I,i,j ^,^.^.^.^, is implied with the constraint that the diagrams I,i,j ..., in the disconnected diagram [I][i][j] ... have no common particles. Thus the product [I] times [i] != [I][i], since the product contains terms in which I and i have common particles.

The denominator is expanded in powers of h only, and we obtain

X_{0}mark == ^int^ PSI_{0^{2}(r}sub 1 ^,^.^.^.^, ^ r_{N)}d^{3}r_{1}... d^{3}r_{N

}

lineup = 1+[i]+ cf12 [i][j]+ ... . (2.6)

On performing the divisionsup 9 we get

n_{0}( r_{{11}prime} )= rho "{" mark [ I] -[I][i]-[I][i]- ... + [I][i][j] L

lineup + cf12 [I][i][j]+[I][i][j]+ ... - ... "}" . (2.7)

Here overhead bars denote common particles. For example [I][i]^{{}sup ^{}}is the sum of the products of all diagrams I and i that have one common particle. The diagrams I and i contributing to [I][i]^{{}sup ^{}}must have two common particles, while only the diagrams I, i, and j that have one common particle contribute to [I][i][j]^{{}sup ^{},}etc. The order of magnitude of the contribution of a term in Eq. (2.7) is given by N to power {the number of [ ] -1 -number of square lines -2 timesnumber of square ^square lines - ... "}". Thus [I][i]^{{}sup ^{}}has a contribution of order N^{-1,}and it is neglected while calculating the n_{0}(p)_.

Compact cluster expansions are obtained by noting that the connected diagrams [I] and [i] can be expressed as

[I] = mark [A]+ [A a prime ] + cf12 [A a prime b prime ] + cf12 [ A a prime b prime ]

lineup + [A a prime b prime ] + ... , (2.8)

[i]=[a]+ cf12 [ab] + ... , (2.9)

where A denotes irreducible diagrams which contain both points 1 and 1 prime; and a prime ,b prime ^,^.^.^.^, denote irreducible diagrams that may contain one or none of the points 1 and 1 prime. The term {[A a prime ]}^{{}sup ^{}}represents all the diagrams in [I] with one articulation point. {[A a prime b prime ]}^{{}sup ^{}}is the sum of diagrams that can be broken into three pieces at one articulation point, while {[A a prime b prime ]}^{{}sup ^{}}and {[A a prime b prime ]}^{{}sup ^{}}are sums of diagrams with two articulation points etc. The irreducible diagrams a,b^,^.^.^.^, that occur in the expansion of [i] are identical to the diagrams a prime ,b prime ^,^.^.^.^, when they do not include the point 1 or 1 prime. Diagrams a prime , b prime ^,^.^.^.^, that include the point 1 or 1 prime, are formed with zeta lines starting from 1 or 1 prime; whereas diagrams a,b^,^.^.^.^, which include the point 1 can come only from the [i] in denominator, and they have only h-lines. The a,b^,^.^.^.^, cannot have point 1 prime.

On substituting Eqs. (2.8) and (2.9) in Eq. (2.7) we obtain

1 over rho n_{0}( r vec_{{11}prime} ) = mark [ A] + "{" [ A a prime ] - [A][a] "}" + "{" cf12 [ A a prime b prime ] - [A a prime ][b] - cf12 [A][ab] + [A][a][b] "}"

lineup + "{" cf12 [A a prime b prime ] - [A a prime ][b] + cf12 [A][a][b] "}" + "{" [A a prime b prime ] - [A a prime ] [b] - [A][ab] + [A][a][b] "}" + ... . (2.10)

It can be easily verified that all the terms in the { } are zero unless the common points are 1 and/or 1 prime. Let [a_{1^{sprime}]_}and [a_{1}]_ be the sums of irreducible a prime and a diagrams with points 1, and [a_{{1}prime}^{sprime}]_ be the sum of irreducible diagrams with point 1 prime. The R_{w_}and R_{d_}of Ref. 3 are defined as

R_{w}= [a_{1^{sprime}]}=[a_{{1}prime}^{sprime}] , (2.11)

R_{d}= [a_{1}] . (2.12)

The sum [A] can be factored out of the n_{0}(r_{{11}prime} )_ to obtain

n_{0}(r_{{11}prime} ) = rho [A] [1+2 R_{w}-R_{d}+ cf12 (2R_{w}-R_{d})^{2}+ ... ] . L

The higher terms simply complete the exponential series, and we obtain the well knownsup 3 expression: (2.14) n_{0}(r_{{11}prime} ) = rho [A] ^ exp (2R_{w}-R_{d}) == rho [A] n_{c} , (2.13)

where n_{c_}is the fraction of particles in the p=0 condensate.

The cluster expansion of delta n_{{k}vec} ( r vec_{{11}prime} )_ is calculated in the same way. We express delta n_{{k}vec} ( r vec_{{11}prime} )_ as

delta n_{{k}vec} ( r vec_{{11}prime} ) = X_{0}over {X_{{k}vec}} ( {Y_{{k}vec}} over X_{0}- Y_{0}over X_{0}{X_{{k}vec}} over X_{0}right ) == X_{0}over {X_{{k}vec}} U_{{k}vec} , L (2.15)

Y_{{k}vec} =N ^int sum from m,n mark ^ e^{{i}k vec cdot r vec_{mn}}PSI_{0}( r vec_{{1}prime} ^,^.^.^.^,^ r vec_{N}) L

lineup times PSI_{0}( r vec_{1}^,^.^.^.^,^ r vec_{N)}d^{3}r_{2}... d^{3}r_{N} , (2.16)

X_{{k}vec} = ^int sum from m,n ^ e^{{i}k vec cdot r vec_{mn}}PSI_{0^{2}(}r vec_{1}^,^.^.^.^,^ r vec_{N}) d^{3}r_{1}... d^{3}r_{N} . L (2.17)

In Eq. (2.16), m is summed from 1 to N and n from 1 prime to N, while in Eq. (2.17) for X_{{k}vec}_ both m and n are summed from 1 to N.

There are N terms with m=n in X_{{k}vec}_, and their contribution is NX_{0_.}The m != n terms are calculated by cluster expansion. As in the theory of elementary excitations, the exp(i k vec cdot r vec_{mn})_ is represented in the cluster diagrams by an exchange line from m to n. The sum of irreducible diagrams containing this line is denoted by [B]. The ratio X_{{k}vec} /X_{0_}is easily calculated to be

X_{{k}vec} /X_{0}=N+[B]=NS(k) , (2.18)

where S(k) is the familiar static structure function. Since this ratio is of order N, we have to calculate the U_{{k}vec}_ in Eq. (2.15) to order 1.

The contribution of terms with m=n to Y_{{k}vec}_ is (N-1) Y_{0_,}and to U_{k_}is -n_{0}( r vec_{{11}prime} )_. The contribution of m != n terms has to be calculated with cluster expansion. The m != n terms of Y_{{k}vec}_ can be written as

Y_{{k}vec} (m != n) = rho "{" mark [ L]+[L][i]+ cf12 [L][i][j]+ ...

lineup + [I][J]+[I][J][i] + ... "}" , (2.19)

where L denotes connected diagrams that contain points 1 and 1 prime and the line mn, while J denotes connected diagrams that contain the line mn but not the points 1 and 1 prime. The ratio is found to be

Y_{{k}vec} (m != n) / X_{0}= rho "{" mark [ L]- [L][i] + [L][i][j]+ cf12 [L][i][j]]+[L][i][j]+ ...

lineup + [I][J]- [I][J][i]-[I][J][i]-[I][J][i]+ ... "}" . (2.20)

We also obtain

Y_{0}X_{{k}vec} ( m != n) / (X_{0})^{2}= rho "{" [I][J]+[I][J] -[I][J][i]-[I][J][i] -2[I][J][i] - [I][J][i]^{{}sup ^{}}- [I][J][i]^{{}sup ^{}}-2 [I][J][i]^{{}sup ^{}}+ ... "}" . L

and thus, (2.22) U_{{k}vec} (m != n ) = rho "{" [L]- [L][i]+ ... - [I][J] + [I][J][i]+[I][J][i]+[I][J][i]+2[I][J][i] + ... "}" . (2.21)

Let C denote irreducible diagrams containing points 1 and 1 prime and the exchange line mn. We then have

[L] = [C] +[C a prime ] + cf12 [ C a prime b prime ] + cf12 [ C a prime b prime ] + [C a prime b]+ ... + [A B prime ] + [A B prime a prime ] + [ A B prime a prime ] + [ A B prime a ] + [A B a prime ] + ... , L

(2.24) [J]=[B]+[Ba]+ ... . (2.23)

The irreducible B prime diagrams are identical to the B diagrams when they do not contain points 1 or 1 prime. B prime diagrams containing 1 or 1 prime have zeta line starting from 1 or 1 prime. Substituting Eqs. (2.8), (2.9), (2.23), and (2.24) in (2.22) we obtain

1 over rho U_{{k}vec} (m != n ) = mark [ C] + "{" [C a prime ] -[C][a] "}" + ... + "{" [AB prime ] -[A][B] "}"

lineup + "{" {[A B prime a prime ]}^{{}sup ^{}}- {[A a prime ][B]}^{{}sup ^{}}- [A][Ba]^{{}sup ^{}}- {[A B prime ][a]}^{{}sup ^{}}+2[A][B][a]^{{}sup ^{}}"}"

lineup + "{" {[A B prime a prime ]}^{{}sup ^{}}- {[A a prime ][B]}^{{}sup ^{}}- {[A B prime ] [a]}^{{}sup ^{}}+ [A][B][a]^{{}sup ^{}}"}" + "{" {[A B prime a prime ]}^{{}sup ^{}}- {[A B prime ][a]}^{{}sup ^{}}- [A][Ba]^{{}sup ^{}}+ [A][B][a]^{{}sup ^{}}"}"

lineup + "{" {[AB a prime ]}^{{}sup ^{}}- {[A a prime ][B]}^{{}sup ^{}}- [A][Ba]^{{}sup ^{}}+ [A][B][a]^{{}sup ^{}}"}" + ... . (2.25)

The terms in { } are nonzero only when the common particles are 1 and/or 1 prime. All the terms with irreducible diagrams C can be summed up to obtain [C] exp (2R_{w}-R_{d})_. One of the [A][B][a]^{{}sup ^{}}cancels [A][Ba]^{{}sup ^{},}and the { }'s starting from {[AB prime a]}^{{}sup ^{}}and {[AB a prime ]}^{{}sup ^{}}give zero contribution because they must have a common particle other than 1 and 1 prime. The two common particles in the terms of the { } starting with {[A B prime a prime ]}^{{}sup ^{}}must be 1 and 1 prime, otherwise this { } is zero. We denote by B_{1_,}B_{1^{sprime_}and}B_{{1}prime}^{sprime_}the irreducible diagrams containing particles 1 and 1 prime. Note that there are no B_{{1}prime}_ diagrams. The terms in U_{{k}vec}_ containing A diagrams are reordered in the following two series:

- [A][B_{1}] - [A a_{1^{sprime}]}[B_{1}] - [A a_{{1}prime}^{sprime}] [B_{1}] + [A] [B_{1}] [a_{1}] + ... , L

+ [ AB_{1^{sprime}]}+[AB_{{1}prime}^{sprime}] + [AB_{1^{sprime}a}sub 1^{sprime}] + [A B_{{1}prime}^{sprime}a_{{1}prime}^{sprime}] + [AB_{1^{sprime}a}prime_{{1}prime} ] + [AB_{{1}prime}^{sprime}a_{1}] -[ A B_{1^{sprime}]}[a_{1}] - [AB_{{1}prime}^{sprime}] [a_{1}] + ... . R

These series are summed to obtain the total U_{{k}vec}_ and delta n_{{k}vec} ( r vec_{{11}prime} )_ including m=n contribution as

U_{{k}vec} = rho [C] ^ exp (2R_{w}-R_{d}) -n_{0}( r vec_{{11}prime} ) (1+[ B_{1}] -2[ B_{1^{sprime}]) , }(2.26)

delta n_{{k}vec} ( r vec_{{11}prime} ) =U_{{k}vec} / [NS(k)] . (2.27)

The second term of Eq. (2.26) represents particles taken out of the n_{0}(p)_ by the excitation. The structure of 1+[B_{1}]_ is shown in Fig. 1. We use the diagrammatic notation of Ref. 4 in which dashed and wiggly lines represent generalized g_{dd}-1_ and g_{wd}-1_ bonds. These bonds, respectively, represent sums of all possible correlations with h-lines at both ends and with h-lines at one end and zeta lines at the other. The B_{1_}diagrams have only g_{dd}-1_ bonds. From Fig. 1 we obtain

1+ [B_{1}] =S^{2}(k) [1+ LAMBDA_{d}(k) ] , (2.28)

LAMBDA_{d}(k) = rho^{2}^int mark ^ d^{3}r_{12}d^{3}r_{13}e^{{i}k vec cdot r vec_{23}}[g_{dd}(12) -1] [g_{dd}(13)-1] [g_{dd}(23)-1]

lineup times (1+ A_{ddd}(123) "{" 1+ [g_{dd}(12)-1]^{-1}+ [g_{dd}(13) -1]^{-1}+[g_{dd}(23)-1]^{-1}"}" ) , (2.29)

where A_{ddd_(123)}is the contribution of Abe diagrams.

The [B_{1^{sprime}]_}is shown in Fig. 2. We obtain

[B_{1^{sprime}]}=[B_{{1}prime}^{sprime}] =g_{wd}(k) [1+g_{wd}(k) ] +S^{2}(k) LAMBDA_{w}(k) , (2.30)

g_{wd}(k) = rho ^int^ d^{3}r [g_{wd}(r)-1] e^{{i}k vec cdot r vec} , (2.31)

LAMBDA_{w}(k) = rho^{2}^int mark ^ d^{3}r_{12}d^{3}r_{13}e^{{i}k vec cdot r vec_{23}}[g_{wd}(12)-1] [g_{wd}(13)-1] [g_{dd}(23)-1]

lineup times (1+A_{wwd}(123) "{" 1+[g_{wd}(12)-1]^{-1}+[g_{wd}(13)-1]^{-1}+[g_{dd}(23)-1]^{-1}"}" ) . (2.32)

We define t_{1}(k)_ as

t_{1}(k) =(2 [ B_{1^{sprime}]}-1-[ B_{1}] )/S(k) = 2 over S(k) g_{wd}(k) [1+g_{wd}(k) ] -S(k) [1+ LAMBDA_{d}(k) -2 LAMBDA_{w}(k) ] , (2.33)

so that the contribution of this term to delta n_{{k}vec} ( r vec_{{11}prime} )_ is simply t_{1}(k) n_{0}(r_{{11}prime} ) /N_.

The contribution of C diagrams [Eq. (2.26)] can be divided into three parts. The diagrams belonging to the first part are shown in Fig. 3(a), and their contribution is given by

1 over NS(k) n_{0}(r_{{11}prime} ) e^{{i}k vec cdot r vec_{{11}prime}} [1+g_{wd}(k) ]^{2}== 1 over N t_{2}(k) n_{0}( r vec_{{11}prime} ) e^{{i}k vec cdot r vec_{{11}prime}} . (2.34)

The contribution of diagrams included in the second part [Fig. 3(b)] is obtained as

1 over NS(k) n_{0}(r_{{11}prime} ) e^{{-i}k vec cdot r vec_{{11}prime}} [g_{wd}(k)]^{2}== 1 over N t_{3}(k) n_{0}(r_{{11}prime} ) e^{{-i}k vec cdot r vec_{{11}prime}} . (2.35)

The third part includes all other C diagrams, whose contribution depends upon the directions of k vec and r vec_{{11}prime}_ in a nontrivial fashion. We express it as T( k vec , r vec_{{11}prime} ) n_{0}(r_{{11}prime} )/N_,

T ( k vec , r vec_{{11}prime} ) =[1+g_{wd}(k) ] D ( k vec , r vec_{{11}prime} ) +g_{wd}(k) D^{star}( k vec , r vec_{{11}prime} ) +S(k) D prime ( k vec , r vec_{{11}prime} ) . (2.36)

The D( k vec , r vec_{{11}prime} )_ term is from diagrams shown in Fig. 4(a), while the second D^{star}( k vec , r vec_{{11}prime} )_ term is from diagrams of Fig. 4(b). We have

D ( k vec , r vec_{{11}prime} ) = rho ^int mark ^ d^{3}r_{12}(e^{{i}k vec cdot r vec_{12}}+e^{{i}k vec cdot r vec_{{21}prime}} ) [g_{wd}(12)-1] [g_{wd}(1 prime 2)-1]

lineup times (1+ A_{wwd}(11 prime 2) "{" 1+ [g_{wd}(12)-1]^{-1}+[g_{wd}(1 prime 2)-1]^{-1}"}" ) . (2.37)

The D prime ( k vec , r vec_{{11}prime} )_ term represents the contribution of four point and higher order diagrams shown in Fig. 5.

In all, we have

delta n_{{k}vec} ( r vec_{{11}prime} ) = 1 over N n_{0}(r_{{11}prime} ) [t_{1}(k) +t_{2}(k) e^{{i}k vec cdot r vec_{{11}prime}} +t_{3}(k) e^{{-i}k vec cdot r vec_{{11}prime}} +T( k vec , r vec_{{11}prime} ) ] , (2.38)

from which we obtain

delta n_{{k}vec} ( p vec ) = 1 over N [ mark t_{1}(k) n_{0}(p) +t_{2}(k) n_{0}(| k vec - p vec |)

lineup + t_{3}(k) n_{0}(| k vec + p vec | ) +t_{4}( k vec , p vec ) ] , (2.39)

where

t_{4}( k vec , p vec ) = ^int^ d^{3}r_{{11}prime} n_{0}(r_{{11}prime} ) T( k vec , r vec_{{11}prime} ) e^{{-i}p vec cdot r vec_{{11}prime}} . (2.40)

The delta n_{{k}vec} ( p vec )_ takes a rather simple form in the limits k ->0 and k -> inf. At small values of k we have the relations

S(k -> 0) = hbar over 2mc k , (2.41)

g_{wd}(k -> 0)=- cf12 + ... , (2.42)

where the ellipsis represents a term of O(k). Here c is the velocity of sound, and the above relations follow from the asymptotic behavior of the pair correlation:sup 10

f(r -> inf )=1- mc over {2 pi^{2}hbar rho} 1 over r^{2} . (2.43)

In the limit k ->0, t_{1_,}t_{2_,}and t_{3_}have terms of order 1/k, while T is of order 1, and hence negligible. We find by using the limits (2.41) and (2.42) in Eqs. (2.33)ndash(2.35)

t_{1}(k -> 0)=-1/[2S(k)] , (2.44)

t_{2}(k -> 0)=1/[4S(k)] , (2.45)

t_{3}(k -> 0)=1/[4S(k)] . (2.46)

Thus a long wavelength phonon removes 1/[2S(k)] particles from the ground state n_{0}(p)_ and divides them equally into two distributions n_{0}(| p vec - k vec |)_ and n_{0}(| p vec + k vec |)_ centered at p vec = k vec and p vec =- k vec. The n_{0}(p)_ has the singular term Nn_{c}delta_{p,0_,}and hence a phonon state has n_{c}/[4S(k)]_ particles each in the states with p vec = +- k vec. The change in the kinetic energy (KE) of the liquid on exciting a phonon is given by

delta (KE)= OMEGA over {(2 pi )^{3}}hbar^{2}over 2m ^int^ d^{3}p delta n_{{k}vec} ( p vec ) p^{2} , (2.47)

and in the long wavelength limit it is

delta (KE)(k -> 0) = {hbar^{2}k^{2}}over 2m 1 over 2S(k) = cf12 hbar omega (k) . (2.48)

The phonon in Bose liquids is, indeed, a pure harmonic vibration with half of its energy from kinetic and the other half from potential terms.

The k -> inf limit is only of mathematical interest, since we do not expect the Feynman wave function of the excitation to be realistic in this limit. In this limit the excitation has a single particle character. The S(k -> inf )=1, and g_{wd}(k -> inf )_, LAMBDA (k -> inf ), and T( k vec -> inf , r vec ) are all zero. Thus, we have

t_{1}(k -> inf )=-1 , (2.49)

t_{2}(k -> inf ) =1 , (2.50)

t_{3}(k -> inf ) =t_{4}( k vec -> inf , p vec ) =0 , (2.51)

a single particle is removed from the ground state, and put in a distribution n_{0}(| p vec - k vec |)_ centered at p vec = k vec. In this limit the energy of the Feynman excitation is hbar^{2}k^{2}/2m, and it equals the change in the kinetic energy.

The pair functions g_{xy_,}xy=dd, and wd, and the Abe functions A_{xyz_,}xyz=ddd, ddw, and wwd have been calculated in Ref. 4 with the hypernetted chain scaling (HNC/S) method in which the contribution of elementary diagrams is approximated by scaling that of the four point diagrams. We use these functions to calculate the delta n_{{k}vec} ( p vec )_. The S(k) and g_{wd}(k)_ are shown in Fig. 6 for easy reference. The g_{dd}(r)_ and g_{wd}(r)_ are shown in Fig. 2 of Ref. 4. The quantities t_{1_,}t_{2_,}and t_{3_}are relatively simple integrals of the g and A functions. We find that the LAMBDA_{d}-2 LAMBDA_{w_}term contributes less than 5% of the t_{1_,}and hence the Abe corrections to the LAMBDA's [Eqs. (2.29) and (2.32)] are neglected. The calculated values of the t_{i_'s}are tabulated in Table I.

The t_{4}( k vec , p vec )_ is generally a function of k, p, and the angle between k vec and p vec. It is difficult to calculate it, but it may be reasonably approximated as follows. We define

D_{1}( k vec , p vec ) = ^int^ d^{3}r_{{11}prime} n_{0}(r_{{11}prime} ) D ( k vec , r vec_{{11}prime} ) e^{{-i}p vec cdot r vec_{{11}prime}} . L (2.52)

This integral appears in the contribution of the D( k vec , r vec_{{11}prime} )_ term to t_{4}( k vec , p vec )_. We can rewrite it as

D_{1}( k vec , p vec ) = rho ^int^ d^{3}r_{{11}prime} n_{0}(r_{{11}prime} ) e^{{-i(}p vec - k vec /2) cdot r vec_{{11}prime}} ^int ^ d^{3}r_{12}(e^{{i}k vec cdot r vec_{c2}}+e^{{-i}k vec cdot r vec_{c2}}) [g_{wd}(12)-1] [g_{wd}(1 prime 2)-1] (1+ scrA ) , L (2.53)

where scrA represents Abe terms, and

r vec_{c}= cf12 ( r vec_{1}+ r vec_{{1}prime} ) . (2.54)

It is now clear that D_{1}( k vec , p vec )_ is invariant under the transformation p vec -> k vec - p vec, and hence it is convenient to consider D_{1_}as a function of k, | p vec - k vec /2| and the polar angle CTHETA of | p vec - k vec /2| using k vec to define the Z axis. This function is invariant under the transformation CTHETA -> pi - CTHETA. We neglected the Abe terms in D, and calculated D_{1}(k, | p vec - k vec /2| , CTHETA )_ for chosen values of k, | p vec - k vec /2|, and CTHETA. The D_{1}(k,| p vec - k vec /2|, CTHETA )_ is peaked at | p vec - k vec /2|=0. At small values of | p vec - k/2| it has negligible dependence on CTHETA. For k=1 A ang^{-1}and | p vec - k vec /2|=1 A ang^{-1,}the dependence on CTHETA is app15%. The D_{1_}at k=1 A ang^{down}20 -1 has decreased to app half its peak value at | p vec - k vec /2|=1 A ang^{-1.}Thus it appears that in the first approximation we may neglect the CTHETA dependence of D_{1_.}

The angle average value,

D bar_{1}(k,| p vec - k vec /2|)= cf12 ^int_{-1^{1}^}d^ cos CTHETA D_{1}( k,| p vec - k vec /2|, CTHETA ) , (2.55)

is much simpler to calculate. It is given by

D bar_{1}(k,| p vec - k vec /2|) = 2 rho ^int ^ d^{3}r_{{11}prime} n_{0}(r_{{11}prime} ) j_{0}(| p vec - k vec /2| r_{{11}prime} ) ^int^ d^{3}r_{12}j_{0}(kr_{c2}) [g_{wd}(12)-1] [g_{wd}(1 prime 2)-1] (1+ scrA ) ,

and should be a good approximation to D_{1}(k,| p vec - k vec /2|, CTHETA )_. The results for k=0.6, 1.0, and 2.0 A ang^{-1}are shown in\p Fig. 7. This function does not have any singularities. The singularities in n_{0}(p)_ come from the long range (r_{{11}prime} -> inf )_ part of n_{0}(r_{{11}prime} )_, which is cut out in the D by the product of short range functions g_{wd}(12)-1_ and g_{wd}(1 prime 2)-1_.

The second integral in the t_{4}( k vec , p vec )_ involving D^{star}( k vec , r vec_{{11}prime} )_ is calculated in a similar way. It is easy to show that (2.57) int ^ d^{3}r_{{11}prime} n_{0}(r_{{11}prime} ) D^{star}(k,r_{{11}prime} ) e^{{-i}p vec cdot r vec_{{11}prime}} =D_{1}( k vec ,- p vec ) approx D bar_{1}(k,| p vec + k vec /2|) . (2.56)

The contribution of D prime ( k vec , r vec_{{11}prime} )_ contains the integral

D_{2}( k vec , p vec ) = ^int^ d^{3}r_{{11}prime} n_{0}( r_{{11}prime} ) D prime ( k vec , r vec_{{11}prime} ) e^{{-i}p vec cdot r vec_{{11}prime}} . (2.58)

It depends upon the polar angle CTHETA_{p_}of p; however, it is invariant under the transformation CTHETA_{p}-> pi - CTHETA_{p_.}We neglect the dependence on CTHETA_{p_}and approximate D_{2_}by its angle average

D bar_{2}(k,p) = cf12 ^int_{-1^{1}^d^}cos CTHETA_{p}D_{2}(k,p, CTHETA_{p}) . (2.59)

The Abe corrections to D prime are neglected; we then have (from Fig. 5):

D bar_{2}(k,p) = rho^{2}^int^ d^{3}r_{{11}prime} n_{0}(r_{{11}prime} ) j_{0}(pr_{{11}prime} ) ^int mark ^ d^{3}r_{12}d^{3}r_{13}j_{0}(kr_{23})

lineup times (2[g_{wd}(12)-1] [g_{wd}(1 prime 3)-1] "{" 1+[ g_{wd}(1 prime 2)-1] +[g_{wd}(13)-1] "}" [g_{dd}(23)-1]

+ [g_{wd}(12)-1] [g_{wd}(1 prime 2) -1] [g_{wd}(13)-1] [g_{wd}(1 prime 3) -1] g_{dd}(23) ) . L (2.60)

The calculated values are shown in Fig. 8. The T_{4}( k vec , p vec )_ is thus approximated with

t_{4}( k vec , p vec ) = mark [ 1+ g_{wd}(k) ] D bar_{1}(k,| p vec - k vec /2|)

lineup + g_{wd}(k) D bar_{1}(k,| p vec + k vec /2|) + S(k) D bar_{2}(k,p) .

The T_{4}(k_=1^Aang^{-1},^p,^ CTHETA_{p_=0)}is shown in Fig. 9.

The conservation of particles and momentum requires that (2.63) OMEGA over {(2 pi )^{3}}^int^ d^{3}p delta n_{{k}vec} ( p vec ) =0 , (2.61)

OMEGA over {(2 pi )^{3}}^int^ d^{3}p delta n_{{k}vec} ( p vec ) p vec = k vec . (2.64)

This implies that

t_{1}(k) +t_{2}(k)+t_{3}(k) +t_{4n}(k) =0 , (2.65)

t_{2}(k) -t_{3}(k) +t_{4m}(k)=1 , (2.66)

where

t_{4n}(k) = 1 over {rho (2 pi )^{3}}^int^ d^{3}pt_{4}(k,p, CTHETA_{p}) , (2.67)

t_{4m}(k) = 1 over {rho (2 pi )^{3}}1 over k ^int^ d^{3}pt_{4}(k,p, CTHETA_{p}) p^ cos CTHETA_{p} . (2.68)

The calculated values of the t's are listed in Table I. The identities 2 and 2 are satisfied with an accuracy of app15% of the largest t.

In this section we calculate the delta n_{{k}vec} ( p vec )_ with the wave function (1.2) for the ground state and the Feynman excitation operator (1.4), using the three body correlation determined in Ref. 11. The cluster expansion for the delta n_{{k}vec} ( p vec )_ has the form given in Sec. II^A; however, we must now consider three body bonds of the type [f_{3}(ijk)]^{2}-1_ and f_{3}(1jk)-1_. The effect of these bonds on the two point functions g_{xy}-_1 is taken into account as discussed in Refs. 4 and 11. Their effect on the Abe functions is generally given by

A_{wwd}(11 prime 2) =A_{wwd^{g}(11}prime 2) + A_{wwd^{t}(11}prime 2) , L (3.1)

A_{wdd}(123)= f_{3}(123) [ 1+ A_{wdd^{g}(123)}+ A_{wdd^{t}(123)]-1 , }L

L A_{ddd}(123) =[f_{3}(123)]^{2}[ 1+ A_{ddd^{g}(123)}+ A_{ddd^{t}(123)]-1 , }(3.2)

where A_{xyz^{g_}are}four- or more body Abe corrections from two body bonds g_{xy}-1_, and A_{xyz^{t_}are}four- or more body Abe corrections that include three body bonds.

The t_{2_}and t_{3_}[Eqs. (2.34) and (2.35)] contain only two body integrals, and they are calculated as described in the last section. In the Jastrow calculations and here also we neglect all Abe corrections to the LAMBDA's [Eqs. (2.29) and (2.32)]. This corresponds to the following approximations for the LAMBDA's: LAMBDA_{d}(k) = rho^{2}^int^ d^{3}r_{12}d^{3}r_{12}d^{3}r_{13}e^{{i}k vec cdot r vec_{23}}([g_{dd}(12) -1] [g_{dd}(23) -1] [g_{dd}(31)-1] + "{" [f_{3}(123)]^{2}-1 "}" g_{dd}(12) g_{dd}(23) g_{dd}(31) ) , (3.3)

LAMBDA_{w}(k) = rho^{2}^int^ d^{3}r_{12}d^{3}r_{13}e^{{i}k vec cdot r vec_{23}}"{" [ g_{wd}(12)-1] [g_{wd}(13) -1] [g_{dd}(23)-1] + [f_{3}(123)-1] g_{wd}(12) g_{wd}(13) g_{dd}(23) "}" . (3.4)

The three body correlations do not have a large effect on the n_{0}(p)_ as discussed in Ref. 4. The calculated values of the t_{{1}ndash 3}_ with the wave function (1.2) are listed in Table II. These differ from the t_{{1}ndash 3}_ for the Jastrow PSI_{0_}by <15%. The effect of the triplet correlation on the t_{4}( k vec , p vec )_ has not been calculated.

In this section we calculate the t_{{1}ndash 3}_ with wave functions (1.2) for the ground state and (1.5) and (1.6) for the excited state. The cluster expansion of Sec. II^A is still valid; however, we now have many more terms containing the backflow correlation bonds eta (r_{ij}) k vec cdot r vec_{ij_.}There are no two body backflow correlations in the ground state, and so the g_{xy_}do not contain two body backflow effects. The triplet correlation, however, can be thought of as a backflow effect.sup 12

We have developed, in Ref. 8, two approximations to sum backflow terms. These approximations must be used with the short range backflow function eta_{S}(r)_ of Ref. 8. The simpler of these is called the two body (TB) factorizable approximation in which only those terms whose contributions can be expressed as products of two body integrals are retained. The second, called extended TB or ETB, considers the modification of the backflow correlation eta (r) due to Jastrow and triplet correlations in calculating the TB integrals. For the sake of clarity we first calculate the t_{{1}ndash 3}_ in the TB approximation, and later give expressions for the ETB.

The ratio X_{{k}vec} / X_{0_,}given by NS(k) in the Jastrow theory [Eq. (2.18)], becomes {X_{{k}vec}} over X_{0}mark == {langle PSI_{0}| rho_{B^{dag}(}k vec ) rho_{B}( k vec ) | PSI_{0}rangle} over {langle PSI_{0}| PSI_{0}rangle} == NX_{B}(k) (3.5)

lineup = N "{" S(k)[1+ cf12 I bar_{9,2}(k) ]^{2}+ I bar_{10,2}(k) "}" , (4.1)

in the TB approximation.sup 8 The two body integrals I bar_{9,2}(k)_ and I_{10,2}(k)_ are

I bar_{9,2}(k) =2 rho ^int^ d^{3}re^{{i}k vec cdot r vec} eta (r)i k vec cdot r vec g_{dd}(r) , (4.2)

I bar_{10,2}(k) = rho ^int^ d^{3}r(1-e^{{i}k vec cdot r vec} ) [ k vec cdot r vec eta (r) ]^{2}g_{dd}(r) . (4.3)

We have to replace the S(k) in the denominator of Eq. (2.27) for delta n_{{k}vec} ( r vec_{{11}prime} )_ by the X_{B}(k)_.

The terms included in this calculation of X_{B}(k)_ are illustrated with diagrams in Fig. 10. The dashed dotted line in these diagrams represents the backflow correlation i eta (r) k vec cdot r vec (in Ref. 8 the backflow correlations are shown by wiggly lines which we have used here for denoting g_{wd}-1_ bonds). The backflow is around the particle in momentum state k vec. The exchange line starts and ends on the vertices representing this particle in PSI_{{k}vec}_ and PSI_{{k}vec}^{star_.}Thus all backflow correlations in PSI_{{k}vec}_ must start from the beginning of the exchange line, while those in PSI_{{k}vec}^{star_}must start from the end of the exchange line. A mark between the exchange and the backflow line is used to differentiate the backflow correlations from PSI_{{k}vec}_ and PSI_{{k}vec}^{star_.}If the mark is at the beginning of the exchange line the correlation is from PSI_{{k}vec}_, and if it is at the end then the correlation is from PSI_{{k}vec}^{star_.}The backflow correlation between two points i and j is +- i eta (r_{ij}) k vec cdot r vec_{ij_,}where i is the vertex at which the lines are marked, and the sign is positive (negative) when it is from PSI_{{k}vec}_ ( PSI_{{k}vec}^{star})_. We note that in the numerator diagrams with points 1 and 1 prime, the point 1 can only have backflow correlations from PSI_{{k}vec}_, and 1 prime from PSI_{{k}vec}^{star_.}

The diagrams of Fig. 11 illustrate the backflow contribution to [B_{1}]_ in the TB approximation. The terms of Fig. 11.1 give S_{B}(k) I bar_{9,2}(k)_, where

S_{B}(k) =S(k)[1+ cf12 I bar_{9,2}(k) ] . (4.4)

The terms of Fig. 11.2 give

S^{2}(k) "{" I bar_{9,2}(k) + case 1 over case 4 [ I bar_{9,2}(k) ]^{2}"}" , (4.5)

to which we add S^{2}(k) from the contribution with no backflow lines [Eq. (2.28)] to obtain [S_{B}(k)]^{2_.}The terms of Fig. 11.3 give 2I bar_{10,2}(k)_, while those of Fig. 11.4 together with the S^{2}(k) LAMBDA_{d}(k)_ from Eq. (2.28) give [S_{B}(k)]^{2}LAMBDA_{d}(k)_. Thus the 1+[B_{1}]_ in TB approximation is given by

1+ [B_{1}] = mark S_{B}(k) I bar_{9,2}(k)

lineup + S_{B^{2}(k)}[1+ LAMBDA_{d}(k) ] +2 I bar_{10,2}(k) . (4.6)

The backflow contribution to [B_{1^{sprime}]_}is illustrated in Fig. 12. The diagrams of Fig. 12.1 give cf12 S_{B}(k) I bar_{wd,2}(k)_, where

I bar_{wd,2}(k) =2 rho ^int^ i k vec cdot r vec eta (r) g_{wd}(r) e^{{i}k vec cdot r vec} d^{3}r . (4.7)

On adding the contribution of diagrams 12.2 to the g_{wd}(k) [1+g_{wd}(k)]_ from Eq. (2.30) we obtain

g_{wd}(k) [1+ cf12 I bar_{9,2}(k) ] L

times "{" 1+g_{wd}(k) [1+ cf12 I bar_{9,2}(k) ] + cf12 I bar_{dw,2}(k) "}" , R

In the TB approximation I bar_{dw,2}= I bar_{wd,2_}[Eq. (4.7)]. Diagrams of Fig. 11.4, with wiggly instead of the dashed lines from point 1, also contribute to [B_{1^{sprime}]_.}They convert the S^{2}(k) LAMBDA_{w}(k)_ of Eq. (2.30) to [S_{B}(k) ]^{2}LAMBDA_{w}(k)_. In all we obtain t_{1}(k) = 1 over {X_{B}(k)} (mark 2 g_{wd}(k) [1+ cf12 I bar_{9,2}(k)] "{" 1+g_{wd}(k) [1+ cf12 I bar_{9,2}(k) ] + cf12 I bar_{dw,2}(k) "}" (4.8)

lineup + S_{B}(k) [ I bar_{wd,2}(k) - I bar_{9,2}(k) ] -S_{B^{2}(k)}[1+ LAMBDA_{d}(k) -2 LAMBDA_{w}(k)] -2 I bar_{10,2}(k) ) . (4.9)

The backflow [C] diagrams that contribute to t_{2_}are shown in Fig. 13. By adding their contribution to that of diagrams without backflow lines [Eq. (2.34)] we obtain

t_{2}(k) = 1 over {X_{B}(k)} "{" 1+g_{wd}(k) [1+ cf12 I bar_{9,2}(k)] + cf12 I bar_{dw,2}"}"^{2

}

Similarly by adding the contribution of [C] diagrams in Fig. 14 to the t_{3_}of Eq. (2.35), we get (4.11) t_{3}(k) = 1 over {X_{B}(k)} "{" g_{wd}(k) [1+ cf12 I bar_{9,2}(k)] "}"^{2 }(4.10)

In the ETB approximation we add the contribution of diagrams with elements shown in Fig. 15.1 to all diagrams that contain the element shown in Fig. 15.2. The contribution of the elements of Fig. 15.1 is denoted by delta eta_{dd}(r)_:

delta eta_{dd}(r_{ij}) = rho over {r_{ij^{2}}^int}mark ^ d^{3}r_{ik}r vec_{ij}cdot r vec_{ik}eta (r_{ik}) g_{dd}(ik)

lineup times [g_{dd}(jk)-1- scrA ] . (4.12)

The elements delta eta_{dw}(r)_ and delta eta_{wd}(r)_ are shown in Figs. 15.3 and 15.5 We obtain

delta eta_{dw}(r_{i1}) = rho over {r_{il^{2}}^int}mark ^ d^{3}r_{ik}r vec_{il}cdot r vec_{ik}eta (r_{ik}) g_{dd}(ik)

lineup times [g_{wd}(lk) -1+ scrA ] , (4.13)

delta eta_{wd}(r_{li}) = rho over {r_{li^{2}}^int}mark ^ d^{3}r_{lk}r vec_{lk}cdot r vec_{li}eta ( r_{lk})g_{wd}(r_{lk})

lineup times [g_{dd}(ik)-1+ scrA ] . (4.14)

These are added to diagrams that have elements in Figs. 15.4 and 15.6, respectively.

The two body integrals I bar_{9,2}(k)_, I bar_{wd,2}(k)_, I bar_{dw,2}(k)_, I bar_{10,2}(k)_ modified by the delta eta contributions are, respectively, given by

I bar_{9}(k) =2 rho ^int^ d^{3}r i k vec cdot r vec e^{{i}k vec cdot r vec} "{" eta (r) g_{dd}(r) + delta eta_{dd}(r) [g_{dd}(r) -1 ] "}" , (4.15)

I bar_{wd}(k) =2 rho ^int^ d^{3}r i k vec cdot r vec e^{{i}k vec cdot r vec} "{" eta (r) g_{wd}(r) + delta eta_{wd}(r) [g_{wd}(r)-1] "}" , (4.16)

I bar_{dw}(k) =2 rho ^int mark ^ d^{3}r i k vec cdot r vec e^{{i}k vec cdot r vec} "{" eta (r) g_{wd}(r) + delta eta_{dw}(r) [g_{wd}(r)-1] "}" , (4.17)

I bar_{10}(k) = rho ^int^ d^{3}r g_{dd}(r) ( k vec cdot r vec )^{2}"{" eta (r) [ eta (r) + delta eta_{dd}(r)] -e^{{i}k vec cdot r vec} [ eta (r) + delta eta_{dd}(r) ]^{2}"}" . (4.18)

The calculated values of t_{{1}ndash 3}_ in the ETB approximation are listed in Table II. We see that the backflow correlations have a significant effect on the t_{2_}in the maxon region (k app 1 A ang^{-1}). We have not calculated the effect to backflow correlations on t_{4}( k vec , p vec )_.

At low (<1 K) temperatures liquid sup 4He can be thought of as a gas of noninteracting elementary excitations. Thus the momentum distribution of atoms in the liquid at low temperatures can be expressed as

n(T,p)= n_{0}(p)+ delta n(T,p) , (5.1)

delta n(T,p)= OMEGA over {(2 pi )^{3}}^int^ {d^{3}k} over {exp [ beta epsilon (k)]-1} delta n_{{k}vec} ( p vec ) , (5.2)

where epsilon (k) is the energy of the excitation of momentum k, and beta is the inverse temperature. In this temperature range only the excitations with k<0.2 A ang^{-1}are important, and the t_{4_}term of the delta n_{{k}vec} ( p vec )_ is unimportant. Neglecting it we obtain

delta n(T,p) = delta n_{1}(T,p)+ delta n_{2}(T,p) , L (5.3)

delta n_{1}(T,P)= 1 over {(2 pi )^{3}rho} n_{0}(p) ^int^ {d^{3}k} over {exp [ beta epsilon (k)]-1} t_{1}(k) , L (5.4)

delta n_{2}(T,p)= 1 over {(2 pi )^{3}rho} ^int mark ^ {d^{3}k} over {exp [ beta epsilon (k)] -1} L

lineup times [t_{2}(k) +t_{3}(k)] n_{0}(| p vec - k vec |) . (5.5)

We can further approximate the delta n(T,p) at small T by using the small-k limits discussed in Sec. II^B. This gives

delta n_{1}(T,P) mark = -1 over {(2 pi )^{3}rho} n_{0}(p) ^int^ {d^{3}k} over {exp ( beta hbar kc)-1} mc over {hbar k} L

lineup = -n_{0}( p vec ) T^{2}langle2 m over {12 rho hbar^{3}c} rangle2 == -n_{0}(p) (T/ T_{0})^{2} ,

L delta n_{2}(T,p) = 1 over {(2 pi )^{3}rho} ^int^ {d^{3}k} over {exp ( beta hbar kc)-1} mc over {hbar k} n_{0}(| k vec - p vec |) . (5.6)

The effect of temperature on the fraction of particles n_{c_}in the p=0 state is given by Eq. (5.6) as (5.8) n_{c}(T)= n_{c}(T=0)[1-(T/ T_{0})^{2}] , (5.7)

where T_{0_}is app7.6 K. The above equation has also been obtained by a phenomenological approach,sup 13 and from the structure of the perturbation theory at finite temperature.^{14}The results for the condensate fraction obtained with the epsilon (k) and the t_{1}(k)_ [Eq. (5.4)] are given in Table III. These are quite close to the asymptotic (T ->0) form (5.8), and fairly independent of the choice of the wave functions. They mostly depend upon the experimentally known S(k) and epsilon (k).

The delta function term rho n_{c}(0) delta (p)_ in n_{0}(p)_ gives rise to terms in delta n_{2}(T,p)_ that have singular behavior at p ->0. We denote by delta n_{s}(T,p)_ the contribution of this term to delta n_{2}(T,p)_:

delta n_{s}(T,p)= {n_{c}(0)mc} over {hbar p} 1 over {exp ( beta hbar pc)-1} . (5.9)

At small p ( beta hbar pc <<1) we can expand the Bose factor in powers of p:

1 over {exp ( beta hbar pc)-1} = 1 over {beta hbar pc} - 1 over 2 + ... , (5.10)

and obtain

delta n_{s}(T,p)= n_{c}(0) m over {hbar^{2}beta p^{2}}- cf12 n_{c}(0) mc over {hbar p} + ... . (5.11)

Recall that the n_{0}(p)_ has exactly the same 1/p term^{4}with positive sign. Thus at beta p hbar c <<1, the 1/p term in the n_{0}(p)_ is exactly cancelled by the 1/p term in delta n(T,P). Thus the total n(T,p) can be expected to exhibit the 1/p singularity only for beta hbar pc>1. This cancellation has recently been noted by Griffin.sup 15 The 1/p^{2}term in Eq. (5.11) has been discussed in Refs. 13 and 14.

The calculated values of the delta n(T,p !=0) are shown in Table IV. The Eqs. (5.3)ndash(5.5) are used to obtain these results, but they are not too sensitive to the deviations of t_{1-3}(k)_ or epsilon (k) from their low-k limits. The delta n(T,p !=0) is positive in this temperature range. Thus it appears that at low temperatures, atoms are removed from the p=0 condensate and put in states with epsilon (p) <= pi T. The momentum distribution at values of p such that epsilon (p) > pi T is unaffected by the thermal effects.

This work was supported by the U. S. Department of Energy, Division of Materials Sciences, under Grant No. DE AC02 767ER01198.

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- FIG. 1. Diagrammatic calculation of 1+ [B_{1}]_. The dashed lines represent g_{dd}-1_ bonds, the dots, internal points, and the little circles, the external point 1. The directed line represents the exp(i k vec cdot r vec_{mm})_.
- FIG. 2. Diagrams contributing to [B_{1^{sprime}]_.}The wiggly lines represent g_{wd}-1_ bonds, and the small circle denotes point 1. Note that in B_{1^{sprime_}diagrams,}the exchange lines can start from point 1 but not end in point 1.
- FIG. 3. Diagrams that contribute to t_{2_}and t_{3_.}The two small circles denote points 1 and 1 prime. Equations (2.34) and (2.35) are obtained by factorizing the diagrams as illustrated in (b).
- FIG. 4. Diagrams that give D( k vec , r vec_{{11}prime} )_ and D^{star}( k vec , r vec_{{11}prime} )_ contributions.
- FIG. 5. Some of the diagrams that contribute to D prime ( k vec , r vec_{{11}prime} )_.
- FIG. 6. S(k) and g_{wd}(k)_.
- FIG. 7. D bar_{1}(k,| k vec / 2- p vec |)_.
- FIG. 8. D bar_{2}(k,p)_.
- FIG. 9. t_{4}(k,p, CTHETA_{p_=0).}Total contributions of the D and D^{star}terms is shown by the dashed dotted line, and the dashed line gives that of the D prime term. The full line gives t_{4}(k,p, CTHETA_{p_=0).
- FIG. 10. Diagrammatic illustration of the calculation of X_{k}/ X_{0_}in the TB approximation.
- FIG. 11. Backflow diagrams in [B_{1}]_.
- FIG. 12. Backflow diagrams in [B_{1^{sprime}]_.
- FIG. 13. Backflow diagrams contributing to t_{2_.
- FIG. 14. Backflow diagrams contributing to t_{3_.
- FIG. 15. Elements 15.1, 15.3, and 15.5 represent the delta eta's. They can substitute the eta elements 15.2, 15.4, and 15.6 in TB\p diagrams. The exchange lines in elements 15.1, 15.2, and 15.3 can have any direction.

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