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Optimised Jastrow correlations for Fermi liquids
Volume 89B, number 3,4
PHYSICS LETTERS
28 January 1980
OPTIMISED JASTROW CORRELATIONS FOR FERMI LIQUIDS J.C. OWEN 1 The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen (~, Denmark Received 17 September 1979
An Euler-Lagrange equation for optimising the Jastrow correlations describing the ground state of a Fermi liquid is obtained and solved by preserving the long wavelength properties due to the exclusion principle (Fermi cancellations). For liquid 3He the calculated structure function is in reasonable agreement with experiment. Although the effect of the Fermi statistics is fairly small in this case it is suggested it will be much more significant for nuclear matter.
The Jastrow product ansatz for the ground state wavefunction for liquid 3He and liquid 4He is a useful nonperturbative starting point for the description of the ground states of these liquids [1-3]. For the Bose liquid 4He a functional variation of the energy evaluated in the hypernetted chain approximation (HNC) gives reasonable values for the energy and compressibility as well as providing an indication of instability against density fluctuations at low density and evidence for solid-like structure at high density [4,5]. Progress for liquid 3He has been slower, however, since the exclusion principle has been shown to lead to an important long wavelength cancellation phenomenon which cannot be included within the HNC approximation [6]. Nevertheless results have been obtained for a functional variation of the FermiHNC equations by ignoring the Fermi cancellations [7,8]. In this letter we report the results of solving an Euler-Lagrange equation for liquid 3He which includes both the many-body effects of HNC structure which are important for 4He and the Fermi cancellations due to the exclusion principle. It is shown that the equation has solutions only for a restricted region in the density centered around the equilibrium density for the liquid. At low densities the point at which the equation ceases to have solutions corresponds closely to the point at which the calculated in1 Present address: Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, UK.
compressibility would become negative. The fact that the optimisation of the correlation function cannot be extended smoothly into the low density region is a nice illustration of the fact that a self-bound liquid should not be regarded as a weakly interacting system. The Euler-Lagrange equation may be divided into boson-like and fermion-like terms. For liquid 3He at its experimental equilibrium density the boson-like terms dominate which explains why the neglect of the Fermi cancellation phenomenon has led to reasonable results at this density [8]. As the density is lowered, however, the fermion-like terms become increasingly important. Preliminary calculations for nuclear matter using simple nucleon-nucleon potentials indicate that here the boson-like and fermionlike terms are of comparable magnitude and opposite sign and so a correct treatment of the Fermi cancellations may be essential. A more realistic treatment of nuclear matter, however, will demand a generalisation to include spin-dependent correlations [9]. Let us define a trial variational wavefunction of the form I~) = I-] f(rij)l~) , i<]
(1)
where I¢) is the many-body wavefunction for the noninteracting system. A simple integration by parts in the kinetic energy allows us to write the energy expectation value in the so-called Jackson-Feenberg form [1,2,10] 303
Volume 89B, number 3,4
PHYSICS LETTERS
3h 2
28 January 1980
where
--lOre
~
_
X'(k)=X'dd(k)+2 +5
~
(h2/4m)k2(S(k) - 1) + S'(k) = 0 ,
(3)
where S(k) (the liquid structure function) is unity plus the ouner transform ofg(r) - 1 and S (k) is the result of varying In f2(r) in the distribution functions [6] (the present derivation differs from that of ref. [6] only in that we use the JF form for the energy). The quantity S'(k) is most easily constructed using diagrammatic expansion techniques [1,2]. We take any diagram which contributes to the energy and replace in turn each correlation line f2(r) - 1 joining a pair of points by f2(r) and take these points as external. This is equivalent to taking all diagrams contributing to S(k) and replacing in turn exactly one correlation line by the effective interaction ~t
Veff(r) = (-(h2/4m)V21n f2(r) + V(r))f2(r),
o
(4)
plus a corresponding replacement of exchange lines for the final term in eq. (2). The quantity S"(k) is a sum of both nodal diagrams (also called chain or ring diagrams) and non-nodal diagrams. Using a s~aightforward diagrammatic analysis the quantity S '(k) may be expressed in terms of S(k) and purely non-nodal diagrams. Taking care of the rules for constructing diagrams which involve exchange lines we obtain the following result
"ff'(k) = S2(k)X'(k) ,
304
+X--~) Xde(k)
(5)
(6)
1 + Xee(k ) ]
(2)
where g(r) is the two-body radial distribution function, g(l)(r) is the one-body distribution function and in the final term it is understood that the V2 operator acts only on the exchange lines which occur in the diagrammatic expansion ofg(t)(r) and not on the correlation lines. Now a functional variation of In f2(r) will produce two terms, one from the explicit occurrence of In if(r) in the second term of eq. (2) and one from the implicit occurrence of In f2(r)in the distribution functions g(r) and g(l)(r). Performing this variation and Fourier transforming leads immediately to the Euler-Lagrange equation
'
1
In ]'2(r) + V(r g(r) dr
~2 f v2g(l)(rl)drl + 8--m-~
F
(1 - Xde(k)~ ~,
Here ~'dd, "Yde and "~ee are non-nodal diagrams with exchange lines joined to neither, one and both of their external points, respectively. Xdd , Xde and Xee are corresponding non-nodal diagrams containing the effective interaction of eq. (4) exactly once. If we ignore he lmpllcxt dependence of X (k) on S(k) then eqs. (3) and (5) constitute a quadratic equation for S(k). The usual condition for the existence of real roots implies the restriction t
'
•
•
~W
16mX'(k)/hk 2 + 1 > 0 ,
(7)
which together with the condition that S(k) ~ 1 for k -+ oo implies 0 < S(k) < 2. Krotscheck [6] shows that due to the Fermi cancellations ~'de(k), X'de(k), Q.,+.~ee(k)) ~ k and ~'~e(k) "~ k 2 for k ~ 0. Thus X (k) is finite for k -+ 0 and S()k ~ k(h 2/4m.~'()_k)1/2 for k -'- 0 Of the three contributions to X', the term Xdd is present for both bosons and fermions whereas the remaining two terms represent pure fermion effects. If the quantities Xde and Xee are evaluated m the FHNC approximation of Fantoni and Rosati [10], they remain finite for k ~ 0 and the effect of the Fermi statistics on the structure function is determined by the error in the FHNC approximation, rather than by the ratio of the leading terms. Thus it is important to maintain the Fermi cancellations if the effect of the Fermi statistics on the structure function is to be studied. The Euler-Lagrange equation (3), (5) and (6) was solved by evaluating X~d in the hypernetted chain approximation as a function ofgB(r) (where gB(r) is the contribution to g(r) of those diagrams which have no exchange lines attached to their endpoints), and "~de, X~e etc. The quantities Xde, X~Ie, Xee, X~e , were in turn evaluated self-consistently in terms of gB (r) so as to ensure the correct small k behaviour due to the Fermi cancellations. Thus it is necessary to include diagrams which have an elementary (i.e. non-hypernetted chain) structure. The structure of the diagrams which were included in Xee is shown in fig. 1. They •
.
~¢
!
.
Volume 89B, number 3,4
PHYSICS LETTERS
28 January 1980 I
r
I
1.2 /
~
0.8
(~2/m)k2(~B/2 --
1) + SF(k)Vgl/'~2(k) = D(k)
,
(8)
where SF(k ) is the structure function for the gas of non-interacting fermions and D(k) contains only terms which are at least quadratic in gl/2. Eq. (8) has the form of a state averaged Bethe-Goldstone equa. tion where S F plays the role of a state averaged Pauli exclusion operator. The essential point is that both SF(k ) "-" k and D(k) "" k for small k and so the exclusion principle term alone can ensure the "healing" g~/2 _ 1 ~ k - 1 for small k. The driving term represents the many-body effects due to interactions with other particles in the medium. Although small in the low density limit [11 ] it is large in the region of the saturation density. It depends only weakly on the precise form ofglB/2,'however, and iterations of this equation converge rapidly. In practice, it is convenient to write eq. (8) in the form of a linear integral equation for gl/2(r) which is solved by matrix inversion at each iteration step. The results of solving eqs. (3), (5) and (6) for liquid 3 He using the Lennard-Jones potential are shown in fig. 2 and table 1. Fig. 2 shows the calculated liquid structure function at the density p = 0.0164 A - 3 as compared with the structure function as measured ex. perimentally at a temperature of 0.56 K [12]. The agreement between the slopes of the calculated and measured functions as k ~ 0 is reasonable. The first peak in the calculated S(k) is rather broader and less
i
J
I
i
"" ...
~/.//
O.4
0.0
were evaluated by numerical integration in momentumspace. Diagrams of a corresponding structure were included m Xde, Xde and Xee. Eqs. (3), (5) and (6) are thus cast into the form o f a non-linear equation for gB(r).This was solved by picking out the linear terms in g"~T2(r)- 1 (but always keeping a factor ofgl/2(r) to cancel the repulsive core in the potential) and writing the equations in the following form
"-.
I
7
U')
Fig. 1. The diagrams which were included in the evaluation of Xee. The dashed line represents a renormalised correlation line gB(r) - l.
!i ~
P
L
J~
o
'
(~K' k ) Fig. 2. The liquid structure function S(k) for liquid 3He at p = 0.0164 A -3. The calculated function (solid curve) is compared with the function as measured experimentally at T = 0.56 K (dashed curve). pronounced than in the experimental function. The binding energy o f - 0 . 9 4 K at a density o f p = 0.012 A - 3 is close to other variational calculations [8] but rather far from the experimentally measured numbers of - 2 . 5 3 A - 3 at p = 0.0164 A - 3 . Part of this discrepancy is due to the neglected elementary diagrams in our formalism but mostly it is due to the inadequacy of the Jastrow ansatz [3]. Table 1 also shows the incompressibility (calculated as C = (d/dp)(p2dE/dp)), the value of.~'(0) which determines the slope of S(k) as k ~ 0 and the value o f . ~ d ( 0 ) which gives the contribution of boson-like terms to X'(0). At p = 0.0164 A - 3 , 70% of the value of.~'(0) ~t is contributed by Xdd(0 ). This explains why the mass3 boson system gives a reasonable approximation to the distribution function for 3He fermions and also why a variation of the FHNC equations, ignoring the Fermi cancellations, gives reasonable results [8]. As the density is lowered the relative contribution from Xdd(0 ) becomes rapidly smaller and It is clear that a Table 1 Calculated properties of liquid 3He as a function of density. p
(A -3)
0.0080 0.0100 0.0120 0.0140 0.0164
E/N(K) C (K) (h2/m)~'(0)
(l~2/m)X'dd(O)
-0.74 0.08 -0.04
-0.89 1.5 0.18 0.03
-0.94 4.0 0.32 0.13
-0.88 8.4 0.51 0.30
-0.63 0.81 0.56
305
Volume 89B, number 3,4
PHYSICS LETTERS
reliable treatment of the fermion-like contributions is essential. For p < 0.008 A -3, X'(0) becomes negative and the Euler-Lagrange equation no longer has a solution. This happens in the region where the incompressibility would become negative. At high densities, the first peak in the calculated S(k) becomes larger and larger and passes through 2 for p ~ 0.034 ~ - 3 . For larger densities the Euler-Lagrange equation once again ceases to have solutions. In this region the numerical methods and the HNC-type of approximations which were employed become unreliable, however, and this result should be interpreted merely as a qualitative indication for the existence of a density about which the Euler-Lagrange equation does not possess solutions. A similar phenomenon has been observed for Bose systems [5]. The equations which we have solved reduce to the optimised HNC equations in the case of Bose statistics. However, in order to incorporate the effects of the Fermi statistics correctly it is necessary to abandon the "clean" approach of writing down an approximate energy functional (for example in the I INC approximation) and performing an exact variation on this approximate functional. Instead, a variation is performed on the exact energy functional and then approximations are introduced into the Euler-Lagrange equation. There is therefore an inconsistency between the energy functional and the Euler-Lagrange equation which will be small provided that the diagrams which are negelcted in the fermion-like terms Xde etc. are small. This inconsistency will probably get worse with increasing density and so, for example, the evaluation of the incompressibility by differentiating the approximated energy may become unreliable. One approach might be to invent approximate methods to
306
28 January 1980
treat the elementary diagrams involved in the Fermi cancellations through all orders [2,13]. Alternatively, we could perform the differentiations on the exact energy functional and then introduce approximations. This leads us directly into the evaluation of Landau's Fermi liquid parameters and work in this direction is in progress. Many thanks are due to E. Krotscheck and A. Jackson for stimulating discussions. The financial support of the Royal Society and the Danish National Science Research Council in the form of a fellowship within the European Science Exchange Program, and the hospitality of the Niels Bohr Institute, are gratefully acknowledged.
References [ 1] J.W. Clark, Progressin particle and nuclear physics (Pergamon, Oxford, 1978), to be published. [2] J.G. Zabolitzky, Adv. Nucl- Phys. (1979), to be published. [3] V.R. Pandharipande and R.B. Wiringa, Rev. Mod. Phys. (1979), to be published. [4] A.D. Jackson, A. Land4 and L.J. Lantto, Nucl. Phys. A317 (1979) 70. [5] L. Castillejo, A.D. Jackson, B.K. Jennings and R.A. Smith, Phys. Rev. B., to be published. [6] E. Krotscheck, Phys. Rev. A15 (1977) 397. [7] L.J. Lantto and P.J. Siemens, Nucl. Phys. A317 (1979) 55. [8] L.J. Lantto, Nucl. Phys. A, to be published. [9] J.C. Owen, Ann. Phys. 118 (1979) 373. [10] S. Fantoni and S. Rosati, Nuovo Cimento 25 A (1975) 593. [ll] J.C. Owen, Phys. Lett. 82B (1979) 23. [12] E. Achter and L. Meyer, Phys. Rev. 188 (1969) 29l. [13] E. Krotscheck, Nucl. Phys. A317 (1979) 149.
PHYSICS LETTERS
28 January 1980
OPTIMISED JASTROW CORRELATIONS FOR FERMI LIQUIDS J.C. OWEN 1 The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen (~, Denmark Received 17 September 1979
An Euler-Lagrange equation for optimising the Jastrow correlations describing the ground state of a Fermi liquid is obtained and solved by preserving the long wavelength properties due to the exclusion principle (Fermi cancellations). For liquid 3He the calculated structure function is in reasonable agreement with experiment. Although the effect of the Fermi statistics is fairly small in this case it is suggested it will be much more significant for nuclear matter.
The Jastrow product ansatz for the ground state wavefunction for liquid 3He and liquid 4He is a useful nonperturbative starting point for the description of the ground states of these liquids [1-3]. For the Bose liquid 4He a functional variation of the energy evaluated in the hypernetted chain approximation (HNC) gives reasonable values for the energy and compressibility as well as providing an indication of instability against density fluctuations at low density and evidence for solid-like structure at high density [4,5]. Progress for liquid 3He has been slower, however, since the exclusion principle has been shown to lead to an important long wavelength cancellation phenomenon which cannot be included within the HNC approximation [6]. Nevertheless results have been obtained for a functional variation of the FermiHNC equations by ignoring the Fermi cancellations [7,8]. In this letter we report the results of solving an Euler-Lagrange equation for liquid 3He which includes both the many-body effects of HNC structure which are important for 4He and the Fermi cancellations due to the exclusion principle. It is shown that the equation has solutions only for a restricted region in the density centered around the equilibrium density for the liquid. At low densities the point at which the equation ceases to have solutions corresponds closely to the point at which the calculated in1 Present address: Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, UK.
compressibility would become negative. The fact that the optimisation of the correlation function cannot be extended smoothly into the low density region is a nice illustration of the fact that a self-bound liquid should not be regarded as a weakly interacting system. The Euler-Lagrange equation may be divided into boson-like and fermion-like terms. For liquid 3He at its experimental equilibrium density the boson-like terms dominate which explains why the neglect of the Fermi cancellation phenomenon has led to reasonable results at this density [8]. As the density is lowered, however, the fermion-like terms become increasingly important. Preliminary calculations for nuclear matter using simple nucleon-nucleon potentials indicate that here the boson-like and fermionlike terms are of comparable magnitude and opposite sign and so a correct treatment of the Fermi cancellations may be essential. A more realistic treatment of nuclear matter, however, will demand a generalisation to include spin-dependent correlations [9]. Let us define a trial variational wavefunction of the form I~) = I-] f(rij)l~) , i<]
(1)
where I¢) is the many-body wavefunction for the noninteracting system. A simple integration by parts in the kinetic energy allows us to write the energy expectation value in the so-called Jackson-Feenberg form [1,2,10] 303
Volume 89B, number 3,4
PHYSICS LETTERS
3h 2
28 January 1980
where
--lOre
~
_
X'(k)=X'dd(k)+2 +5
~
(h2/4m)k2(S(k) - 1) + S'(k) = 0 ,
(3)
where S(k) (the liquid structure function) is unity plus the ouner transform ofg(r) - 1 and S (k) is the result of varying In f2(r) in the distribution functions [6] (the present derivation differs from that of ref. [6] only in that we use the JF form for the energy). The quantity S'(k) is most easily constructed using diagrammatic expansion techniques [1,2]. We take any diagram which contributes to the energy and replace in turn each correlation line f2(r) - 1 joining a pair of points by f2(r) and take these points as external. This is equivalent to taking all diagrams contributing to S(k) and replacing in turn exactly one correlation line by the effective interaction ~t
Veff(r) = (-(h2/4m)V21n f2(r) + V(r))f2(r),
o
(4)
plus a corresponding replacement of exchange lines for the final term in eq. (2). The quantity S"(k) is a sum of both nodal diagrams (also called chain or ring diagrams) and non-nodal diagrams. Using a s~aightforward diagrammatic analysis the quantity S '(k) may be expressed in terms of S(k) and purely non-nodal diagrams. Taking care of the rules for constructing diagrams which involve exchange lines we obtain the following result
"ff'(k) = S2(k)X'(k) ,
304
+X--~) Xde(k)
(5)
(6)
1 + Xee(k ) ]
(2)
where g(r) is the two-body radial distribution function, g(l)(r) is the one-body distribution function and in the final term it is understood that the V2 operator acts only on the exchange lines which occur in the diagrammatic expansion ofg(t)(r) and not on the correlation lines. Now a functional variation of In f2(r) will produce two terms, one from the explicit occurrence of In if(r) in the second term of eq. (2) and one from the implicit occurrence of In f2(r)in the distribution functions g(r) and g(l)(r). Performing this variation and Fourier transforming leads immediately to the Euler-Lagrange equation
'
1
In ]'2(r) + V(r g(r) dr
~2 f v2g(l)(rl)drl + 8--m-~
F
(1 - Xde(k)~ ~,
Here ~'dd, "Yde and "~ee are non-nodal diagrams with exchange lines joined to neither, one and both of their external points, respectively. Xdd , Xde and Xee are corresponding non-nodal diagrams containing the effective interaction of eq. (4) exactly once. If we ignore he lmpllcxt dependence of X (k) on S(k) then eqs. (3) and (5) constitute a quadratic equation for S(k). The usual condition for the existence of real roots implies the restriction t
'
•
•
~W
16mX'(k)/hk 2 + 1 > 0 ,
(7)
which together with the condition that S(k) ~ 1 for k -+ oo implies 0 < S(k) < 2. Krotscheck [6] shows that due to the Fermi cancellations ~'de(k), X'de(k), Q.,+.~ee(k)) ~ k and ~'~e(k) "~ k 2 for k ~ 0. Thus X (k) is finite for k -+ 0 and S()k ~ k(h 2/4m.~'()_k)1/2 for k -'- 0 Of the three contributions to X', the term Xdd is present for both bosons and fermions whereas the remaining two terms represent pure fermion effects. If the quantities Xde and Xee are evaluated m the FHNC approximation of Fantoni and Rosati [10], they remain finite for k ~ 0 and the effect of the Fermi statistics on the structure function is determined by the error in the FHNC approximation, rather than by the ratio of the leading terms. Thus it is important to maintain the Fermi cancellations if the effect of the Fermi statistics on the structure function is to be studied. The Euler-Lagrange equation (3), (5) and (6) was solved by evaluating X~d in the hypernetted chain approximation as a function ofgB(r) (where gB(r) is the contribution to g(r) of those diagrams which have no exchange lines attached to their endpoints), and "~de, X~e etc. The quantities Xde, X~Ie, Xee, X~e , were in turn evaluated self-consistently in terms of gB (r) so as to ensure the correct small k behaviour due to the Fermi cancellations. Thus it is necessary to include diagrams which have an elementary (i.e. non-hypernetted chain) structure. The structure of the diagrams which were included in Xee is shown in fig. 1. They •
.
~¢
!
.
Volume 89B, number 3,4
PHYSICS LETTERS
28 January 1980 I
r
I
1.2 /
~
0.8
(~2/m)k2(~B/2 --
1) + SF(k)Vgl/'~2(k) = D(k)
,
(8)
where SF(k ) is the structure function for the gas of non-interacting fermions and D(k) contains only terms which are at least quadratic in gl/2. Eq. (8) has the form of a state averaged Bethe-Goldstone equa. tion where S F plays the role of a state averaged Pauli exclusion operator. The essential point is that both SF(k ) "-" k and D(k) "" k for small k and so the exclusion principle term alone can ensure the "healing" g~/2 _ 1 ~ k - 1 for small k. The driving term represents the many-body effects due to interactions with other particles in the medium. Although small in the low density limit [11 ] it is large in the region of the saturation density. It depends only weakly on the precise form ofglB/2,'however, and iterations of this equation converge rapidly. In practice, it is convenient to write eq. (8) in the form of a linear integral equation for gl/2(r) which is solved by matrix inversion at each iteration step. The results of solving eqs. (3), (5) and (6) for liquid 3 He using the Lennard-Jones potential are shown in fig. 2 and table 1. Fig. 2 shows the calculated liquid structure function at the density p = 0.0164 A - 3 as compared with the structure function as measured ex. perimentally at a temperature of 0.56 K [12]. The agreement between the slopes of the calculated and measured functions as k ~ 0 is reasonable. The first peak in the calculated S(k) is rather broader and less
i
J
I
i
"" ...
~/.//
O.4
0.0
were evaluated by numerical integration in momentumspace. Diagrams of a corresponding structure were included m Xde, Xde and Xee. Eqs. (3), (5) and (6) are thus cast into the form o f a non-linear equation for gB(r).This was solved by picking out the linear terms in g"~T2(r)- 1 (but always keeping a factor ofgl/2(r) to cancel the repulsive core in the potential) and writing the equations in the following form
"-.
I
7
U')
Fig. 1. The diagrams which were included in the evaluation of Xee. The dashed line represents a renormalised correlation line gB(r) - l.
!i ~
P
L
J~
o
'
(~K' k ) Fig. 2. The liquid structure function S(k) for liquid 3He at p = 0.0164 A -3. The calculated function (solid curve) is compared with the function as measured experimentally at T = 0.56 K (dashed curve). pronounced than in the experimental function. The binding energy o f - 0 . 9 4 K at a density o f p = 0.012 A - 3 is close to other variational calculations [8] but rather far from the experimentally measured numbers of - 2 . 5 3 A - 3 at p = 0.0164 A - 3 . Part of this discrepancy is due to the neglected elementary diagrams in our formalism but mostly it is due to the inadequacy of the Jastrow ansatz [3]. Table 1 also shows the incompressibility (calculated as C = (d/dp)(p2dE/dp)), the value of.~'(0) which determines the slope of S(k) as k ~ 0 and the value o f . ~ d ( 0 ) which gives the contribution of boson-like terms to X'(0). At p = 0.0164 A - 3 , 70% of the value of.~'(0) ~t is contributed by Xdd(0 ). This explains why the mass3 boson system gives a reasonable approximation to the distribution function for 3He fermions and also why a variation of the FHNC equations, ignoring the Fermi cancellations, gives reasonable results [8]. As the density is lowered the relative contribution from Xdd(0 ) becomes rapidly smaller and It is clear that a Table 1 Calculated properties of liquid 3He as a function of density. p
(A -3)
0.0080 0.0100 0.0120 0.0140 0.0164
E/N(K) C (K) (h2/m)~'(0)
(l~2/m)X'dd(O)
-0.74 0.08 -0.04
-0.89 1.5 0.18 0.03
-0.94 4.0 0.32 0.13
-0.88 8.4 0.51 0.30
-0.63 0.81 0.56
305
Volume 89B, number 3,4
PHYSICS LETTERS
reliable treatment of the fermion-like contributions is essential. For p < 0.008 A -3, X'(0) becomes negative and the Euler-Lagrange equation no longer has a solution. This happens in the region where the incompressibility would become negative. At high densities, the first peak in the calculated S(k) becomes larger and larger and passes through 2 for p ~ 0.034 ~ - 3 . For larger densities the Euler-Lagrange equation once again ceases to have solutions. In this region the numerical methods and the HNC-type of approximations which were employed become unreliable, however, and this result should be interpreted merely as a qualitative indication for the existence of a density about which the Euler-Lagrange equation does not possess solutions. A similar phenomenon has been observed for Bose systems [5]. The equations which we have solved reduce to the optimised HNC equations in the case of Bose statistics. However, in order to incorporate the effects of the Fermi statistics correctly it is necessary to abandon the "clean" approach of writing down an approximate energy functional (for example in the I INC approximation) and performing an exact variation on this approximate functional. Instead, a variation is performed on the exact energy functional and then approximations are introduced into the Euler-Lagrange equation. There is therefore an inconsistency between the energy functional and the Euler-Lagrange equation which will be small provided that the diagrams which are negelcted in the fermion-like terms Xde etc. are small. This inconsistency will probably get worse with increasing density and so, for example, the evaluation of the incompressibility by differentiating the approximated energy may become unreliable. One approach might be to invent approximate methods to
306
28 January 1980
treat the elementary diagrams involved in the Fermi cancellations through all orders [2,13]. Alternatively, we could perform the differentiations on the exact energy functional and then introduce approximations. This leads us directly into the evaluation of Landau's Fermi liquid parameters and work in this direction is in progress. Many thanks are due to E. Krotscheck and A. Jackson for stimulating discussions. The financial support of the Royal Society and the Danish National Science Research Council in the form of a fellowship within the European Science Exchange Program, and the hospitality of the Niels Bohr Institute, are gratefully acknowledged.
References [ 1] J.W. Clark, Progressin particle and nuclear physics (Pergamon, Oxford, 1978), to be published. [2] J.G. Zabolitzky, Adv. Nucl- Phys. (1979), to be published. [3] V.R. Pandharipande and R.B. Wiringa, Rev. Mod. Phys. (1979), to be published. [4] A.D. Jackson, A. Land4 and L.J. Lantto, Nucl. Phys. A317 (1979) 70. [5] L. Castillejo, A.D. Jackson, B.K. Jennings and R.A. Smith, Phys. Rev. B., to be published. [6] E. Krotscheck, Phys. Rev. A15 (1977) 397. [7] L.J. Lantto and P.J. Siemens, Nucl. Phys. A317 (1979) 55. [8] L.J. Lantto, Nucl. Phys. A, to be published. [9] J.C. Owen, Ann. Phys. 118 (1979) 373. [10] S. Fantoni and S. Rosati, Nuovo Cimento 25 A (1975) 593. [ll] J.C. Owen, Phys. Lett. 82B (1979) 23. [12] E. Achter and L. Meyer, Phys. Rev. 188 (1969) 29l. [13] E. Krotscheck, Nucl. Phys. A317 (1979) 149.