REFE1 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS IC/T3/91 THEORY OF A DENSE STRONGLY INTERACTING FERMI LIQUID AT 0 K. R.

Transcription

1 REFE1 IC/T3/91 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THEORY OF A DENSE STRONGLY INTERACTING FERMI LIQUID AT 0 K R. Rajaraman INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1973 MIRAMARE-TRIESTE

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3 IC/73/91 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOE THEORETICAL PHYSICS THEORY OF A DENSE STRONGLY INTERACTING FERMI LIQUID AT 0 K * E. Rajaraman ** International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT A beginning is made towards constructing a microscopic theory for the ground-state energy of a Fermi liquid, which has "both high density as veil as strong interparticle forces. No such microscopic theory, starting from the interparticle potential, exists at present, although 3 h such important systems as neutron star interiors, liquids He and He form examples of such liquids. Since expansions neither in powers of the density nor the potential will converge fast, n-body clusters must be evaluated for all n and to all orders in the potential. Working within a simplified model, we provide a method for evaluating n-body energies for any general n, Our method is far simpler than more direct methods employed in the literature which can be used only for low values of n. We also verify and illustrate our method by comparison with these existing calculations for n = 2, 3, and k. A discussion is given of the remaining steps to be taken. The paper is only concerned with the normal ground state - possible "pairing" and "super" properties are outside its scope. * To be submitted for publication. MIRAMARE - TRIESTE August 1973 ** Address after 15 September 1973: Institute for Advanced Study, Princeton, NJ 0851*0, USA. On leave of absence from University of Delhi, Delhi 7, India.

4 I, INTRODUCTION Considerable progress has been made in the 1 last two decades in calculating the properties of certain quantum liquids starting from "first principles", i.e. the interparticle potential and the many-body Schrodinger equation. Such success has so far been limited to systems which have low density, weak forces, or both. Thus, work on quantum gases with hard 2) sphere interactions has been possible because the density there is low. Similarly, the nuclear matter system, which involves a strong potential but whose density is not too large, has been satisfactorily dealt with The repulsive core potential is handled by iterating it to all orders and obtaining a "reaction matrix" in closed form. The resulting sequence of two-body energy, three-body energy, etc. can then be expected to converge fast since nuclear matter *) 2r density is not too large. Numerical calculations 3) support this expectation. Turning to systems with high density but weak forces, as in a dense electron gas, here again there exist satisfactory microscopic theories. The so-called "random phase approximation" (ring-diagram BUtmnation) can be employed. ' This is effectively an expansion in powers of Tp, where T is the interaction strength, and this expansion is summed in closed form. Terms in higher order in T for a given power of p may be neglected if F is small (weak forces). Finally, let us come to systems of interest in this paper, viz, those with both strong forces and high density. Important examples of such systems are pulsar interiors, which are widely considered to be dense hadron matter with densities as high as ten times nuclear density, and also 3 k liquids He J and He. A great deal of theoretical analysis can and has been done for liquid helium without starting from the interparticle potential. Landau's theory and the large body of work along similar lines comprise one example ; so do variational calculations which have yielded very good equations of state both for liquid helium and the neutron star interior. However, *) By the phrase "density", we refer here not to the actual density p 3 but to the factor pr where R is a typical range of forces. Similarly, the terms "weak" and "strong" forces are not meant in the technical sense of elementary-particle physics, but in the sense of whether the potential is small.enough compared to typical excitation energies for perturbation theory to converge, or not. Thus, the interatomic force in liquid helium, while electromagnetic in nature, will be termed "strong" by us

5 with due deference to the success of theae theories, they are relatively phenomenological In comparison to a theory which, can derive tha vave function, energy and other properties from the Schrb'dinger equation. attempt to evolve even approximately the latter' type of theory from first principles can, apart from its theoretical importance and challenge, also lead to the derivation and improvement of the parameters and trial functions which go into the former type of theories. Indeed, this was the "basic motivation behind the study of nuclear matter vhen the shell model had already explained so many features of nuclei. However, unlike the cases of nuclear matter or the electron gas, the calculation of the energy of a dense and strongly interacting system starting from the Hamiltonian presents serious difficulties even at 0 K. For such systems, neither a perturbation expansion, nor a density expansion, can a priori be expected to converge. The strong repulsive core can, of course, he treated by using Brueckner's reaction matrix. and other nuclear matter techniques, some workers An Using this method have calculated tvoand three-body energies of liquid He at 0 K. But there is no justification for the neglect of n-body clusters of larger n (higher powers of p ), given that the density is high. In fact, even the partial sum of ring (polarization-loop) diagrams implied in the "random phase approximation" is not sufficent when the interaction is strong. In the language of Goldstone diagrams, which provides a systematic way of enumerating perturbation contributions to the vave function, all n-hole line diagrams for large n will have to be included. It is very difficult even to enumerate the different classes of n-body diagrams,let alone sum them. This is clear from Day's painstaking analysis T) of four-hole line diagrams, where he identifies 16 classes of diagrams, each class consisting of infinite members and having to be summed separately. It is estimated that five-hole line diagrams may come in over 100 classes. Such complexity will increase manyfold as n increases further. Consequently ft \ there is no systematic method at present for evaluating the ground state energy or the vave function of dense systems with strong forces, in terms of the potential or, equivalently, Brueckner's reaction matrix. The purpose of this paper is to make a beginning towards dealing with this problem, using techniques already developed in nuclear matter theory. These techniques have yielded, after considerable effort, satisfactory estimates of two-, three- and four-body energies. We suggest a method of estimating n-body contributions for arbitrary n. We do this by working within a simpler model, designed to eliminate some secondary complications - 3 -

6 that arise in real Fermi liquids, vhile at the same time retaining the eceential difficulty of high. density coupled with etreng foreeb. Sugh a model has been used "by other people in the past. ' 9) Our method is "based on an important result proved by Day. He shows that certain sequences of Goldstone diagrams can be added up to yi eld a simple Jastrow type of contribution to the vave function. Unfortunately, the diagrams not present in these sequences outnumber those that are, so that Day's Jastrow function is hot manifestly a good approximation to the correct ground state wave function. This is discussed in Sec.III. We find, however, that when Day's idea is suitably adapted to the energy expansion rather than the wave function, one can sum a vast majority of diagrams into integrals over Jastrow functions. There are only three exceptional classes to this rule, which must be estimated separately. Considering the hundreds of types of diagrams that exist for n > 5, this is a great simplification. In Sec.II we present the model and its advantages. In Sec.Ill our method for summing n-body clusters is discussed. In Sec.IV we illustrate and verify our result by explicitly checking it with known contributions of two-,three- and four-body clusters. Even within the model, our evaluation of the total BE from the sum of n-body clusters is far from complete. Sec.V discusses the remaining steps to be taken and the inadequacies of our method. We employ Goldstone s linked cluster expansion in diagrams as our theoretical foundation and freely use some successful techniques developed in nuclear matter theory. Familiarity on the reader's part with these methods is assumed. II. THE MODEL We work in a well-known model used for simulating bosonic behaviour in fermion systems. The model is defined as follows: i ) There are N f ermions, with N + OO. ii) The spin s of these fermions is very large (s >: CN-l)/2) SO that each fermion can have a spin component s different from the others. Thus, in the absence of any forces, they can all have zero momentum in the ground state. - k -

7 iii) The interparticle forces are apin-independent. this, there are no restrictions on the potential v(r.»l. Apart from iv) The systems we are interested in (liquid He, He, neutron matter) all have a strong repulsive core followed by an attractive well. The longrange tail of these potentials is weak and poses no problems. It can in principle be treated perturbatively, or by RPA. Therefore, let us take for simplicity a potential v(r) with a hard core up to r a c, followed by an attractive well cut off at r - d, (see Fig.la). The exact value of d is not critical, as long as the potential that exists beyond r = d in real systems is weak enough. neglected. v) The actual density of the particles can be high, i.e. pd > 1. vi). Three-body forces and more complicated many-body forces are These are not expected to be significant in atomic systems such as liquid helium. In neutron matter, they are large, but it has been shown 12) that even here the presence of tvo-body forces suppresses their effect vii) The Hamiltonian H i3 divided into hi - y < *>. *- v.. ' H. * y v.. -. r v. ;*i i * (1) The single-particle potential V. added and subtracted from H can be chosen at will. We take it as V, = 0 for "particle states" - - A for "hole states", (2) where particles and holes are as defined by Goldstone. We work strictly at 0 K, which is hard enough, and are therefore interested in the ground state of the Hamiltonian. Let us examine the consequences of these specifications. Except for ii) and iii), the other assumptions are clearly valid even for the real systems of interest. The problem we are trying to solve - of high density on the one hand and a strong potential on the other - certainly exists, and all n-body cluster terms for large n will have to be included. However, some simplifications arise because of ii) and iii), as can be seen below

8 The techniques of the Brueckner-Goldstone-Bethe theory of nuclear matter are clearly applicable here - some even more BO. In the reference spectrum, method used in nuclear matter, integration over particle state momenta is formally extended all the way down to zero, therby neglecting the Pauli exclusion operator. This approximation is even "better in our model, since the Fermi momentum of the -unperturbed ground state $) is zero here, thanks to the large value of the "spin" s. The energy gap A between particles and holes prevents zero energy denominators as in the reference spectrum method. The "binding energy (BE) and the exact ground state Y} of H can "be expanded into the usual linked Goldstone diagrams. Uninterrupted ladders of the potential v{r) "between a pair of particle lines can "be summed into the Brueckner reaction matrix g defined by y (3) where the exclusion operator has been dropped, as mentioned earlier. The resulting Goldstone diagrams involve only g and V,. [Fig. 2a shows the example of a four-body diagram, to fifth order in g, contributing to, while Fig. 2b shows a corresponding contribution to the BE.] However, assumptions ii) and iii) greatly simplify the complexity of many-body diagrams. Since each particle starts out in a different spin state from the others, and further-the interaction cannot flip spin, exchange diagrams do not exist. Thus, of the nl diagrams which can exist corresponding to a given diagram, by exchanging the final-state particles, only the direct diagram needs to be considered. Further, no non-diagonal hole-hole transitions can occur, as this would require a spin flip. Thus in Fig.3a the case k - I and m = n is the only one that can contribute. But this restraint clearly reduces the contribution of the diagram by l/n, where N->», Of course, there are some diagrams (Fig.3b) where momentum-conservation forces a hole-hole or a hole-particle interaction to be diagonal. These diagrams will survive. Similarly, a term like Fig.3c, where the two hole-lines labelled m are the same, is permissible. This diagram manifestly violates the Pauli principle, but Goldstone'a proof requires that such violation in the intermediate states must be permitted in order to factor out the linked cluster expansion. Note that, while exchange diagrams are forbidden in our model, Fig.3c is not truly an exchange term in that the two hole-lines m are the same body and no spin flip is needed. Diagrams like Fig.3b and 3c - 6 -

9 do survive in our model. All other types of diagrams involving either exchange or hol«-hola traditions ar«forbidden, This greatly reduces the number of classes of n-body diagrams. For instance, of the l6 classes of four hole-line diagrams discussed by Day only five survive, as will he pointed out in Sec.IV, We emphasize that this model and some of its advantages have been discussed by others '*"', What is new here is our proposal for evaluating n-body cluster energies in this model. III. EVALUATION OF n-body CLUSTERS 9) Day has recently shown that certain n-body sequences of Goldstone diagrams add up to give a Jastrow type of contribution to l^) - a result lit) which was noticed earlier "by Mozkowski for the special case of three "bodies. Day's result is an important ingredient for our method and is stated below. Consider the two-body correlation function T (r. } defined "by where \ ) is the ground state of H fl consisting of zero-momentum particles. [in Eq.(U) and all subsequent ones, we suppress the spin indices of the N particles which in any case remain unaltered.] For the hard core potential in Fig.l, n(r) has the form shown in Fig.lb. For r < c, r (r) = 1. For r > c, r)(r) falls off to zero by a distance comparable to d. (For simplicity, we have set the total volume ft = 1, so that p = W/fi = N. ) Day first generalizes Eq.(U) to functions other than \$), i.e. r (5) It is clear that Eq.(5) certainly contains (h) for F tr 12,r 2...)=$= but is more generalized. For this generalization to be valid, one must ignore off-energy-shell effects on the operator (l/(e 0 -HQ))g... These - T -

10 off-energy-she11 effects make the correlation function fall off more steeply for r > c, but do not affect it for r < e. Day neglects this difference as a starting point and so do ve. Some useful special cases of Eq.. (5) are (6) j; Using these ideas, Day then sums a sub-class of diagrams contributing to JY/. Consider n-body diagrams with n-hole lines, -where a pairs of particles (a 5 n(n-l)/2) interact with each other once or more than once in all possible orders. Further, take only those diagrams of this class where a hole line, once created, does not interact again. Then the sum of tin infinite sequence of Goldstone diagrams so defined is (-1) rkp ri^ ri.,, where there is one factor of T\ for each of the O pairs of particles that interact with each other. By varying n, the number of bodies in the diagram, and 0, the number of pairs among them that interact, one can thus generate the corresponding terms contained in the Jastrow function, TT (7) This is Day's result and it gives an underlying theoretical framework to Jastrow functions often used empirically in many-body theories. Of course, linked Goldstone diagrams can yield only linked terms in the expansion of \f* T. To illustrate the class of diagrams included in Day's result, consider the sequence of which some members are shown in Fig.U. They satisfy the condition that no hole line interacts again after being created. Four bodies are involved where three pairs (12, 23 and 3*0 interact an arbitary number - 8 -

11 of times in all possible orders. The sum of all such, terms in Day's approximation is C-l) *L I^o Tu^. By contrast,.the diagrams in Fig.5 are random examples of linked diagrams which do not belong to the class of Jastrov diagrams. A term like Fig.5a, for instance, has an off-diagonal hole-hole transition violating the rule that hole-lines, once formed, should not interact again if the diagram is to be included in if T Such diagrams are present in real systems such as nuclear matter or He, but are absent in our model. On the other hand, diagrams in Figs. 5b, 5c and 5d are allowed even in our model, since here the hole-lines that interact again annihilate the same "particles" that were created from them, therefore requiring no "spin" flip; but they are not included in the Jaetrow sequence. Diagrams in Fig.5 have been deliberately chosen to yield to the same final state as those in Fig.lv. This Illustrates the fact that for every diagram which contributes to the Jastrow sequence (say, diagram (a) in Fig.U), there are infinitely many (such as 5b, 5c and 5d) which exist in the jv expansion but which are not included in if T. Diagrams like 5b-5d are similar to "self energy" and "radiative corrections", in the language of relativistic Feynman diagrams, to the diagram in Fig. ka. While a diagram like 5b may be absorbed into the self energy of particle a, the same cannot be easily done to the other diagrams. Thus, Day's elegant result notwithstanding, one cannot prove that the correct wave function V can be approximated by the ^_, given in Eq.(T). This statement should be distinguished from the empirical result that Jastrow functions, with a correlation T){r) not necessarily given by Eq.(U) but adjusted by hand, can yield good results for certain quantities like the energy. Our point is that this does not follow from Schrb'dinger's equation in any obvious way, despite Day's result. Similarly, if the energy is evaluated either from the equation = Oi Bf E with T T used in the place of the correct V, it is not clear what the contribution of the large number of terms left out would be. However, we now show that when Day's ideas are directly applied to the BE expansion instead of to the JV/ expansion, the vast majority of the linked Goldstone diagrams contributing to the BE can be approximated by integrals over Jastrov functions. 9

12 We begin by paraphrasing a technique in nuclear matter theory for evaluating th«g-matrlx lunant. Tht axaot two-body correlation function is given by As mentioned earlier, this can be veil approximated, thanks to the energy gap A, "by extending the "particle state" momenta below km down to zero. (k_, itself is zero in our model, but that does not affect the arguments below.) Particles can now have the same momenta as holes, although their energy will be different by A, because of the single particle potential in Eq.(2). Let us use the same capital and small letters respectively to denote particles and holes of the same momenta. Eq.(8) becomes «2. A y However, since MJj)> and mn/ have the same momenta {and spin), (9) This well-known result has been presented in some detail, as we are going to generalize it to n-body clusters. Consider, for example, the diagram in Fig.2a (call it [)$ ) which contributes to Y) and Fig.2b (.call it D) which contributes to BE. Extending the above arguments to this case, we can easily see that e =

13 while so that do) Since in our model holes have zero momentum, = A* U 3 r,tfr^jl V, <( ^ X C^,<V, r l, ^ J (11) In deriving (10) and (11), off-energy-shell effects have been implicitly neglected as was done in Day's result for the Jastrow function. Now, it may be noticed that Fig.2a, which represents x), is a term satisfying Day's conditions and hence belongs to that "Jastrow sequence" which adds up to (+ri _rio-a ^ % Ho!,) The binding energy contribution D is related to the integral over X through Eq.(ll). In this vay it can be checked that: a) Every n-body cluster diagram contributing to BE (with two exceptions, to which we shall come later) is just na times the integral over a corresponding contribution to the wave function. Further, these contributions are all of the Jastrow type. b) The-converse is true, i.e. that every linked Goldstone diagram contributing to the Jastrow wave function, when integrated over all coordinates and multiplied by na, will give the corresponding diagram in the BE expansion, with only one exceptional class of diagrams. Barring the three exceptional classes mentioned, which we shall discuss individually, this means that the sum of n-body cluster diagrams contributing to the BE is simply related to integrals over the Jastrow expansion. For instance, consider all fourbody BE diagrams where only the pairs 12, 23, 13 and 3 1 * interact an arbitrary number of times (greater than zero), and in arbitrary order. Their sum is just (12)

14 Bumming this over.all sets of pairs which, exist among four-bodies, we obtain the total four-te^y oontribution to the BE as where if). is the sum of all linked four-body terms in the Jastrow expansion. By "unlinked" we mean here a term such as i"l(r,g) ^^l^» wnere ^>lae tvo P airs are unconnected. Similarly, the sura of n-body clusters contributing to the E E i s ' - ' (11+) To give an example of ^, lk is given by Eqs.(l2) to (16) must, of course, be corrected for the three classes of exceptional diagrams mentioned earlier. Two of these classes belong in the Goldstone series for the BE, but are not included in the sum in Eq.(l^). The third is a set that is included in Eq.[lk) but does not belong to the allowed Goldstone BE series. These exceptions are: P) All diagrams involving the single-particle potential V are clearly not included in Eq.(l*O. Since V = 0 for particle lines, such V insertions can be attached only to hole lines. Q) All diagrams involving hole-hole or hole-particle interactions are not included in (lu). In the binding-energy diagrams included in (lu), any hole line interacts only twice - once at its creation and once at its annihilation * but never in between

15 However, ab noted earlier, the assumptions ii) and iii) of our model Most Of tk* diagrams involving hole-hole or bol«*particle interactions in the intermediate states. Since each hole line has a different spin index which. ib conserved, only diagonal hole-hole or hole-particle interactions are alloved. This reduces the contribution of terms like Fig.3a "by a factor l/n as N -»». This K dependence has been pointed out "by Brandow. The only such diagrams to survive are exemplified "by Fig.3b, where perforce momentum ib conserved for the hole line. These diagrams are characterized "by the fact that, if the diagonal hole-hole or hole-particle interaction in question were removed, the diagram would split into two unlinked parts. Also to be included in class Q are the pseudo "exchange diagrams" as exemplified in Fig.3c Here again, if the hole lines labelled m were de-exchanged, one would obtain a pair of unlinked diagrams. class Q diagrams could be exploited in their summation. This property of R) This is a class of diagrams present in Eq. (l*+), but not present in the parent linked Goldstone expansion for the BE. Consider for example Fig.6a, which exists and contributes to the term r^r..^) n(r p _) n(r^) in the Jastrow wave function. But its integral, which is given by Fig.6b, is not an allowed Goldstone diagram in the BE expansion. Fig.6b would be strictly zero if the Pauli exclusion operator were retained, since it would violate momentum conservation at the lowest interaction! * ( [p > p while p is unchanged). & IT SI However, in deriving Eq.(13) or (lu) we have been working in the approximation -+ + where the exclusion operator is neglected, i.e. p can be equal to p. In a n that case, Fig,6b can be redrawn as Fig.6c, which is an unlinked term not to be included in the BE. *^ Generalizing from this example, it can be seen that the right-hand side of Eq.(l^) contains spurious contributions corresponding to those diagrams in the Jastrow wave function (like Fig.6a) which when integrated give "momentum violating" diagrams (like Fig.6b -6c). separately and subtracted from Eq.(lU). These spurious diagrams must be summed Our final result then is that the sum of n-body clusters in the BE Goldstone expansion is given by * ) Note that the energy denominator at the level A in Fig.6c is not zero, but 2A in our reference spectrum approximation. Thus, Fig,6c doeb not diverge, but gives a finite contribution proportional to Np

16 A few remarks about this result are due. Eq. (l7) does not explicitly give the full result for (BE) n in that classes P, Q and R must still be evaluated separately. (it might be possible to cancel off class Q, which may be converted to self energy insertions in hole lines, with class P which involves V insertions in hole lines. This is discussed in the concluding section.) Nevertheless, the variety of n-body cluster diagrams is much greater than that contained in classes P, Q and R. All the other diagrams have been summed into the integral in Eq.. (17 )> where ^ (r. r ) is easy to enumerate for any given n from the Jastrow product wave function. This is a great simplification and renders the evaluation of the BE much more practicable than earlier more direct methods in the literature which have so far only been worked out, with great difficulty, up to n * k. This becomes evident from the examples in the next section. IV. n-body CLUSTERS FOR n = 2, 3 AND h In this section we illustrate and verify Eq.(17) for some low values of n. By "verify" we mean that for n k the left-hand side (l.h.s.) of Eq.(17) has been evaluated directly in the literature by classifying the diagrams into different groups such that coupled equations can be written for some of them. The contributions of the diagrams in each group are then summed separately. Of these groups, a few belong to classes P and Q. We are not concerned with them in this section and consider instead (18) The quantity (B E) n is available in the literature for n ^ It. In this section we explicitly evaluate (class R) and show that within our model. The cases of n = 2 and 3 are simple enough so that evaluating the right-hand side of Eq.(19) Is no easier than the direct

17 evaluation of ibe),. However, for n «* k, the former is much simpler an^ onowi itt incren»ing utifttlnm» n inoro****. i) ' -a «2 The situation is trivial here. Barring diagrams involving V insertions.(class P), there is only one diagram here, corresponding to the diagonal matrix element. There are no exchange diagrams in our model nor any terms belonging to classes Q or R, for n = 2. Thus g from Eq.4(9)- This agrees vith the right-hand side of Eq.. (l9)j since no class E diagrams exist for. n = 2. fi» 1, so that N = p.) ii) n => 3 (Note that we are using a normalization volume The three-tody energy in nuclear matter has been evaluated "by the Bethe- Faddeev theory. In our language, it can be seen that this theory sums all diagrams except those classes P and Q which must be separately evaluated. In other words, the Bethe-Faddeev theory gives (B E) O in terms of three-body (i) correlation functions Z which in turn have been evaluated from coupled equations. The well-known result is ftr) = ( where Ik) in the approximation of ignoring off-energy-shell effects. Applying the operators (l/(e Q -H 0 ))g.. as per Eq,(5)» this gives (written in a symmetrical form) (20) On the other hand, the Jastrow contribution here is

18 (21).V which is different from (BE)_. This is because of class R diagrams, six of which exist for n = 3. These are shown in Fig.7 and clearly add up, in our approximations, to N V J d y ^ (22) It is clear that together Eqs.(20) to (22) satisfy Eq.(l9) exactly. Again, we have evaluated only direct diagrams, since exchange termb are forbidden in our model,. iii) n m k Only a few people have worked on four-body clusters. Of these we 7) use the work of Day, which allows a direct evaluation of (BE), and also reveals how difficult that task is. Day exhaustively identifies all four-body clusters and divides them into l6 classes labelled A1-A9, B1-B6 and C. He then evaluates each class separately. In particular, bis class C involves "genuine" four-body correlation functions for which he writes coupled equations and solves. a hundred terms (Eq.(3.102) of Hef.7). The resulting function has over This function has to be multiplied by yet another one involving two- and three-body correlations, and integrated to obtain the contribution of class C. Day also evaluates classes Al to A9 and Bl to B6, although these are much simpler. Our model system, because of its assumptions ii) and iii), is simpler than nuclear matter. Thus, of the l6 classes of diagrams, only five survive, namely Al, A^, Bl, B3 and C. The others involve transitions from one hole to another. Further, since exchange diagrams are not allowed in our model, the correlation functions need not be antisymmetrized. since we are ignoring off-energy effects and are using Eq.(5) for the operator (l/(e 0 -H Q ))g.,, some of Day's expressions become less lengthy. His Eq.(3.102) in Ref.7 has only 48 terms in our modell Finally, Despite such reduction of complexity, the evaluation of (BE)^ by Day's methods still involves much algebra,, particularly for class C. On

19 going through, this algebra we find that Day's results, when applied to our model, yield the following contributions to, -4A N H JXT: (~^ V^IVP'HJ (23a) where dx«^ r, (( a r.rf^rf^ and ^.. - 4Y.. (23b) Day's Al and B3 "belong to our class Q and do not contribute to (Kl).. The sum of these contributions gives (23c) (in Eqs.(23) and (2^), equal contributions such as n Ho-a loh ^T anii dt have I ^een 6 rou P e(a together.) This same result can be arrived at much more simply by using the righthand side of Eq.(l9). The linked four-body part ^ of the Jastrow function can easily be obtained from the form 7T(l-'0. i i) to give We see that the expression in Eq.(25) agrees with Eq.(2U) except for the third term, whose sign is different. This disparity, according to Eq.(19) must come from the (class E), diagrams, which we now identify and evaluate. Class E diagrams are those terms which exist in the integral of the Jastrow function, but are not allowed in the parent Goldstone series. When drawn in the Goldstone particle-hole notation, they appear to violate momentum

20 conservation. It can "be seen that they have the following characteristics through vhloh they can be exhaustively identified. Let two particle-hole pairs (bodies'1 and 2) be created at some stage due to the interaction between them. Then let the two particles interact with each other an arbitrary number of times until one of the pairs (body l) is annilated. In the meantime, the other bodies present (other than 1 and 2) can interact amongst themselves arbitrarily. Once pair 1 is annihilated, the particle 2 can and must interact with the others, or else the diagram will become unlinked. Fig.8a shows such an example. If the Paul! principle is strictly enforced, then it can be seen that the interaction immediately below level D in Fig.8a violates momentum conservation, so that the diagram would vanish. However, in our co-ordinate space method, the exclusion principle is neglected so that the "particle" b can have the same momentum as the hole n. If one sets b = n, Fig.8a will look like Fig.8b, an unlinked term, which should not be included and hence belongs to class R. Note that, although Fig.8b is unlinked, the energy denominator at level D is not zero, but 2A. Also the contribution of Fig.8b is It- 5 proportional to N and not H, since the two hole lines labelled n are the same. Another example is shown in Fig. 8c, where the pair 3l+ also interacts, interspersed "between 12 interactions. ordering (GTO), it can be seen However, upon using generalized time that all diagrams like Fig.8c are contained in Fig,8a, provided we put the interactions "on the energy shell", which in oar theory we have been doing anyway. Further, for every diagram like Fig.Oa, its time-inverted analogue (Fig.8d) also belongs to class R. Using these criteria, all class R diagrams involving four-bodies can be summed easily. Since in terms such as Fig.8a, bodies 2, 3 and four can interact an arbitrary number of times above level D, the sum of such interactions is the three-body correlated wave function for bodies 2, 3 and h. These functions, as noted earlier, are already available. Thus, Fig.9a contains all terms like Fig.8a where the shaded rectangle corresponds to the three-body correlation function Similarly, Fig.o/b contains all terms like Fig.8d. Note that the "bubble" body can be chosen in four ways for any set (123V) and the body with which it interacts chosen from the remaining three. But care must be taken not to double-count diagrams such as Fig.10a, which is symmetrical under timereversal, or Fig.10b, which is the same as Fig.10c under GTO

21 Taking care to avoid aucb..double counting, the contribution of distinct in *%g.9 I** v wi, C26) When (class Rk is subtracted from the Jastrow term in Eq.(25)> it is seen that the directly evaluated (BEk in Eq. (2U). results, as per our "basic Eq. (19). This is the sense in which Eq.. (19) is verified here. V. DISCUSSION In the previous sections it was seen that an expression for (BE) can "be found merely "by calculating class R diagrams and subtracting them from Jastrow-type contributions. 'While for n» k we saw that even class R diagrams require some effort to compute, this task is much simpler.than the work involved in Day's direct method. This is not to denigrate Day's excellent work in any way; his lengthy and careful analysis is unavoidable if one wishes to work at his level of accuracy. Not only does he work with a real system (nuclear matter) where hole-hole transitions can occur, but also distinguishes between two types of correlation functions ri and <y, depending upon whether a g matrix is "on" or "off" the energy shell. He also evaluates four-body correlation wave functions instead of just the binding energy. However, the resulting complexity is such that extension of Day's direct evaluation method to n 5 seems forbidding. By contrast, in our simplified method, this task reduces to calculating only class P, Q and R diagrams and using Eq.(IT). This seems to us to be closer to the bounds of practicability for large values of n as well. Our method is, of course, still incomplete and has its deficiencies, which warrant discussion. We present, in Eqs.(20) to (26), results only in the form of integrals, which we do not evaluate. We also do not attempt to sum over n to obtain the full BE from all clusters. We do not provide a general prescription for evaluating classes P, Q and R. These are the questions we hope to tackle next. Even at this stage some useful remarks can be made about these steps of the calculation yet to be undertaken. It can be seen that class Q diagrams can be considered as self-energy corrections to hole lines in a part of the

22 diagram. For instance, Figs.3h and 3c can "be absorbed into the self-energy of the hole m, Since eaeh linked diagram can be connected to any hole state of another to form class Q diagrams, it might be possible to calculate the sum of class Q implicitly in terms of the BE, thereby obtaining a consistency equation for the BE. Alternatively, if class Q is summed into a self-energy contribution for the hole lines, then this can be set, on average, equal to the single-particle potential at least on average. method. V, so that class P and class Q cancel one another, This is simply a generalization of the Hartree-Fock In this way class P diagrams need not be separately evaluated, although class Q will have to be summed,in order to determine V. Brandow's extensive discussion of class Q diagrams can be exploited in this connection. ' Regarding class E, although we evaluated them only for the examples n = 3 and k, their identifying characteristics have been described in the last section.. A closed expression for their sum for arbitrary characteristics, remains to be found. n, given these We again emphasize that the vast majority of diagrams do not belong to P, Q or R and that their sum is contained in the integral of the Jastrow function ifi (r^n)" ') Once P, Q and R are summed for arbitrary n, one has (BE) in the form of a sum of integrals. Each integrand is a product of the form n(r lp ) T)(rp_), so that these are relatively simple multidimensional integrals. If the number of n(r..) 's involved in a term is less than 3n, then once 3 3 dr = d r d r is written in terms of Ildr.., the integration is trivial. -L But for large n, the number of r.. 's ^ (n/2)(n-l), while the dimensionality = 3n, so that the problem is not so simple for n 7 Finally, after integration, the final series must be summed over. Both these problems of integration and summation over n are common to many theories. For instance, variational calculations for the energy of liquid helium, or neutron stars, involve similar integrals and sums. The techniques developed there, such as the use of hypemetted chains, may be adaptable here. * All this remains to be done. Apart from these aspects, our method has used some approximations. Some, such as the neglect of the Pauli principle in the presence of the energy gap A, are part of the state of the art and are used even in simpler situations like two- and three-body correlations in nuclear matter. This approximation is essential if the problem is to be transferred to co-ordinate space in a simple way. We have further ignored the variation of ncr) off the energy shell. This again is essential to keep the problem within control, or else there will be (n-l) different correlation functions in a n-body problem. Even Day in his

23 four-body work uses only two functions r\(x) and *^ (r), /whereas his problem really needs at least three, corresponding to two-, three- and four-body excitations. Ignoring off-shell effects on ncrl is quite a good starting approximation. For a hard-core potential, these effects do not change T\(T) for r < c and only effect the rate at which n(r) falls to zero for r > c. Finally, we have worked in a model system with an artificial assumption of distinguishability. This was to avoid complications due to the finiteness of k_, exchange diagrams, etc. To adapt the final results of our approach to real Fermi liquids, one must put in the required antisymmetry. It should, however, be noted that even in real systems high density is not necessarily high k, provided there are enough species present. There are reasons to believe that the interior of neutron stars might contain a variety of hadronic species. The above paragraphs clearly show the tentative and approximate nature of this work. It is, nevertheless, a good starting point - the only one so far available - for studying a dense strongly interacting fluid from first principles. The insight gained from BOlving such a model system could then be used to suggest approximate solutions to real systems. ACKNOWLEDGMENTS It is a pleasure to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste

24 REFERENCES 1) A.L. Fetter and J.D. Walecka, ' Quantum Theory of Many-Partiele Systems (Me Graw-Hill, New York 1971). 2) T.D. Lee and C.U.Yang, Fhys. Rev. 105_ (1957) 1119; N.M. Hugenholtz and D. Pines, Phys. Rev. Il6 (1959) b&9-3) H.A. Bethe, Ann. Rev. Nucl. Sci. 21. (l97l) 93. h) M. Gell-Mann and K.A. Brueokner, Phys. Rev. 106_ (1957) 36U. 5) V. Pandharipande, Nucl. Phys. TftA (1971) 6Ul; ibid 178A (1971) 123; V. Pandharipande and H.A. Bethe, Phys. Rev. CJ_ (1973) ) E. Ostgaard, Phys. Rev. 1J0. (1968) 257; ibid 171 (1968) 2^8; J. Burkhardt, Ann. Phys. (NY) kj_ (1968) ) B.D. Day, Phys. Rev. l8j_ (1969) ) B.H. Brandow, Phys. Rev. Letters 22_ (1969) 173; Ann. Phys. (NY) 6^ (1971) 21. 9) B.D. Day, Phys. Rev. M (l97l) 68lj K.R. Lassey and D.W.L. Sprung, Nucl. Phys. A177 (1971) ) B.D. Day, Rev. Mod. Phys. 39_ (1967) 719; R. Rajaraman and H.A. Bethe, ibid 39 (1967) 7^5. 11) R. Rajaraman, Fhys. Rev. 151 (1967) ) B.H.J. McKellar and R. Rajaraman, Phys. Rev. C3_ (l9tl) ) H.A. Bethe, B.H. Brandov and A.G. Petschek, Phys. Rev. Ig9_ (1963) 225. Ik) S.A. Moszkowski, Phys. Rev. luob (1965) ) J.M.J. Van Leeuwen, J. Groenveld and J. De Boer, Fhysica 25. (1959)

25 Fig.l (a) a) The potential vcr..) is used, with, a hard core up to r = c i,) * and a cut-off at r - d. The cut-off is such, that v(r > d) is veak. b) The two-body correlation function corresponding to such a potential. -23-

26 Fig.2 (a) (b) a) A four-tody Goldstone diagram contributing to the wave function. It is of fifth order in the Brueckner reaction matrix g. b) A corresponding contribution to the binding energy. The "correspondence" is discussed in Sec.Ill,

27 ik mt 1 kamaa- \..(a) Some diagrams that are not part of a natural ladder series. Figure a) is negligible in our model since k * I and m # n is forbidden, but figures b) and c) are present. -25-

28 Fie (b)! no Some early terms of a sequence of Goldstone diagrams leading to a Jastrow type of term n 12 n 23 ^h in the wave

29 ,. (*») (c) Some diagrams that do not "belong to the Jastrov series. -27-

30 Fig. 6 (a) (b) m n Equivalent representations of a typical class R diagram. -28-

31 Fig.T (e) (0. All class E diagrams for n = 3-29-

32 Fig. 8 n D (b) (c) Some class R diagrams for n = h -30-

33 TJ-B-S 3 4 (a) (b). A representation of all class R diagrams for n «k. The shaded rectangles, described in the text, correspond to the full three-body vave function. -31-

34 Fig.10 (a) (c) Some (class R). diagrams regarding the care which must be taken to avoid double counting t K ±i