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Validity of many-body approximation methods for a solvable model
1.C
[
Nuclear Physics 62 (1965) 211--224; (~) North-Holland Publishing Co., Amsterdam
1
Not to be reproduced by photoprint or microfilm without written permission from the publisher
VALIDITY OF M A N Y - B O D Y APPROXIMATION M E T H O D S FOR A SOLVABLE M O D E L (IlI). Diagram Summations A. J. GLICK t
University of Maryland, College Park, Maryland H. J. LIPKIN
Weizmann Institute of Science, Rehovoth, Israel and N. MESHKOV
University of Maryland, College Park, Maryland tt Received 18 February 1964 Abstract: The model N-fermion system studied in two previous papers is reconsidered in order to investigate the range of validity of diagrammatic many-body perturbation theory and various selective summations of higher order graphs. The familiar random phase approximation or bubble graph sum is shown to be the first term in an expansion in powers of 1/N for which the coefficients are arbitrary functions of NV/e, where V is the interaction strength and e is the level separation of the non-interacting system. However, the 1IN expansion converges very slowly (if at all) for N < 40. Other corrections to the bubble sum which do not fit into such an expansion are shown to be much more important than the 1/N corrections for small N. These are the exchange and self-energy graphs which correct for counting errors and. violations of the exclusion principle when enumerating the states that a given particle can occupy. With these terms a rather good approximation to the excitation energy can be obtained, though the results are not quite as good as that obtained with fourth-order perturbation theory for small N, or for the whole range of N with the self-consistent linearization procedure described in a previous paper.
1. Introduction In I and II, we have employed various approximations in studying a model manybody system for which exact solutions can be obtained. This investigation was carried out in an attempt to gain a better understanding of the range of validity of the approximation procedures and to develop better methods for treating more realistic systems with a finite number of particles. In order to further clarify these questions and to search for systematic ways of improving the results, we now utilize a diagrammatic and a Green function formalism. Such methods have been extensively applied in recent years 3) to studies of many-body systems. By consideration of the equations t N.S.F. Postdoctoral Fellow 1959-1961. tt Work supported in part by the Air Force Office of Scientific Research, OAR, through the European Office, Aerospace Research, United States Air Force, and through AFOSR-62--46, and in part by the Office o f Naval Research. 211
212
A.J.
GLICK et
al.
satisfied by Green functions and/or summing selective sets of high order graphs it has been possible to introduce approximations valid for different ranges of the parameters characterizing the systems. In some cases, these approximations can also be obtained with other, and sometimes simpler methods, but the Green function techniques often provide new insights into the nature of the approximations and indicate systematic corrections to the calculations. The model we employ might be considered an idealization of more realistic models used to calculate the energy of vibrational levels in finite nuclei 4, 5). A detailed description of the model was given in I. In essence, it consists of N fermions distributed in two levels separated by energy 5. The lower level is denoted by a quantum number a taking the value a = - 1 and in the upper level ¢r = + 1. Each particle also has other quantum numbers represented by p which can take N values; however, the energy is degenerate in p. If all the particles were placed in the a = - 1 level, that level would be full and hence the model can be viewed as a system with a "closed shell". The particles interact via a "monopole-monopole" force which scatters pairs of particles from one level to the other without affecting the other quantum numbers p o f the particles being scattered. Thus the Hamiltonian can be written in the form U = ½~Etra~,%,,+½ V Z p, tY
apaav'aav'.-cap. + + -a,
(I.1)
p, p', ¢
where the ap, are the usual annihilation operators for a particle in state (p, a) and they satisfy fermion anticommutation relations. Here, as in I and II, we are primarily concerned with calculations of the energy difference between the first-excited state and the ground state A E = E 1 - E o . We determine this quantity indirectly through a study of the linear response function 6) R+(o~) = ~., [(nlJ+lO>lZf(co-E,+ Eo),
(1.2)
n
where s ÷ = E a ; +1 a., -1
(1.3)
P
is an operator introduced in I, and the states and energies are characteristic of the total Hamiltonian as given by eq. (1.1). The response function R+(og) is non-zero only for those frequencies ~o which are equal to the energy differences between the excited states n and the ground state, and provided that not all matrix elements (nlJ+10> vanish. F r o m symmetry considerations and the nature of J+ it is evident that only states with odd numbers of particles in the upper level will contribute to R + (co). The first peak in R + (co) is the one that interests us and the frequency at which it occurs gives E 1 - - E o. In the next section the algebra of computation is summarized with a prescription for drawing graphs and rules for calculating contributions appropriate to the model. Sect. 3 contains a rederivation of the results of perturbation theory obtained earlier in I. For such a calculation the Green function formulation is quite cumbersome
MANY-BODY APPROXIMATION METHODS (IlI)
213
compared to more direct methods and the number of contributing graphs multiplies very rapidly with increasing order of perturbation theory. It is only when alternative approximation schemes are sought that the present formalism shows its value. The random phase approximation (RPA) is easily obtained in sect. 4 and it is shown to be in effect the first term of an expansion in powers of 1IN in which the coefficients are arbitrary functions of (NV/s). The first correction term to the RPA in this series is also calculated. However, it is found that the convergence of the series is very poor unless N is quite large and of the order of 50 particles for the range of parameters described here. A completely different set of corrections to the RPA is considered in sect. 5. These are the exchange and self-energy corrections, and they are shown to be much more important than 1/N corrections in the region N < 50. Inclusion of the self-energy corrections to second order in V/8 results in a quite accurate determination of AE which is very similar to that obtained in II by using "empirical" singleparticle energies. The final section contains a brief summary and conclusions.
2. The Algebra of Computation The response function R+(og) can be expressed in terms of the real part of an integral 6) over the Green function G(t): R+(og) = _1Re
dt e'°"G(t),
(2.1)
7~
where
G(t) =,
(2.2)
and J_ (t) is the Heisenberg operator corresponding to J_ = ~
a p+ ,_lap,+l
•
(2.3)
p
As in refi 6), a set of rules can be formulated for constructing a set of graphs and determining the contribution of any given graph to G(t). In the present case, however, many graphs give a vanishing contribution due to the nature of the interparticle interaction which can either create or destroy two particle-hole pairs, but which cannot scatter particles, holes, or particle-hole pairs between p states. As a result, each closed particle loop in a graph must have an even number of vertices. In addition, only graphs with an even number of interaction lines can contribute to R+(o~). Nevertheless, even with a pure particle-hole interaction, many graphs still contribute in addition to the familiar simple bubble graphs. Typical contributing graphs are shown in fig. 1. These graphs represent all possible time orderings of the interaction lines. However, it is convenient in this case to require that the particles (upward) or hole(downward) nature of the solid lines should not be altered in the time reordering,
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A.J. GLICKet al.
The Green function can be constructed f r o m the graphs labelled with frequencies according to the rules (i) a factor V/(2~i) for a dashed interaction line, (ii) a factor i toi + ½~-+ i~
(2.4a)
(2.4b)
for a particle (upper sign) or hole (lower sign) line labelled by frequency cot, (iii) an overall factor N l (-)
e -''~'
4~2
(2.4e)
where N is the n u m b e r o f particles in the system, I is the n u m b e r o f closed particle loops in the graph and to' is the frequency carried by the wiggly J+ line, (iv) integration t over all frequencies tot and to' from - m
J
(o1
(d)
(b)
(c)
(e]
(f)
JY
to oo.
(g) (hl (i) Fig. 1. Typical graphs which contribute to Green's function G(t). These graphs represent all possible time ordering of the interaction lines, but the particle (upward) or hole (downward) nature of the solid lines must remain unaltered in the time reordering. The two wavy lines are separated by time t with time flowing upward. These rules were constructed in such a w a y that the n u m b e r o f "particle" lines in contributing graphs are in no way limited by, and can exceed, the n u m b e r N o f actual particles in the system. The N in rule (iii) arises from the summation over the degenerate p q u a n t u m number: p is the same for all particle lines in a given closed loop, b u t can take on any one o f N values. Further consequences o f this rule will be discussed in sect. 5. t The integrations over to and t are most easily performed as indicated in appendix B of ref. ~).
MANY-BODY APPROXIMATION METHODS (III)
215
3. Perturbation Theory As a first application of the present formalism, the perturbation theory results of I can be rederived. For this purpose one must combine graphs according to the number of dashed interaction lines they contain, so that the only zero order contribution is that shown in fig. 1(a). An elementary calculation gives
R+(CO)= N6(co-e).
(3.1)
The argument of the 6 function gives the expected excitation energy characteristic of the system without interactions: E 1 - E o = e. In second order, the non-vanishing contributions are given by the remaining graphs in fig. 1. The corresponding result for R+(co) is
R~(co) = N(~(co-e)+ N(N- I)(N- 3)V2 6(co-8)+ N(N- I)(N- 3)V2 6'(co-8), 482
2e .
(3.2)
containing a term with a derivative of a 6 function. These terms give R + (co) to second order in Vie. However, we need the argument of the 6 function to second order. We can infer it from eq. (3.2) by recognizing that this expression arises from an expansion of an R~+ (co) of the form
Rr2+(co)= (N=FaV2] bY2-).8:2 ] ~ (co--8+ Expanding the 6 function in a "Taylor Series" about eq. (3.2), we obtain
R2+((-D) IN--I-N(/~'l)(/~-3)V2] 6482 j :
(3,3)
Vie =
0 and comparing with
(-0.~-~-[-(N-1)(N'3)V2"~28 .] ,
(3.4)
so that
E1-Eo) 8
=
I_½(N_ I)(N_3)'(V'I 2,
/2
(3:5)
\8/
in agreement with eq. (,4.3) I, of the perturbation theory result in I to second order. In like fashion we can extend the calculation to fourth order. However, the number of graphs multiplies very rapidly with increasing order and such a scheme rapidly becomes-impractical.
4. The 1IN Expansion A different approximation method which has frequently been used to calculate excitation energies is the random phase approximation. This method was originally developed for treating the interacting electron gas a), and it gives very good results
216
A.J. GLICKet al.
provided the particle density is high. It has also been applied to a variety of other systems including the Heisenberg ferromagnet 9), superconductors lo) and finite nuclei 4, s), though perhaps the physical justification for using the approximation in these cases is not quite as apparent as for the high density electron gas.
y.j (o)
(.b) Fig. 2. The class of graphs which contribute to G(t) in the random phase approximation. (a) Green's function expressed in terms of the effectiveinteraction which is the solution of the integral equation represented in (b). Only odd numbers of bubbles contribute to 3v'°(co)for the model under consideration.
The excitation energy can be obtained in the random phase approximation with the present formalism by combining the contributions due to all graphs consisting of a simple chain of "bubbles", including such terms as shown in figs. l(a) and l(b). The class of these graphs can be represented as in fig. 2(a) where the heavy horizontal line represents an effective interaction which, for the present model, contains only an odd number of bubbles as indicated in fig. 2(b). The integral equation represented by fig. 2(b) can be solved very simply to obtain the effective interaction as (
2hi
=
N V2
.
(4.1)
2hi (ojt_4B2_N2V2..~io~)((Dt-~4~2-N2V2-i~)
Inserting ~/r°(co') into the calculation of R+(co) gives R~-pA(f..0) = N(,~+ 4~- ~ - N2_V2) t~(f,o- 4~~ - 2 ~/e2 _ N 2 V 2
N 2 V2),
(4.2)
for which the corresponding excitation energy is (4.3) / RPA
MANY-BODY
APPROXIMATION
METHODS
217
(lIl)
This result is meaningful provided NV/8 < 1. Note that (4.3) is the same answer as was obtained for the limit of large N with a linearization method discussed in II. The present formalism shows very clearly that the random phase approximation should give accurate results in the large N limit if VIe is not too large. Reference to rule (iii) of eq. (2.4) shows that for large N the graphs with the most closed particle loops should dominate. In any given order of perturbation theory, say the nth order, the maximum possible number of closed particle loops which can be formed with the available lines is n + 1, and only one graph can appear in that order with all n + 1 loops. This graph consists of a simple bubble chain. Thus the RPA gives the same result as would be obtained by summing perturbation theory to all orders, but retaining only the highest power of N in each order.
~
w )-
1.0
~X
I/N
E l~J Z IU
.7
/L
.6 I
2
4
I 8
I 14
I 30
I SO
I
I00
I
200
NUMBER OF PARTICLES
Fig. 3. The excitation energy (E1--Eo)/tplottedas afunction of particle number N for NV/e = 0.8. The crosses indicate exact values; P~ indicates the result obtained with perturbation theory to second order; L is the RPA result; and L~/Nis the RPA with 1IN corrections. In fig. 3 the excitation energy is plotted versus particle number with fixed NV/e. The RPA result is shown by the dashed line marked L (for linearization). As might be expected the RPA is rather poor for small N and is as good or better than the secondorder perturbation theory result only for N > 75. Thus it is of considerable interest to determine corrections to the RPA for smaller N. Note that the response function eq. (4.2) contains the interaction strength V only in the combination NV/e,. Thus this approximation can be interpreted as the first term in an expansion in powers of 1IN in which the coefficients are arbitrary functions of NV/~ (or an expansion in powers o f Vie in which the coefficients are arbitrary functions of NV/e). We next calculate the second term in this expansion in order to see how great an improvement is obtained with the inclusion of 1IN corrections. The relevant graphs are shown in fig. 4 and can be constructed by adding one effective (simple bubble) interaction in
218
A.I. GLICKe t
al.
every possible way to the graphs of fig, 2, In general, additional contributions could arise if the extra interaction connects the parts of previously unlinked graphs. However, for the model under consideration all such contributions vanish.
(o)
(b)
~/(~)m
-
-
-
-
+
(o) Fig. 4. The 1/Ncorrections to Green's function. (a) The contributing graphs (withwavylinesomitted) expressed in terms of the odd and even bubble interactions and effective self-energy corrected ble bubble.. (b) The effective bubble. (c) The effective interaction with even numbers of bubbles. It is now necessary to take into account both the Odd bubble interaction of eq. (4.1) and also the interaction with an even number of bubbles. 3¢'c(o9') as indicated by the heavy broken line in fig. 4(c). Carrying out the integrations impiied in fig. 4(c) one finds oe.~(oy) =
.
V(co'2 - e 2)
( o ; - ~/~2 _ N 2 v ~ + i ~ ) ( o , , + 4 ~ ~ - N ~ v ~ -
(4.4)
~)"
N o w more complicated "bubbles" also appear: There is the bubble which appears in the last for graphs o f fig. 4(a) containing a 3¢re(¢o') interaction between the particle and hole, and also the b~ bubble shown in fig. 4(b) which contains a 3V'°(co') selfinteraction of the particle or the hole. The evaluation of the contributions from the graphs in fig. 4(a) is straightforward but tedious. One obtains terms containing a 5 function a n d terms with the derivative of a 6 function. Using the same argument
MANY-BODYAPPROXIMATIONMETHODS (III)
219
as for obtaining eq. (3.4), from eq. (3.2) we find to order 1/N
R~/N(¢O) = ½N [~ + ~/~2 _ N 2V 2 _ (NV)2e(3e_ x/e2 _ N 2 V2)t t 4~2-- N2V 2
~
~
(4.5)
J
x ~ (03 -- 4 ~ 2 -- N 2 V 2 - ( N V)2e(2e2 + N 2 V 2 + 6e~/e 2 - N 2 V 2)
NC_
N2V2)2
)"
The corresponding excitation energy is thus (El-EO) = 41-(gv/l~)2÷ ', e, /1/~
(N VIe) 2[2 + (NV/8) 2 + 6x/1 -- (N V/e) 2] NEI_(NV/e)2][I+41_(NVI~)2]
(4.6)
2
In fig. 3 the curve marked Lll N shows the behaviour of the excitation energy in this 1/N corrected random phase approximation. It can be seen that the I/N terms give a strong overcorrection for small N so that there is no net improvement over the ordinary RPA unless N > 12. For larger N the 1/N expansion appears to converge more rapidly. However, for N < 40 this result is not as good as second-order perturbation theory, and it only improves fourth-order perturbation theory provided N > 45. The structure of second order perturbation theory (see eq. (3.5)) might lead one to expect that the 1/N terms would lead to a real improvement of the RPA for N > 4; however the 1IN terms of higher order in Vie contribute significantly to eq. (4.6) and actually decrease the accuracy of the result until N is considerably larger. The next correction term in this expansion, consisting of the 1/N 2 terms, would account for all terms of second-order perturbation theory, among others, and presumably would give a better result: Terms up to 1/N4 are required if all terms of fourth-order perturbation theory are to be brought in. However, such a programme is very difficult to carry out owing to the large number and complicated nature of the graphs which must be considered.
5. Exchange and Self-Energy Corrections In dealing with systems of few particles a different set of graphs is often included along with the bubble graphs in order to improve the RPA. These are the exchange counterparts of the bubble graphs containing particle-hole "ladders". Examples of such graphs are shown in fig. l (c), (d) and (e). In general they are included either because the force has a strong exchange part or to take better account of the exclusion principle. If one attempts to interprete any of these graphs in terms of the actual motion o f the particles in the model under consideration moving between ~r levels with the flow of time upward it is easily seen that these graphs do not correspond to physical processes. Indeed, they imply the simultaneous presence of more than one particle with a given p, and hence violate the exclusion principle. They arise in this
A.J. OLICK et al.
220
formalism to compensate for the counting error made in computing a corresponding simple bubble graph. The N which appears in rule (2.4c) arises from a sum over all p values which can label the closed particle loop. However, if more than one closed loop is present simultaneously in a graph, the contribution in which they have the
Fig. 5. The graphs contributing to G(t) in the bubble plus ladder (RPA with exchange) approximation. The cross hatched interaction can be obtained from the integral equation represented by the figure and then used in the bubble on the left to determine G(t).
>. o td
I.C
""" ..
0.9 ,
LSE
f'..
z
tu
Z 0
o.a L E ' ~ ' " " ' . ~ x O.IJ
X bl
o.Q - -
2
" ~
_l_ I
I
I
I
4
8
14
30
I
I
50
I
I00
I 200
Fig. 6. The excitation energy versus N for NV/e = 0.8. The crosses indicate exact solutions; L is the R P A result; Ls was obtained with the R P A plus exchange corrections; Ls~ is thesame as LE, but with additional self-energy corrections; Lc is taken from II and represents the result of linearizing the equations of motion with "experimental" single-particle energies.
Fig. 7. The renormalized single-particle propagator using the self-energy operator to second order in V.
same p is also non-physical and violates the exclusion principle. For large N the redundancy in counting this one state has little effect, but for small N the exchange terms which cancel these errors can give rise to significant corrections.
MANY-BODY APPROXIMATION METHODS (1II)
221
The contribution of both the bubble and the ladder graphs to the response function can be obtained for this model with a technique similar to that used in the last section to obtain the RPA. Now the Green function is expressed in terms of a different effective interaction indicated by the cross-hatched boxes in fig. 5. Solving the integral equation implied by that figure, we can obtain the exchange corrected response function
R+(¢o) = N(e+~/8 2 - ( N - 1)2V2) ~(¢0- x/~--(N- 1) 2V2), 2,/, ~ - ( N - 1)5v 2
(5A)
and excitation energy
E
/E
: E1
1)2
3,
Note that eqs. (5.1) and eq. (5.2) are identical to eq. (4.2) and eq. (4.3) except that improved counting has replaced N by N - 1. Eq. (5.2) was also obtained in II where a normal ordering of operators was chosen prior to linearization. The choice of normal ordering seemed somewhat orbitrary in the linearization procedure, and only the large N limit was uniquely defined. Thus one could claim that the present formalism vindicates that choice. A comparison of the RPA and the RPA with exchange was already carried out in II and is repeated in fig. 6 for NV/e = 0.8. The curve marked L represents the excitation energy according to RPA and LE represents AE/s for RPA with exchange. A further improvement in the calculation was obtained in II by introducing "experimental" single-particle energies in place of ~. The experimental energies contain some self-energy effects due to interactions with the physical vacuum or core states of the system. With the present formalism one need not resort to "experiment". It is possible to calculate self-energy effects on e directly. As is well known such effects appear graphically as corrections to single particle lines which leave the particles in the same state as they were in initially. Figs. 1(f), (g), (h) and (i) illustrate graphs containing such corrections to second order. These corrections can be introduced systematically into the calculation by replacing all single-particle propagators by renormalized propagators which satisfy an integral equation containing the unrenormalized propagators and a self-energy kernel. Keeping the self-energy only to second order, the integral equation can be represented schematically as in fig. 7. Solving this equation, the corrected propagator which replaces the factor in rule (2.4b) is
i
~
~+4~2+(N-1)v 2
24~ ~+ ( N - 1)V~ L(o,,_+½~~-4
~
2_+i~) ~ - 4 ' ~ + ( N - 1 ) V2
(5.3)
^ . J. GLICK e t al.
222
Substituting this factor for each of the particle lines in fig 5, we obtain after a tedious calculation the exchange and self-energy corrected response function
_
RS'E(~)
N
{[(A-1-2x/l+nu2)(A+3+nu2)(A-l+2x/l+nu2)~ 4 l--+nn~2A(A + B)(A - B) 6(m--eA)
-E(B-i-2x/l+nu2)(B+3+nu2)(B-l+2x/l+nu2); 2B(A + B)(A-B)
(5.4)
6(co-eB)}
where we have used n = ( N - 1) and u = (V/e) and A = [2(1+nu2)(5+4nu2)-n2u2-C~ 2(l+nu2) ,
(5.5)
B= [2(l +nu2)(5+4nu2)-n2uZ +C~:r 2(1 + nu 2) '
(5.6)
C = [(211 +nu2][5+4nuZ]-n2u2)2-4(l +nu2)([1
+nu2][3 +4nu2] 2 -n2u213+nu212)]~.
(5.7)
Note that two 6 functions appear in eq. (5.4). As remarked in sect 1, all odd excited states should contribute peaks to R+(cn), but this approximation is the first one sufficiently complete to exhibit more than one peak. By considering the limit as u ~ 0 it is seen that the peaks arise from the first and third excited states of the system. The position of the second peak gives the excitation energy of the third-excited state above the ground state in this approximation as
t 8
= 8
(5.8)
I SE
However, we are primarily interested in the position of the other peak which gives El E _ -_ oI e
= A.
(5.9)
]SE
The exchange and self-energy corrected excitation energy, eq. (5.9) is plotted in fig. 6 as the curve marked LSE. Note that LSE provides a rather good approximation procedure over the entire range of N, though perturbation theory to fourth order (see II) is still somewhat better when N < 50. In practice, the importance of the LsE approximation is enhanced by the fact that it is often more easily obtained than the carrying of perturbation theory to fourth order. Also plotted in fig. 6 is the curve Lc which was obtained in II by using "experimental" energies to correct the linearization procedure. Note the close agreement
MANY-BODY A P P R O X I M A T I O N
METHODS (II1)
223
between the results derived with these two methods. These approximations become identical if the "experimental" corrections to the energy range are retained only to second order and the propagator (5.3) is approximated by letting V --, 0 except that in the cos dependent denominator, the term of order V 2 is retained (see curve Lc2 in fig. 4 of II). However, the more accurate calculation shows that the use of experimental energies is not completely equivalent to the introduction of renormalized single-particle propagators, and for N > 4 it appears that LsE is more accurate than Lc. Nevertheless, for the model under consideration the higher order terms do not play a major role in this approximation and the difference is not very large.
6. Summary and Conclusions In this paper a function R+(co), describing the linear response of a model of a many-body system has been studied in order to determine the excitation energy of the first-excited state above the ground state of the system. The response function was calculated in various approximations by means of a diagrammatic many-body perturbation theory technique and the selective summation of high order graphs. The bubble graph sum or RPA was shown to be the first term in an expansion in powers of 1/N where the coefficients are arbitrary functions of NV/~. However, the 1/N expansion was found to converge very slowly (if at all) for small N. The first correction term gives an overcorrection for N < 40, and actually decreases the accuracy of the absolute value of the excitation energy for N < 12. Correction terms of a quite different type were found to play a much more important role than the 1IN corrections for small N. These are the "exchange" graphs which correct for counting errors and violations of the exclusion principle when enumerating the states that a given particle can occupy. The "ladder" graphs fall into this class and must be included along with the bubbles to account for the unavailability of certain states when more than one particle is simultaneously excited. The single-particle self-energy graphs play a similar role in that they correct for shifts in the single-particle energies and for counting errors in the eigenstate normalization introduced by using a linked cluster expansion t. The ladder and self-energy corrections are important whenever the number of excluded states is comparable to the number o f states available to particles in the system. For the model considered this situation occurs when N is small since there are only as many states as particles. In other cases it can also occur because of a short-range interaction which limits the number of states a particle can reach. With these correction terms a rather good approximation to the excitation energy can be obtained, though the results are not quite as good as that obtained with fourth-order perturbation theory for small iV, or for the whole range of N with the self-consistent linearization procedure discussed in II. An unfortunate aspect of the exchange and self-energy corrected summations is that they do not arise naturally from a consistent expansion in terms of the parameters t See Goldstone s).
224
A.J. GLICKet al.
(here N, V and e), describing the system. Hence, further correction terms are not uniquely defined andit is in general difficult to discuss the convergence of such a procedure. Also any attempt to do better by introducing exchange corrections into a series such as the 1IN expansion is plagued by the presence of many redundant terms. The next approximation which might be attempted to improve upon the self-energy and exchange corrected R P A is one in which the exchange and self-energy corrections of figs. 5 and 7 are introduced using the effective interactions 3e~e(co) and 3¢/'°(co) in place of the bare interaction V. Such a procedure was used in ref. 7) for treating the electron gas and was shown to be a "conserving" approximation for obtaining transport properties of many-body systems by Baym and Kadanoff 11). However, it is not certain that these corrections wiU give a significant improvement for small N since higher order exchange terms are not consistently retained in such a scheme. In addition, the detailed calculation is quite difficult even for the simple model considered here. We are grateful to M. Baranger, G. E. Brown and S. FaUieros for helpful discussions and correspondence.
References 1) H. J. Lipkin, N. Meshkov and A. L Glick, Nuclear Physics 62 (1965) 188 2) N. Meshkov, A. J. Glick and H. J. Lipkin, Nuclear Physics 62 (1965) 199 3) J. Goldstone, Proc. Roy. Soc. A239 (1957) 267; J. Hubbard, Proc. Roy. Soc. A240 (1957) 539; V. Galitskii and A. Midgal, JETP (Soviet Physics) 7 (1958) 96; A. Klein and R. Prange, Phys. Rev. 112 (1958) 994; P. Martin and J. Schwinger, Phys. Rev. 115 (1959) 1342; D. DuBois, Ann. of Phys. 7 (1959) 174 4) S. Fallieros, Ph.D. Thesis, University of Maryland (1959) 5) G. E. Brown, L. Castillejo and J. Evans, Nuclear Physics 22 (1960) 1 6) A. J. Glick, Ann. of Phys. 17 (1962) 61 7) A. J. Glick, Phys. Rev. 129 (1963) 1399 8) D. Bohm and D. Pines, Phys. Rev. 92 (1953) 608 9) S. V. Tyablikov, Ukr. Mat. Zh. 11 (1959) 287 10) P. W. Anderson, Phys. Rev. 112 (1958) 1900 11) G. Baym and L. P. Kadanoff, Phys. Rev. 124 (1962) 287
[
Nuclear Physics 62 (1965) 211--224; (~) North-Holland Publishing Co., Amsterdam
1
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VALIDITY OF M A N Y - B O D Y APPROXIMATION M E T H O D S FOR A SOLVABLE M O D E L (IlI). Diagram Summations A. J. GLICK t
University of Maryland, College Park, Maryland H. J. LIPKIN
Weizmann Institute of Science, Rehovoth, Israel and N. MESHKOV
University of Maryland, College Park, Maryland tt Received 18 February 1964 Abstract: The model N-fermion system studied in two previous papers is reconsidered in order to investigate the range of validity of diagrammatic many-body perturbation theory and various selective summations of higher order graphs. The familiar random phase approximation or bubble graph sum is shown to be the first term in an expansion in powers of 1/N for which the coefficients are arbitrary functions of NV/e, where V is the interaction strength and e is the level separation of the non-interacting system. However, the 1IN expansion converges very slowly (if at all) for N < 40. Other corrections to the bubble sum which do not fit into such an expansion are shown to be much more important than the 1/N corrections for small N. These are the exchange and self-energy graphs which correct for counting errors and. violations of the exclusion principle when enumerating the states that a given particle can occupy. With these terms a rather good approximation to the excitation energy can be obtained, though the results are not quite as good as that obtained with fourth-order perturbation theory for small N, or for the whole range of N with the self-consistent linearization procedure described in a previous paper.
1. Introduction In I and II, we have employed various approximations in studying a model manybody system for which exact solutions can be obtained. This investigation was carried out in an attempt to gain a better understanding of the range of validity of the approximation procedures and to develop better methods for treating more realistic systems with a finite number of particles. In order to further clarify these questions and to search for systematic ways of improving the results, we now utilize a diagrammatic and a Green function formalism. Such methods have been extensively applied in recent years 3) to studies of many-body systems. By consideration of the equations t N.S.F. Postdoctoral Fellow 1959-1961. tt Work supported in part by the Air Force Office of Scientific Research, OAR, through the European Office, Aerospace Research, United States Air Force, and through AFOSR-62--46, and in part by the Office o f Naval Research. 211
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al.
satisfied by Green functions and/or summing selective sets of high order graphs it has been possible to introduce approximations valid for different ranges of the parameters characterizing the systems. In some cases, these approximations can also be obtained with other, and sometimes simpler methods, but the Green function techniques often provide new insights into the nature of the approximations and indicate systematic corrections to the calculations. The model we employ might be considered an idealization of more realistic models used to calculate the energy of vibrational levels in finite nuclei 4, 5). A detailed description of the model was given in I. In essence, it consists of N fermions distributed in two levels separated by energy 5. The lower level is denoted by a quantum number a taking the value a = - 1 and in the upper level ¢r = + 1. Each particle also has other quantum numbers represented by p which can take N values; however, the energy is degenerate in p. If all the particles were placed in the a = - 1 level, that level would be full and hence the model can be viewed as a system with a "closed shell". The particles interact via a "monopole-monopole" force which scatters pairs of particles from one level to the other without affecting the other quantum numbers p o f the particles being scattered. Thus the Hamiltonian can be written in the form U = ½~Etra~,%,,+½ V Z p, tY
apaav'aav'.-cap. + + -a,
(I.1)
p, p', ¢
where the ap, are the usual annihilation operators for a particle in state (p, a) and they satisfy fermion anticommutation relations. Here, as in I and II, we are primarily concerned with calculations of the energy difference between the first-excited state and the ground state A E = E 1 - E o . We determine this quantity indirectly through a study of the linear response function 6) R+(o~) = ~., [(nlJ+lO>lZf(co-E,+ Eo),
(1.2)
n
where s ÷ = E a ; +1 a., -1
(1.3)
P
is an operator introduced in I, and the states and energies are characteristic of the total Hamiltonian as given by eq. (1.1). The response function R+(og) is non-zero only for those frequencies ~o which are equal to the energy differences between the excited states n and the ground state, and provided that not all matrix elements (nlJ+10> vanish. F r o m symmetry considerations and the nature of J+ it is evident that only states with odd numbers of particles in the upper level will contribute to R + (co). The first peak in R + (co) is the one that interests us and the frequency at which it occurs gives E 1 - - E o. In the next section the algebra of computation is summarized with a prescription for drawing graphs and rules for calculating contributions appropriate to the model. Sect. 3 contains a rederivation of the results of perturbation theory obtained earlier in I. For such a calculation the Green function formulation is quite cumbersome
MANY-BODY APPROXIMATION METHODS (IlI)
213
compared to more direct methods and the number of contributing graphs multiplies very rapidly with increasing order of perturbation theory. It is only when alternative approximation schemes are sought that the present formalism shows its value. The random phase approximation (RPA) is easily obtained in sect. 4 and it is shown to be in effect the first term of an expansion in powers of 1IN in which the coefficients are arbitrary functions of (NV/s). The first correction term to the RPA in this series is also calculated. However, it is found that the convergence of the series is very poor unless N is quite large and of the order of 50 particles for the range of parameters described here. A completely different set of corrections to the RPA is considered in sect. 5. These are the exchange and self-energy corrections, and they are shown to be much more important than 1/N corrections in the region N < 50. Inclusion of the self-energy corrections to second order in V/8 results in a quite accurate determination of AE which is very similar to that obtained in II by using "empirical" singleparticle energies. The final section contains a brief summary and conclusions.
2. The Algebra of Computation The response function R+(og) can be expressed in terms of the real part of an integral 6) over the Green function G(t): R+(og) = _1Re
dt e'°"G(t),
(2.1)
7~
where
G(t) =
(2.2)
and J_ (t) is the Heisenberg operator corresponding to J_ = ~
a p+ ,_lap,+l
•
(2.3)
p
As in refi 6), a set of rules can be formulated for constructing a set of graphs and determining the contribution of any given graph to G(t). In the present case, however, many graphs give a vanishing contribution due to the nature of the interparticle interaction which can either create or destroy two particle-hole pairs, but which cannot scatter particles, holes, or particle-hole pairs between p states. As a result, each closed particle loop in a graph must have an even number of vertices. In addition, only graphs with an even number of interaction lines can contribute to R+(o~). Nevertheless, even with a pure particle-hole interaction, many graphs still contribute in addition to the familiar simple bubble graphs. Typical contributing graphs are shown in fig. 1. These graphs represent all possible time orderings of the interaction lines. However, it is convenient in this case to require that the particles (upward) or hole(downward) nature of the solid lines should not be altered in the time reordering,
214
A.J. GLICKet al.
The Green function can be constructed f r o m the graphs labelled with frequencies according to the rules (i) a factor V/(2~i) for a dashed interaction line, (ii) a factor i toi + ½~-+ i~
(2.4a)
(2.4b)
for a particle (upper sign) or hole (lower sign) line labelled by frequency cot, (iii) an overall factor N l (-)
e -''~'
4~2
(2.4e)
where N is the n u m b e r o f particles in the system, I is the n u m b e r o f closed particle loops in the graph and to' is the frequency carried by the wiggly J+ line, (iv) integration t over all frequencies tot and to' from - m
J
(o1
(d)
(b)
(c)
(e]
(f)
JY
to oo.
(g) (hl (i) Fig. 1. Typical graphs which contribute to Green's function G(t). These graphs represent all possible time ordering of the interaction lines, but the particle (upward) or hole (downward) nature of the solid lines must remain unaltered in the time reordering. The two wavy lines are separated by time t with time flowing upward. These rules were constructed in such a w a y that the n u m b e r o f "particle" lines in contributing graphs are in no way limited by, and can exceed, the n u m b e r N o f actual particles in the system. The N in rule (iii) arises from the summation over the degenerate p q u a n t u m number: p is the same for all particle lines in a given closed loop, b u t can take on any one o f N values. Further consequences o f this rule will be discussed in sect. 5. t The integrations over to and t are most easily performed as indicated in appendix B of ref. ~).
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215
3. Perturbation Theory As a first application of the present formalism, the perturbation theory results of I can be rederived. For this purpose one must combine graphs according to the number of dashed interaction lines they contain, so that the only zero order contribution is that shown in fig. 1(a). An elementary calculation gives
R+(CO)= N6(co-e).
(3.1)
The argument of the 6 function gives the expected excitation energy characteristic of the system without interactions: E 1 - E o = e. In second order, the non-vanishing contributions are given by the remaining graphs in fig. 1. The corresponding result for R+(co) is
R~(co) = N(~(co-e)+ N(N- I)(N- 3)V2 6(co-8)+ N(N- I)(N- 3)V2 6'(co-8), 482
2e .
(3.2)
containing a term with a derivative of a 6 function. These terms give R + (co) to second order in Vie. However, we need the argument of the 6 function to second order. We can infer it from eq. (3.2) by recognizing that this expression arises from an expansion of an R~+ (co) of the form
Rr2+(co)= (N=FaV2] bY2-).8:2 ] ~ (co--8+ Expanding the 6 function in a "Taylor Series" about eq. (3.2), we obtain
R2+((-D) IN--I-N(/~'l)(/~-3)V2] 6482 j :
(3,3)
Vie =
0 and comparing with
(-0.~-~-[-(N-1)(N'3)V2"~28 .] ,
(3.4)
so that
E1-Eo) 8
=
I_½(N_ I)(N_3)'(V'I 2,
/2
(3:5)
\8/
in agreement with eq. (,4.3) I, of the perturbation theory result in I to second order. In like fashion we can extend the calculation to fourth order. However, the number of graphs multiplies very rapidly with increasing order and such a scheme rapidly becomes-impractical.
4. The 1IN Expansion A different approximation method which has frequently been used to calculate excitation energies is the random phase approximation. This method was originally developed for treating the interacting electron gas a), and it gives very good results
216
A.J. GLICKet al.
provided the particle density is high. It has also been applied to a variety of other systems including the Heisenberg ferromagnet 9), superconductors lo) and finite nuclei 4, s), though perhaps the physical justification for using the approximation in these cases is not quite as apparent as for the high density electron gas.
y.j (o)
(.b) Fig. 2. The class of graphs which contribute to G(t) in the random phase approximation. (a) Green's function expressed in terms of the effectiveinteraction which is the solution of the integral equation represented in (b). Only odd numbers of bubbles contribute to 3v'°(co)for the model under consideration.
The excitation energy can be obtained in the random phase approximation with the present formalism by combining the contributions due to all graphs consisting of a simple chain of "bubbles", including such terms as shown in figs. l(a) and l(b). The class of these graphs can be represented as in fig. 2(a) where the heavy horizontal line represents an effective interaction which, for the present model, contains only an odd number of bubbles as indicated in fig. 2(b). The integral equation represented by fig. 2(b) can be solved very simply to obtain the effective interaction as (
2hi
=
N V2
.
(4.1)
2hi (ojt_4B2_N2V2..~io~)((Dt-~4~2-N2V2-i~)
Inserting ~/r°(co') into the calculation of R+(co) gives R~-pA(f..0) = N(,~+ 4~- ~ - N2_V2) t~(f,o- 4~~ - 2 ~/e2 _ N 2 V 2
N 2 V2),
(4.2)
for which the corresponding excitation energy is (4.3) / RPA
MANY-BODY
APPROXIMATION
METHODS
217
(lIl)
This result is meaningful provided NV/8 < 1. Note that (4.3) is the same answer as was obtained for the limit of large N with a linearization method discussed in II. The present formalism shows very clearly that the random phase approximation should give accurate results in the large N limit if VIe is not too large. Reference to rule (iii) of eq. (2.4) shows that for large N the graphs with the most closed particle loops should dominate. In any given order of perturbation theory, say the nth order, the maximum possible number of closed particle loops which can be formed with the available lines is n + 1, and only one graph can appear in that order with all n + 1 loops. This graph consists of a simple bubble chain. Thus the RPA gives the same result as would be obtained by summing perturbation theory to all orders, but retaining only the highest power of N in each order.
~
w )-
1.0
~X
I/N
E l~J Z IU
.7
/L
.6 I
2
4
I 8
I 14
I 30
I SO
I
I00
I
200
NUMBER OF PARTICLES
Fig. 3. The excitation energy (E1--Eo)/tplottedas afunction of particle number N for NV/e = 0.8. The crosses indicate exact values; P~ indicates the result obtained with perturbation theory to second order; L is the RPA result; and L~/Nis the RPA with 1IN corrections. In fig. 3 the excitation energy is plotted versus particle number with fixed NV/e. The RPA result is shown by the dashed line marked L (for linearization). As might be expected the RPA is rather poor for small N and is as good or better than the secondorder perturbation theory result only for N > 75. Thus it is of considerable interest to determine corrections to the RPA for smaller N. Note that the response function eq. (4.2) contains the interaction strength V only in the combination NV/e,. Thus this approximation can be interpreted as the first term in an expansion in powers of 1IN in which the coefficients are arbitrary functions of NV/~ (or an expansion in powers o f Vie in which the coefficients are arbitrary functions of NV/e). We next calculate the second term in this expansion in order to see how great an improvement is obtained with the inclusion of 1IN corrections. The relevant graphs are shown in fig. 4 and can be constructed by adding one effective (simple bubble) interaction in
218
A.I. GLICKe t
al.
every possible way to the graphs of fig, 2, In general, additional contributions could arise if the extra interaction connects the parts of previously unlinked graphs. However, for the model under consideration all such contributions vanish.
(o)
(b)
~/(~)m
-
-
-
-
+
(o) Fig. 4. The 1/Ncorrections to Green's function. (a) The contributing graphs (withwavylinesomitted) expressed in terms of the odd and even bubble interactions and effective self-energy corrected ble bubble.. (b) The effective bubble. (c) The effective interaction with even numbers of bubbles. It is now necessary to take into account both the Odd bubble interaction of eq. (4.1) and also the interaction with an even number of bubbles. 3¢'c(o9') as indicated by the heavy broken line in fig. 4(c). Carrying out the integrations impiied in fig. 4(c) one finds oe.~(oy) =
.
V(co'2 - e 2)
( o ; - ~/~2 _ N 2 v ~ + i ~ ) ( o , , + 4 ~ ~ - N ~ v ~ -
(4.4)
~)"
N o w more complicated "bubbles" also appear: There is the bubble which appears in the last for graphs o f fig. 4(a) containing a 3¢re(¢o') interaction between the particle and hole, and also the b~ bubble shown in fig. 4(b) which contains a 3V'°(co') selfinteraction of the particle or the hole. The evaluation of the contributions from the graphs in fig. 4(a) is straightforward but tedious. One obtains terms containing a 5 function a n d terms with the derivative of a 6 function. Using the same argument
MANY-BODYAPPROXIMATIONMETHODS (III)
219
as for obtaining eq. (3.4), from eq. (3.2) we find to order 1/N
R~/N(¢O) = ½N [~ + ~/~2 _ N 2V 2 _ (NV)2e(3e_ x/e2 _ N 2 V2)t t 4~2-- N2V 2
~
~
(4.5)
J
x ~ (03 -- 4 ~ 2 -- N 2 V 2 - ( N V)2e(2e2 + N 2 V 2 + 6e~/e 2 - N 2 V 2)
NC_
N2V2)2
)"
The corresponding excitation energy is thus (El-EO) = 41-(gv/l~)2÷ ', e, /1/~
(N VIe) 2[2 + (NV/8) 2 + 6x/1 -- (N V/e) 2] NEI_(NV/e)2][I+41_(NVI~)2]
(4.6)
2
In fig. 3 the curve marked Lll N shows the behaviour of the excitation energy in this 1/N corrected random phase approximation. It can be seen that the I/N terms give a strong overcorrection for small N so that there is no net improvement over the ordinary RPA unless N > 12. For larger N the 1/N expansion appears to converge more rapidly. However, for N < 40 this result is not as good as second-order perturbation theory, and it only improves fourth-order perturbation theory provided N > 45. The structure of second order perturbation theory (see eq. (3.5)) might lead one to expect that the 1/N terms would lead to a real improvement of the RPA for N > 4; however the 1IN terms of higher order in Vie contribute significantly to eq. (4.6) and actually decrease the accuracy of the result until N is considerably larger. The next correction term in this expansion, consisting of the 1/N 2 terms, would account for all terms of second-order perturbation theory, among others, and presumably would give a better result: Terms up to 1/N4 are required if all terms of fourth-order perturbation theory are to be brought in. However, such a programme is very difficult to carry out owing to the large number and complicated nature of the graphs which must be considered.
5. Exchange and Self-Energy Corrections In dealing with systems of few particles a different set of graphs is often included along with the bubble graphs in order to improve the RPA. These are the exchange counterparts of the bubble graphs containing particle-hole "ladders". Examples of such graphs are shown in fig. l (c), (d) and (e). In general they are included either because the force has a strong exchange part or to take better account of the exclusion principle. If one attempts to interprete any of these graphs in terms of the actual motion o f the particles in the model under consideration moving between ~r levels with the flow of time upward it is easily seen that these graphs do not correspond to physical processes. Indeed, they imply the simultaneous presence of more than one particle with a given p, and hence violate the exclusion principle. They arise in this
A.J. OLICK et al.
220
formalism to compensate for the counting error made in computing a corresponding simple bubble graph. The N which appears in rule (2.4c) arises from a sum over all p values which can label the closed particle loop. However, if more than one closed loop is present simultaneously in a graph, the contribution in which they have the
Fig. 5. The graphs contributing to G(t) in the bubble plus ladder (RPA with exchange) approximation. The cross hatched interaction can be obtained from the integral equation represented by the figure and then used in the bubble on the left to determine G(t).
>. o td
I.C
""" ..
0.9 ,
LSE
f'..
z
tu
Z 0
o.a L E ' ~ ' " " ' . ~ x O.IJ
X bl
o.Q - -
2
" ~
_l_ I
I
I
I
4
8
14
30
I
I
50
I
I00
I 200
Fig. 6. The excitation energy versus N for NV/e = 0.8. The crosses indicate exact solutions; L is the R P A result; Ls was obtained with the R P A plus exchange corrections; Ls~ is thesame as LE, but with additional self-energy corrections; Lc is taken from II and represents the result of linearizing the equations of motion with "experimental" single-particle energies.
Fig. 7. The renormalized single-particle propagator using the self-energy operator to second order in V.
same p is also non-physical and violates the exclusion principle. For large N the redundancy in counting this one state has little effect, but for small N the exchange terms which cancel these errors can give rise to significant corrections.
MANY-BODY APPROXIMATION METHODS (1II)
221
The contribution of both the bubble and the ladder graphs to the response function can be obtained for this model with a technique similar to that used in the last section to obtain the RPA. Now the Green function is expressed in terms of a different effective interaction indicated by the cross-hatched boxes in fig. 5. Solving the integral equation implied by that figure, we can obtain the exchange corrected response function
R+(¢o) = N(e+~/8 2 - ( N - 1)2V2) ~(¢0- x/~--(N- 1) 2V2), 2,/, ~ - ( N - 1)5v 2
(5A)
and excitation energy
E
/E
: E1
1)2
3,
Note that eqs. (5.1) and eq. (5.2) are identical to eq. (4.2) and eq. (4.3) except that improved counting has replaced N by N - 1. Eq. (5.2) was also obtained in II where a normal ordering of operators was chosen prior to linearization. The choice of normal ordering seemed somewhat orbitrary in the linearization procedure, and only the large N limit was uniquely defined. Thus one could claim that the present formalism vindicates that choice. A comparison of the RPA and the RPA with exchange was already carried out in II and is repeated in fig. 6 for NV/e = 0.8. The curve marked L represents the excitation energy according to RPA and LE represents AE/s for RPA with exchange. A further improvement in the calculation was obtained in II by introducing "experimental" single-particle energies in place of ~. The experimental energies contain some self-energy effects due to interactions with the physical vacuum or core states of the system. With the present formalism one need not resort to "experiment". It is possible to calculate self-energy effects on e directly. As is well known such effects appear graphically as corrections to single particle lines which leave the particles in the same state as they were in initially. Figs. 1(f), (g), (h) and (i) illustrate graphs containing such corrections to second order. These corrections can be introduced systematically into the calculation by replacing all single-particle propagators by renormalized propagators which satisfy an integral equation containing the unrenormalized propagators and a self-energy kernel. Keeping the self-energy only to second order, the integral equation can be represented schematically as in fig. 7. Solving this equation, the corrected propagator which replaces the factor in rule (2.4b) is
i
~
~+4~2+(N-1)v 2
24~ ~+ ( N - 1)V~ L(o,,_+½~~-4
~
2_+i~) ~ - 4 ' ~ + ( N - 1 ) V2
(5.3)
^ . J. GLICK e t al.
222
Substituting this factor for each of the particle lines in fig 5, we obtain after a tedious calculation the exchange and self-energy corrected response function
_
RS'E(~)
N
{[(A-1-2x/l+nu2)(A+3+nu2)(A-l+2x/l+nu2)~ 4 l--+nn~2A(A + B)(A - B) 6(m--eA)
-E(B-i-2x/l+nu2)(B+3+nu2)(B-l+2x/l+nu2); 2B(A + B)(A-B)
(5.4)
6(co-eB)}
where we have used n = ( N - 1) and u = (V/e) and A = [2(1+nu2)(5+4nu2)-n2u2-C~ 2(l+nu2) ,
(5.5)
B= [2(l +nu2)(5+4nu2)-n2uZ +C~:r 2(1 + nu 2) '
(5.6)
C = [(211 +nu2][5+4nuZ]-n2u2)2-4(l +nu2)([1
+nu2][3 +4nu2] 2 -n2u213+nu212)]~.
(5.7)
Note that two 6 functions appear in eq. (5.4). As remarked in sect 1, all odd excited states should contribute peaks to R+(cn), but this approximation is the first one sufficiently complete to exhibit more than one peak. By considering the limit as u ~ 0 it is seen that the peaks arise from the first and third excited states of the system. The position of the second peak gives the excitation energy of the third-excited state above the ground state in this approximation as
t 8
= 8
(5.8)
I SE
However, we are primarily interested in the position of the other peak which gives El E _ -_ oI e
= A.
(5.9)
]SE
The exchange and self-energy corrected excitation energy, eq. (5.9) is plotted in fig. 6 as the curve marked LSE. Note that LSE provides a rather good approximation procedure over the entire range of N, though perturbation theory to fourth order (see II) is still somewhat better when N < 50. In practice, the importance of the LsE approximation is enhanced by the fact that it is often more easily obtained than the carrying of perturbation theory to fourth order. Also plotted in fig. 6 is the curve Lc which was obtained in II by using "experimental" energies to correct the linearization procedure. Note the close agreement
MANY-BODY A P P R O X I M A T I O N
METHODS (II1)
223
between the results derived with these two methods. These approximations become identical if the "experimental" corrections to the energy range are retained only to second order and the propagator (5.3) is approximated by letting V --, 0 except that in the cos dependent denominator, the term of order V 2 is retained (see curve Lc2 in fig. 4 of II). However, the more accurate calculation shows that the use of experimental energies is not completely equivalent to the introduction of renormalized single-particle propagators, and for N > 4 it appears that LsE is more accurate than Lc. Nevertheless, for the model under consideration the higher order terms do not play a major role in this approximation and the difference is not very large.
6. Summary and Conclusions In this paper a function R+(co), describing the linear response of a model of a many-body system has been studied in order to determine the excitation energy of the first-excited state above the ground state of the system. The response function was calculated in various approximations by means of a diagrammatic many-body perturbation theory technique and the selective summation of high order graphs. The bubble graph sum or RPA was shown to be the first term in an expansion in powers of 1/N where the coefficients are arbitrary functions of NV/~. However, the 1/N expansion was found to converge very slowly (if at all) for small N. The first correction term gives an overcorrection for N < 40, and actually decreases the accuracy of the absolute value of the excitation energy for N < 12. Correction terms of a quite different type were found to play a much more important role than the 1IN corrections for small N. These are the "exchange" graphs which correct for counting errors and violations of the exclusion principle when enumerating the states that a given particle can occupy. The "ladder" graphs fall into this class and must be included along with the bubbles to account for the unavailability of certain states when more than one particle is simultaneously excited. The single-particle self-energy graphs play a similar role in that they correct for shifts in the single-particle energies and for counting errors in the eigenstate normalization introduced by using a linked cluster expansion t. The ladder and self-energy corrections are important whenever the number of excluded states is comparable to the number o f states available to particles in the system. For the model considered this situation occurs when N is small since there are only as many states as particles. In other cases it can also occur because of a short-range interaction which limits the number of states a particle can reach. With these correction terms a rather good approximation to the excitation energy can be obtained, though the results are not quite as good as that obtained with fourth-order perturbation theory for small iV, or for the whole range of N with the self-consistent linearization procedure discussed in II. An unfortunate aspect of the exchange and self-energy corrected summations is that they do not arise naturally from a consistent expansion in terms of the parameters t See Goldstone s).
224
A.J. GLICKet al.
(here N, V and e), describing the system. Hence, further correction terms are not uniquely defined andit is in general difficult to discuss the convergence of such a procedure. Also any attempt to do better by introducing exchange corrections into a series such as the 1IN expansion is plagued by the presence of many redundant terms. The next approximation which might be attempted to improve upon the self-energy and exchange corrected R P A is one in which the exchange and self-energy corrections of figs. 5 and 7 are introduced using the effective interactions 3e~e(co) and 3¢/'°(co) in place of the bare interaction V. Such a procedure was used in ref. 7) for treating the electron gas and was shown to be a "conserving" approximation for obtaining transport properties of many-body systems by Baym and Kadanoff 11). However, it is not certain that these corrections wiU give a significant improvement for small N since higher order exchange terms are not consistently retained in such a scheme. In addition, the detailed calculation is quite difficult even for the simple model considered here. We are grateful to M. Baranger, G. E. Brown and S. FaUieros for helpful discussions and correspondence.
References 1) H. J. Lipkin, N. Meshkov and A. L Glick, Nuclear Physics 62 (1965) 188 2) N. Meshkov, A. J. Glick and H. J. Lipkin, Nuclear Physics 62 (1965) 199 3) J. Goldstone, Proc. Roy. Soc. A239 (1957) 267; J. Hubbard, Proc. Roy. Soc. A240 (1957) 539; V. Galitskii and A. Midgal, JETP (Soviet Physics) 7 (1958) 96; A. Klein and R. Prange, Phys. Rev. 112 (1958) 994; P. Martin and J. Schwinger, Phys. Rev. 115 (1959) 1342; D. DuBois, Ann. of Phys. 7 (1959) 174 4) S. Fallieros, Ph.D. Thesis, University of Maryland (1959) 5) G. E. Brown, L. Castillejo and J. Evans, Nuclear Physics 22 (1960) 1 6) A. J. Glick, Ann. of Phys. 17 (1962) 61 7) A. J. Glick, Phys. Rev. 129 (1963) 1399 8) D. Bohm and D. Pines, Phys. Rev. 92 (1953) 608 9) S. V. Tyablikov, Ukr. Mat. Zh. 11 (1959) 287 10) P. W. Anderson, Phys. Rev. 112 (1958) 1900 11) G. Baym and L. P. Kadanoff, Phys. Rev. 124 (1962) 287