On the summation of generalized ladders for a manyfermion system

I 1.C ]

Nuclear Physics 20 (1960) 533--542; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

O N T H E S U M M A T I O N OF G E N E R A L I Z E D L A D D E R S F O R A M A N Y FERMION SYSTEM M. L. M E H T A t

Centre d'l~tudes Nueldaires de Saclay, Gil-sur-Yvette (S. et 0.), France Received 23 june 1960 A b s t r a c t : The analogy between an interacting fermion s y s t e m and a s y s t e m in an external potential, which m a y be n o n - h e r m i t i a n , is employed to show w h y the generalized ladder approxim a t i o n for an interaction a t t r a c t i v e near the Fermi-surface diverges.

1. I n t r o d u c t i o n

In the study of large interacting systems, several difficulties, such as due to hard cores, were removed b y making a partial summation of all particle-particle interactions 1), giving the so-called "ladder approximation". However, the t-matrix introduced thus m a y have in the case of attractive potentials, one or more singularities below the Fermi-surface yielding divergent integrals in the expression of the binding energy *). The idea 3) of summing all hole-hole as well as particle-particle interactions, hereafter called the "mixed ladder approximation", i~ defining the t-matrix has been shown 4) to lead to the same type of divergence difficulties as the previous one. A still better approximation wilt consist in summing all symmetric double looped graphs, hereafter called the "generalized ladder approximation". A typical example of such a graph is shown in fig. 1. One can systematically write down an integral equation for this class of graphs and solve it exactly. The reason of this simplicity is seen in the presence of the following underlying one-body problem. As noticed b y several authors 5) the problem of a system of fermions in an external field is similar to the problem of a system of fermions interacting via two-body forces. The two problems are, in fact, exactly equivalent only up to the generalized ladder approximation. Fig. 2 shows the graph in the external potential case which corresponds to that of fig. 1. In the external field problem these are all the possible graphs, and their sum must therefore be equal to the exact energy value determined directly, which can easily be checked. The problem of interacting fermions, however, contains many more graphs, such as for t On leave of absence f r o m the T a t a I n s t i t u t e of F u n d a m e n t a l Research, Apollo Pier, B o m b a y . 533

534

M.L.

MEHTA

example those shown in fig. 3, which have no equivalent in the external field case.

. . . . . . . .

Fig. 1. A generalised ladder graph contributing to the ground-state energy of an interacting fermion system.

--][

Fig. 2. The graph corresponding to t h a t in fig. 1 for the fermions in an external field.

\

,

-r

I

+ -

/

+.o.

/

Fig. 3. Some graphs contributing to the ground-state energy of an interacting fermion system. These h a v e no corresponding graphs in the external field case.

We shall take a separable potential and consider only the states of total momentum zero. We will then show that 1) for the problem of fermions in an external field: a) the "ladder approximation" is not sufficient, as in some cases the binding energy is singular for arbitrarily small external potentials; b) the "mixed ladder approximation" is also not sufficient for the same reasons; c) The "generalized ladder approximation", which contains all the possible graphs in this case, gives a finite radius of convergence to the binding energy series, the sum of this series being equal to the exact result;

ON THE

SUMMATION

OF

GENERALIZED

LADDERS

535

2) for the problem of interacting fermions: a), b) the conclusions about "ladder" as well as "mixed ladder" approximations are the same as in the previous problem; c) the "generalized ladder approximation" also has a radius of convergence tending to zero as the unperturbed single-particle levels become more and more dense in momentum space.

2. A S y s t e m of F e r m i o n s in an E x t e r n a l Field This is obviously a one-body problem, as the Pauli principle can be taken into account by putting the fermions one by one in each of the lowest one-body states of the hamiltonian, until all of them are exhausted; the ground state energy being the sum of the energies of the occupied states. We record in sect. 3 the straightforward solution to this problem, in various approximations, for the sake of completeness and for later comparison as the corresponding approximations in the pair interaction problem are obtained by allowing the potential to become imaginary for some states. The hamiltonian is taken in the second quantized form as H = H o + H I + H 2 -~ (Ho++H0-) + (HI++H1-)-I-H2,

(1)

where in terms of the usual creation and annihilation fermion operators, we have

H0+ = X ek$**$,,

Ho- = X emSm*$m,

k

H I + --

*n

X (k[V]k )$k $k,, n t - = I¢, I¢"

X

(m]Vlm')$m*~,,,',

(2)

m , ~'*"

n 2 = - - ~_, { ( k [ V f m ) ~ * * $ , ~ + ( m l V J k ) $ , ~ * $ , } ,

with the convention t h a t k > k F and m < kF (kF being the Fermi-momentum). The ladder approximation consists in taking H :

H o + H 1 +,

and an interaction at each end of the ladder which scatters a particle into a hole state or vice-versa. The mixed ladder approximation is obtained by including H t - also H :

Ho+H1,

and completing the graph, as before, by one interaction at each end connecting a particle and a hole state. We will get the generalized ladder approximation, if we take the full hamiltonian H :

H o + H I + H 2.

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M . L . MEHTA

3. Diagonalization As H0++H1 + and Ho-+H 1- are completely disconnected, we can diagonalize them separately. For Ho++H1+, the Schrodinger equation is (H0++H1 +) X B , k ~ * = e, + • B,~k*, k

k

or

1

Ba, --

~
(3)

E k - - S i + k"

This equation is immediately solved for a separable potential * (ltVll') = V(l)V(V),

(4)

on multiplying both sides of it b y V(k), summing over k and assuming that the common factor does not vanish. Thus we see that e, + are the roots of

~(~+) =

V~(k) X - =

1.

(5)

k 8k - ~ +

Similarly ei-, the eigenvalues of Ho-+H ,- are the roots of

¢(~-)

V2(m) = Z

-

1.

(6)

ra ~m - E -

Let us denote b y ~/~, * ~ the creation and annihilation operators for the eigenstates of H0± + H I ± corresponding to the eigenvalues e±. The orthogonality of the transformations and the normalization of the eigenstates allow us to write simply

V(k) i

$~ =

" e~--e~+

~ (B-~)~,_

= ~

(7)

Y(m) V~n~_,

t Em--~j

J

where ai and b~ are given b y and

bj -1 = ¢ ' ( s f ) =

~

~=,_.

(8)

Changing the representation from ~ to ~, we have H0 + H1

=

~2 ei+~*+~++ ~ i

E ~. --

~ *_ ~ _ ,

j

t,j t F o r t h e p o t e n t i a l w i t h a n e g a t i v e s i g n i n eq. (4), t r i v i a l c h a n g e s i n t h e r i g h t h a n d s i d e s of eqs. (5), (6), (7) a n d (9) a r e n e e d e d .

ON" T H E SUMMATI ON OF G E N E R A L I Z E D L A D D E R S

53~

Now the SchrSdinger equation

(Ho+HI +H2)~r* = E,~,* can be solved in exactly the same way. In the above equation put

fr = ~ L , ~ , + + X L,~j_, to get

(~,+-E.)L, = ~

X V~L,,

(~f--~.lz. = ~'~ X ~ L , .

j

Eliminating the quantities

P+ = X ~/;,L,,

p = X V~L,,

g

J

we see that E, are the roots of the equation

a~b~ = 1. ,,~ (~,+--E) (~F--E)

Y.

We m a y put this equation in a somewhat more convenient form. Note that a_____L__~= I /' 1 dz ~7 z7 e,+--E 2zd ~,(z)--I z - - E

1 ~(E)--I

Y

1

r(E)--I

~(E)--~

~(E) '

where in the middle member above, the contour of integration is large enough to enclose all finite singularities of the integrand, which are simple poles; and this contour m a y be made to tend to infinity for later evaluation. Similarly we have bj _ ¢(E)--I

2~~;---E

¢(E)

Therefore, the eigenvalues E r of H are the roots of ~o(E)+$(E) = 1.

(9)

4. Interacting F e r m i o n s In the approximations where some or all of the symmetrical double looped graphs are taken, i.e. in any of the three approximations considered above, the terms in the perturbation series of the ground state energy for the problem of interacting fermions are equal, except possibly for the sign, to the corresponding terms for the problem of fermions in an external field. The possible change in sign is due to the difference in the number of hole lines as well as the number of closed loops.

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M.L.

MEHTA

Fig. 4. The function w(E)+~b(E) when the potential has the form (12).

Fig. 5. The function t p ( E ) + ~ ( E ) when the potential has the form (10).

ON T H E SUMMATI ON OF G E N E R A L I Z E D L A D D E R S

539

These signs can be properly accounted for, as can be seen from eq. (4) by letting the potential become imaginary in certain states, so that finally there is a relative sign change between V*(m) and V*(k). Let us put V (1) = ,~zvz,

with v~ real. Then if we take 2 k = 1,

2r e = i ,

(10)

we shall be getting terms for an attractive two-body interaction; while if we take ~k----i, 2~----1, (11) we will be getting those for a repulsive two-body interaction. The situation

~ = ~ = 1,

(or ~ = ~= = i),

(1~)

is the real external potential problem. The function ~ ( E ) + $ ( E ) is drawn in figs. 4 and 5 for the cases (12) and (10) respectively. For the case (11) one should turn fig. 5 upside down. The intersections of these curves with the line at a unit distance up give the roots of eq. (9). 5. D i s c u s s i o n It is a well-known fact t h a t for a system of free fermions, the ground state energy is singular in the ladder approximation if the unperturbed states are dense enougb near the Fermi-energy and have a negative external potential 6). The same conclusion is immediately derived for the mixed ladder approximation by drawing a graph corresponding to fig. 1 of ref. 4) for this case and the remarks at the end of sect. 2 there. For attracting fermions also, the ladder as well as mixed ladder approximation lead to similar singularities 2.4). For the generalized ladder approximation, it is shown in the appendix that the perturbation series when summed formally has the proper behaviour if all the roots E r of eq. (9) are real and has a singular behaviour if some of the roots become complex. Now for the case of fermions in an external field, we see from fig. 4 that all the roots E r are real for arbitrary values of the coupling constant. Hence we conclude that the formal sum of the series is always non-singular and this sum is seen to give the exact result by comparison. For repelling fermions, inverting fig. 5, we see that all the roots Er are real for arbitrary values of the coupling constant and hence a proper behaviour of the formal sum is concluded. On the other hand, for attracting fermions, fig. 5 shows that all the roots Er

540

M.L. MEHTA

are real except the two near the Fermi-energy, which eventually become complex if the coupling constant is increased past a certain critical value ge, say. And ge -+ 0 as the levels near the Fermi-energy become more and more dense. Hence a singularity appears for arbitrarily small attractions of levels dense enough near the Fermi-energy. Thus it is seen that in the generalized ladder approximation, the system of interacting fermions is equivalent to a system of fermions in an external non-hermitian potential; and though the problem is exactly solvable, the perturbation treatment diverges due to the appearance of complex energy eigenvalues in the latter problem, giving an exponential behaviour rather than an oscillating one.

I want to thank Professor C. Bloch for suggesting the problem, guidance and helpful critisism throughout the work; to R. Balian, M. Gaudin and J. Lascoux for m a n y interesting discussions, to Service de Cooperation Technique for a maintenance grant during the Summer of 1959 when this work was done and to Commissariat ~t l']~nergie Atomique for the kind hospitality at Saclay.

Appendix In this appendix we shall give briefly the steps and the main consequences of the time-dependent perturbation method. Following the general theory we first write down the integral equation for the particle propagator, i.e. symbolically:

---Taking

~-

--

as new variables

it is easy to solve the linear equation to get F+ (zl)= ~. a, e -`,+~,

with a~ and e, + as defined by the eqs. (5) and (8). The same holds for the hole-propagator

F_(A) = ~ b~e-',-", i

.ON T H E

SUMMATION

OF

GENERALIZED

LADDERS

541

with bt and ej- as given b y eqs. (6) and (8). Incidently we note that b y closing F+ with a hole-line we get the ladder approximation, while b y closing it with an F_ line we get the mixed ladder approximation. Writing the equation for the generalized ladders is now easy. It is symbolicaUy:

T+

--

F+

.~

+

It can be shown that this integral equauon is equivalent to a linear differential equation with constant coefficients. Putting = 81-/53,

7 =/5-/53,

one has then the explicit solution

T+(7, A) --- y~ Ar(y)e-~, 'J, where Er are determined b y eq. (9) and Ar are the roots of 1

~"

¢(E~)

Ar

e-Eta, ~A~

E~--ej-

-- 0.

(A.2)

Closing T+ with an F _ line and integrating over fll and t53 from 0 to 15 we get

1

<01U(/5)[0)c = ~l ~ c, fo ~ ArdT,

(A.3)

where <0[U(fl)10)e is defined after Bloch ~), the subscript c meaning the generalized ladder graphs; we have put

¢(E,) (E,) -- 1 With a little reasoning on the linear dependence or independence of the coefficients in eqs. (A.1) and (A.2), one can prove that 1) if all the roots E, are real, then the limit of the right hand side of eq. (A.3) exists for fl-~ oo.

542

M . L . MEHTA

2) if a pair of roots E, is complex, then the right hand side of eq. (A.3) does not have any limit as/~ -+ oo, because it will then be an integral of a function having poles almost periodically on the positive /5 axis. In the case where the limit exists, we will have for the shift in energy 1

AE ----- -- lim -- (OlV(fl)[O}e #~oo

/5

= (ei++... +

= (E'I+...

e++--E~--...--E~+)

+E'._--~I---...--~L),

if the roots are ordered as

E1> E~>...

> En+ > E' 1 > E ' ~ > . . .

> E'~_,

the integers n+ and n_ being respectively the number of unperturbed levels above and below the Fermi-energy. References 1) R e v i e w s of the method and lists of references are given by H. A. Bethe, Phys. Rev. 103 (1956) 1353 and K. A. Brueckner and J. L. Gammel, Phys. Rev. 109 (1958) 1023 2) L. N. Cooper, Phys. Rev. 104 (1956) 1189; A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936; J. Goldstone, Thesis, Cambridge University; K. Gottfried, CERN report (unpublished); L. v a n Hove, Physica 25 (1959) 849, Lecture at the International summer School of Physics, Naples (1960); C. Bloch, C. R. du Congr~s International de Physique NucMaire, Paris, July 1958 (Dunod, Paris, 1959) 243 3) J. S. R. Chisholm and E. J. Squires, A.E.R.E. Harwell report; also Nuclear Physics 13 (1959) 156 4) M. L. Mehta, Nuclear Physics 12 (1959) 333 5) J. S. Bell, Proc. Phys. Soc. 73 (1959) 118; A. Katz, Int. Summer School of Physics, Naples (1960) 6) J. S. Bell, ref. 5) a b o v e 7) C. Bloch, Nuclear Physics 7 (1958) 451