On model dynamical systems in statistical mechanics

Bogolubov 1966

Physica 32 933-944

Jr., N. N.

.

ON MODEL DYNAMICAL SYSTEMS IN STATISTICAL MECHANICS by N. N. BOGOLUBOV

Jr.

Steklov MathematicalInstitute, Academy of Sciencesof the U.S.S.R., Moscow Some dynamical models generalizing those of the BCS type are investigated. A complete proof is presented that the well-known approximation procedure leads to an asymptotically exact expression for the free energy, when the usual limiting process of statistical mechanics is performed. Some special examples are considered.

Model dynamical systems of the BCS type have been extensively studied in the theory of superconductivity. Their importance lies in the fact that the “model approach” permits us in a relatively simple way to get insight in the fundamental properties of superconductivity. It has been recognized long ago that for the model systems one easily obtains asymptotically exact expressions for the free energy (By asymptotically exact we mean that the corresponding equality becomes exact after the usual limiting process of statistical mechanics (I’ -+ ~9) had been carried out). Let us now consider a model Hamiltonian of the usual BCS type: H =

I;

Tf.ataf

-

(f)

In the papers by N. N. Bogolubov, D. N. Zubarev, Yu. A. Tserkovnikovl) 2) a method was formulated to obtain the asymptotically exact expression for the free energy. In this method one starts by introducing the so-called “approximation Hamiltonian” Ho(C), a quadratic form in the fermi operators, depending upon some arbitrary constant C. A simple diagonalisation procedure permits then to calculate the expressions of the corresponding free energy Fe(C). The approximate value Fe of the real free energy F (for the Hamiltonian (1)) is obtained by finding the values of C which minimise Fa(C). Fe = min Fe(C). (2) (0 In these papers some arguments were given leading to the conclusion that such an Fe becomes asymptotically equal to F as V + 00. -

933 -

934

N. N. BOGOLUBOV

JR.

These considerations were formulated for a model Hamiltonian of the general form (1) but they were not sufficiently rigorous from the mathematical

point of view.

The attempts to present a complete proof of the asymptotic equality Fo N F have shown the existence of many difficult points. Such difficulties were resolved in a paper by N. Bogolubovs)4) but only for the case of zero temperature

and only for a separable kernel

3(f9f’) = 4f) *4f’).

(3)

Both these conditions are essential for the given proof. One may say that a model was considered where “Cooper pairs” interact only in s-states. In recent years interest was attracted to a more complicated situation where the interaction of “pairs” realizes itself in p, d, . . . statess-9). For the study of this situation we consider a Hamiltonian of the form (1) but with a kernel 9(f, f’) being represented by a sum of separable kernels, more specifically as a sum of spherical harmonics with given angular momenta I = 1, 2.. . . It may be stressed that in this case one must take into account the possibility not only of statistical degeneracy with respect to the gauge group (as in (1)) but also of statistical degeneracy with respect to the rotation group. As far as I know, the first complete proof of the asymptotic equality, which is quite rigorous from the mathematical point of view, was given for such a Hamiltonian by E. E. Tareyevaie). Her proof, however, covers also only the case of zero temperature. In my papers I have worked out a method, which permits us to present a complete proof of the asymptotic equality of Fo and F, valid both for 8 > 0 and 0 = 0. We start from the Hamiltonian of the form:

H=T-2V

x

fLW.$$

(4)

lGC$?

If we take here as operators

T =

z

T, $a the usual quadratic expressions:

Tf.at+af;

ja

=

(f)

we obtain the BCS Hamiltonian of s separable kernels :

(1) with the

4(f, f’) represented

by a sum

s

y(f, f’) = lx Mf).qf’). a=1

I wish to point out now that in fact we do not need the specific pressions (5).

ex-

ON MODEL

DYNAMICAL

SYSTEMS

IN STATISTICAL

What is really needed are some general conditions

MECHANICS

935

of the form

Where I... / is the norm of the corresponding operators, and Kl, Kz, KS, are constants as V -+ co .We suppose also that the free energy per unit volume for the “zero” Hamiltonian H = T is bounded by a constant and that the number s is fixed. In the end of this paper we show how the last conditions may be omitted. We thus start from the Hamiltonian (4) with the conditions (7). The approximation Hamiltonian we take in the form H”=T-22V

C

(C,*$,‘+C&)+2l’

Here C, - are complex absolute minimum

z

C&x.

(8)

by the condition

of

1 ef,GS

1
constants which we determine

of the function f(C1, ***C,) = -

+

In Tr e--H0’e

(9)

in the domain

of all complex variables (Cl, . . . C,). This complex set of points (Cl, . . . C,) we shall denote by {Es). The purpose of this paper is to show that the difference fH0- fH tends

to zero as V -+ 00. We shall make now some comments about the general idea of our proof. It begins by considering the auxiliary problem with the Hamiltonian

where vi, . . . vs - are some nonvanishing complex parameters. Their role is to get rid of possible statistical degeneracies. If they were all zero the new Hamiltonian (10) reduces to the original one (4). But we first consider the case when va # 0. (1 < a < s). We construct for this r the approximation Hamiltonian P way as for (4) P=H”-

I’

z 1 =a<8

(va*$a+vz$z).

in the same (11)

We shall obtain estimates for fro - fr, by applying our majoration technique from which will follow that the difference Ifro- fri < &(1/V) is asymptotically small as V -+ 00 Here fro = min fp(C1, . . . C,) and fr - free energy per unit volume for the UP) Hamiltonian F.

936

N.N.BOGOLUBOVJR.

It is easy to show that the problem of the absolute minimum fro(C) = has a solution CK = C&

and this

$-

absolute

of the function

In Tr e-“ie

minimum

(1 < K < s). One can be

is realised

convinced

for finite

values

of this by using the in-

equalities x (lcd2 +

- 4K;s

(Ical+ 2Kd2}

+ y +

fT

>

fro(c)

2

(a)

C =

2

C {/Cal2 (co

b

c ~CCX/~ (a)

(Cl,

Because the function point C = CO

+

(ICal

--2K1)2}

4K;Ls -

y +

. ..C.),

y =

f&C)

is continuously

2K1

-

4K;s -

y +

(1

2

fT

Z Id, (4

differentiable

or, more explicitly

we have in the

i Cl9 = <$s>ro

Therefore,

fT

Tr $b e-r0/0 Tr e-j-0,0

=

(14

from (7) it follows that: [C,J/< KI;

K1 = const.

We shall try now to express the difference

fro -

fr

in terms of the averages

-

CC)

of the type -

2l V

= -2

z

($Ly -

C,)($,+

(13)

1
We keep in mind that

and introduce

an intermediate

Hamiltonian P =P

+ t *$X

(14)

(It is clear that for t = 0, P = P; t = 1, P = F). The constants (Cl, . . . . . . C,) are to be fixed independent of the parameter t. Let us consider the configurational integral Qt = Tr e-Ftle and the free energy for the intermediate hamiltonian P f&l,

**aC,) =

-

-

8 V

In Qt.

(15)

ON MODEL DYNAMICAL

SYSTEMS IN STATISTICAL MECHANICS

937

Differentiating (15) twice with respect to the parameter t by using operator differentiating we have: _e

V

Pfrt atz+s(?!$)2

= ~~rr~~e-~/eue-(r~/ex’-T)~d~. 0

In view of

aft

_-

at-

1

Tr% ePFle

V

Tr e-l’le

= -+

we can obtain

a2ft

- -

at2

= ~

1

1

0VQ s

Tr(% e -P(r)/@‘$j e-r’(i-r)/e} d7

0

Let us use a matrix representation in which the Hamiltonian I’t is diagonalised ; we have

1 = 0V.Q

s

-[(_w-iwvel

C l‘&m12e

T-mw

2

0.

n,m

0

Hence it follows in particular

and thus

aft -- <%I>t atv decreases with increasing parameter t. Continuing these calculations one can obtain the inequalities *) 0


fr 62.

(I<~

-

V,‘>r)>r.

(16)

Here as always

fro = min fro(C). W’l

Let us remember our main problem. We want to show that the difference *) It is convenient for us not to indicate explicitly the statistical averaging, corresponding to the Hamiltonian r, so that we shall write <...>r = (...>.

938

N. N. BOGOLUBOV

JR.

fr0 - fr is asymptotically small as V --+ co. It follows from (16) that we should have solved our problem if we could show the asymptotically smallness of the average

on the right-hand

side of (16).

Taking into account the main idea of an earlier paperii) right-hand side in terms of

Carrying

out the differentiation

1 a2f ---------_~ 0 av,av;

we express this

we have

v2 ‘Tr(Dla, e-(T/o)r D+(a)e-[(l-“/“lr) & Tr ePr10 82 s 0

where J!)(a) = $a

-

Turning to the matrix diagonalised we get

representation

Using Holder’s inequalities

we have:

Then after a simple transformation

(1
<$a>,

in which

the

we get

where M = Ka + 4KlK3.s Hence we obtain

+ 2K3 i

a=1

lvn[

Hamiltonian

is

ON MODEL

DYNAMICAL

SYSTEMS

IN STATISTICAL

MECHANICS

939

It is to be noted that

+

v.

c

Q nm

wn

ID(a) 12. I,--@n/O

_

,-Em/el

where 2 DtAls ewE@ = $ %m

s

Tr D(a) .D+(a) e-r/e =

= V = V. <($a -

<$a>). (f+a _ <$+a>)>

thus we obtain finally (V”

-

)($‘”

Substituting

-

<$+a>)>

this inequality

<

in (16) we find

(18) Hence we see that our problem would be solved if we could show that the second derivatives are bounded by a constant. Unfortunately we are not able to prove such a statement. We have at our disposal only the more trivial inequalities :

and we must therefore use a more sophisticated First let us introduce

argument.

instead of va, vc the corresponding

polar variables:

then

(20) Denote

:

940

N. N. BOGOLUBOV

and take into consideration 1471,

**. 7s;

q%

*a* @)I

<

--a(El,

the following

1471, **. ts;

*** 7s; y1, 71,

JR.

inequality

*a* fps) -

+*a +)I

+

I&

*** ts;

q1,

*** rs)l

<

and

Here

- denote the maximum Since ll-31;2K’:

values of the derivatives

//$+2Klra;

li-3~<4x1;

in the above intervals.

I/~()
(23)

it is clear that

< 4Kls.U

+ 4Kl.l.s

= 12sK1.1

(24)

The expression ~(51, . . . ts; 71, . . . qs) may be majorated from the formulae (22) and (18). After simple calculations we obtain the estimate for the difference considered 471,

Osfro-fr~12K1Est~

. . . 7s;

p1,

0.2Kl

.** ps)

C 1G%s

=

fro -

fr;

(b + 2) + 1.60

ON MODEL

DYNAMICAL

SYSTEMS

IN STATISTICAL

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941

Choose here R

so that

Now with the help of the obvious inequality (1 + 2R)“‘” < 2”‘“(1”‘”+ (2R)“‘“) from (25) we find 0
-

20K1

fr Q 12K1*Zs + p.vp

J,p”

.2”,S I

(4K1)“’ -7 1 Ia

+ -yp’ In all these formulae so that

s(l + 2R) + Ma’” . s. (4K1)“” . 3”“. R”,” pa V ‘Is

1 was an arbitrary

positive

quantity.

(26)

Choose it now

M”‘“(4Ki)“‘” . 22,8 j,Pla. I”!”

12Kll = or

I= By substituting

0
- fr <

this expression 24.Ki.s

M%

P Ir’l”;

P=

3*/s. K“”

= const.

for I in (26) we have

P

T/“I” +

28.Ki.s I/%.p

4%K1-s-R +

T/““.p2

M”l” +

T/“/l” . p’h

+

. s . (4Ki)% .2’/3 . R”l”

We see thus that the difference fp - fr tends to zero as V tends to infinity. But for finite V both functions fro, fr are continuous with respect to vu, and vu does not enter in the denominator. We can therefore put here vu = 0 and obtain 0 <

fHo - fH G

24Kl.s

P

vK

+

28.Kl.S vll,5.p

(27)

In a similar way we see that we can put here 0 = 0 so that this inequality holds also for the zero-temperature situation. Because we have not fixed the concrete expressions for our $Fa, T, the obtained result (27) holds not only

942

N. N. BOGOLUBOV

for the Bardeen

type

Hamiltonian

JR.

with the kernel

.Y(f, f’) presented

by

a sum of separable terms, but also for some other model Hamiltonians. It is to be noticed that so far we have considered only the case when s was fixed.

equalto fH f~0is aSymptOtiCally

We have also extended our result that to the situation where s = 00~2). Let us put

H=T-2V

C

$a.,$,+.

(28)

16%<-

We shall consider the case when the convergence strong enough so that the norm:

of the operator

series is

(29) uniformly where

in regard to V -+ co,

We suppose also that the inequalities (7) are fulfilled for all the indexes. We show that in the case considered we also can state \fH -

Here fH0 is the fw(... C,...) of the

aS

fHO\ --f 0

v

--f co

minimum value of the free energy per unit volume corresponding approximated Hamiltonian (see (8) when

s = oo). It is easy to prove that IfH -

fHo/

<

E (+)

+

(30)

s-+co

0

But since fH we

0 d

<

fH0

<

fH,%

get in view of (30) fH0 -

fH

<

fH.0 -

fH

=

(fH,o -

fH,)

+

(fH, -

fH)

Notice, that in this inequality the value fH,O(see 27). Here we merely transform our notations

H-tH,;

((s, l/V) - is the right-hand

fH,”-

(fH.0 -

fH,)

fH,

<

5

(

s,

we have from (27) +

side of the inequality.

)

+

& f (

also has been estimated

H”-+H,”

because the index s now is not fixed. Therefore 0 <

fH,

<

;

)

ON MODEL

DYNAMICAL

SYSTEMS

IN STATISTICAL

943

MECHANICS

As for any fixed s the ((s, l/V) --f 0 as V --f CXJthus

Since the evaluated

difference

fH0- fH does

not depend on s we can choose

s arbitrary. Choose s so that

then finally we have

And this inequality proves our assertion. As an application of our method we consider the following example. We take the Hamiltonian (1) with the kernel represented by

(31)

Y(f>f’) = 5 k(f) *C(f’) a=1

in which the functions the series

A,(f)satisfy m I’

Xi is uniformly

convergent

\v

all the conditions

z

(7). Let us suppose that

Mf)l 2 1

with regard to I’ + 00 then it is easy to see that

in this case (29) is satisfied

and because

of this our method

is valid for the Hamiltonian

(1) with a

kernel of the form (31). In conclusion we notice that the obtained technique may be applied to the dynamical model of a crystal considered by B azarovia) 14) H =

x (f)

1 T~*ar+a~+- 2V

where pq = C af+,q*af. (f)

(5;;,hl'Pcl"P-q

944

ON MODEL DYNAMICAL

SYSTEMS

IN STATISTICAL

MECHANICS

The vector q takes discrete values with the periods of the reciprocal Q = h(mrbl

lattice

+ mzbz + m&s)

where bl, b2, b3 are basis vectors of the reciprocal lattice and ml, ms, ms are integers (Im\ < const). It is easy to see that such a model also may be included in our schemer5) Acknowledgements. The author is thankful and D. ter Haar for useful conversations.

to N. N. Bogolubov

Received 5-7-65

REFERENCES

1) Bogolubov, 2) Bogolubov, 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

N. N., Zubarev, D. N., Tserkovnikov, YU, A. DAN U.S.S.R. 117 (1957) 788. YU. A., Zh. eksper. tear. Fiz. N. N., Zubarev, D. N., and Tserkovnikov, (U.S.S.R.) 30 (1960) 120. Bogolubov, N. N., Dubna preprint 1960. Bogolubov, N. N., Physica 26 (1960) S 1. Thouless, D. J., Ann. Phys. 10 (1960) 553. Brueckner, K. A., Soda, A. M., Anderson, I’. W. and Morel, I’., Phys. Rev. 118 (1960) 1442. Anderson, P. W. and Morel, P., Phys. Rev. 123 (1961) 1911. Gor’kov, L. P. and Galitskii, V. M., Zh. eksper. teor. Fiz. (U.S.S.R.) 40 (1961) 1124. Cooper, L. N. Mills, R. L. and Sessler, A. M., Phys. Rev. 114 (1959) 1377. Tareeva, E. E., Thesis 1965. Bogolubov Jr., N. N., Vestnik Moscow State Univ. 1966, No. 1. Bogolubov Jr., N. N., Ukrayin. mat. Zh. No. 3 1965. Bazarov, I. P., Physica 28 (1962) 479. Bazarov, I. P., DAN (U.S.S.R.) 140 (1961) 1. Bogolu bov Jr., N. N., Contribution to the Novosibirsk Symposium 1965.