Number projected statistics and the pairing correlations at high excitation energies

Nuclear Physics A552 ( 1993) 205-23 Nosh-Holland

Number

projected

1

NUCLEAR PHYSICS A

statistics and the pairing correlations high excitation energies* C. Esebbag

~eparta~enta

and J.L. Egido

de Fisiea Teiirica, Universidad Received

at

~ut~~v~a

12 December

de Madrid, 28049 Madrid, Spain 1991

Abstract:

We analyze the use of particle-number projected statistics (PNPS) as an effective way to include the quantum and statistical fluctuations, associated with the pairing degree of freedom, left out in finite-temperature mean-field theories. As a numerical application the exact-soluble degenerate model is worked out. In particular, we find that the sharp temperature-induced superfluid-normal phase transition, predicted in the mean-field approximations, is washed out in the PNPS. Some approximations as well as the Landau prescription to include statistical fluctuations are also discussed. We find that the Landau prescription provides a reasonable approximation to the PNPS.

1. Introduction With the development of new detection systems at high excitation energy and angular momentum

a large amount of data on nuclei has become available in recent

years. This has led to a series of interesting discoveries (phase transitions, superdeformation, rotational damping and many others) which again have stimulated new theoretical studies as well as the construction of new crystal ball detectors. These will provide new and detailed information about the nuclear quasicontinuum in the next years. The great handicap in the theoretical study of the quasicontinuum is the level density which is already very large being a few MeV above the yrast line. This makes impossible a realistic calculation of this region within the microcanonical ensemble. The success of mean-field theories CHFB) ‘? and cranked random phase

(as cranked approximation,

~a~ree-Fock-Bogoliubov, CRPA 4-h) to describe

the

states near the yrast line gave rise to the natural extension of these theories to finite temperature ‘-@‘) (FTCHFB). Another approximation has been worked out by Alhassid et al. I(‘). These theories, based on the grand-canonical ensemble, are very attractive since their application is not much more difficult than the zero-temperature CHFB case. TO go beyond mean field is, however, a little harder. In the zerotemperature case we just have quanta1 fluctuations, which have been usually treated Correspondence to: Dr. J.L. Egido, Departamento Madrid, 28049 Madrid, Spain, * Work supported in part by DGICyT, Spain under 037S-9474/93~$0~.00

@I 1993 - Elsevier

Science

de Fisica project

Publishers

Tehrica,

Universidad

PBgX-0177.

B.V. All rights reserved

Aut6noma

de

C. Esebbag,

206

J. L. Egido

/ Number

projected

statistics

within the CRPA approximation. In the finite-temperature case we have both quanta1 and statistical fluctuations, the quanta1 ones being more important at lower excitation energies

and the statistical

The incorporation standing

ones at higher

of correlation

of the quasicontinuum.

have been considered Quantum fluctuations

to study

excitation

is very important Thus, pairing

for instance, phase

energies. for the description statistical

transitions

and

and under-

fluctuations shape

“-‘3)

transitions.

14-“) have been proposed within the frame of the RPA theory. Only recently Puddu et ~1.“) have applied the RPA+SPA (static path approximation), including both statistical and quanta1 fluctuations to calculate level densities in model cases. An additional complication introduced by the thermal averaging in the grandcanonical ensemble is the admixture of quantum numbers. At this point one has to distinguish between the symmetries spontaneously broken in the mean-field approximation - as the particle number in BCS or the rotational invariance in deformed nuclei - and those inherently broken in the thermal averaging - as the spatial parity, number parity, etc. The inclusion of correlations is strongly related with the restoration of symmetries. Thus the angular momentum is related with quantum shape fluctuations and the particle-number symmetry with the pairing correlations. From the above-mentioned studies 9*‘9)we learned - at least for the studied nuclei - that pairing correlations are very sensitive to temperature effects (FTHFB calculations predict a sharp phase transition at 0.5 MeV) while the shape degree of freedom starts to soften at somewhat higher temperature. From the inclusion of statistical fluctuations, furthermore, we know that the pairing degrees of freedom are also important at higher energies. That means, it seems necessary to include both quanta1 (low-temperature effects) and statistical fluctuations in a selfconsistent way for a proper treatment of the pairing degree of freedom. To achieve this, in this paper we propose the use of particle-number projected statistics as a good candidate to fulfill these requirements. To our knowledge parity projected statistics for the first time was proposed by Tanabe ef al. “) to separate, in the thermal averaging, the multiquasiparticle states with the wrong parity. The multiquasiparticle states used for the thermal averaging are eigenstates of the parity operator, i.e. the mean-field hamiltonian, used to determine the density operator, and the parity operator do commute. In this paper we go a step further on with the particle-number statistics because our mean-field hamiltonian and the particle-number operator do not commute, a fact that as we will see in subsect. 3.4 in the evaluation of the entropy, amounts to work with a real many-body density operator. To illustrate the theory we apply it to a simple model. In sects. 2 and 3 we present the theory and show how to evaluate the different quantities. In sect. 4 we discuss the degenerate model and the different approaches. In sect. 5 we discuss the results and the conclusions are presented in sect. 6. Finally,

C. E.vehhag, J.L. Egido / Numher p~~je~~e~statistics

in appendices

A and B we calculate

theory, and in appendix

some matrix

elements

C we discuss the temperature

needed

207 to elaborate

limits of the number

the

projected

statistics. 2. Statistical

description of a quantum system

The statistical approach “) to a physical system is usually adopted when the knowledge of the state of the system is not complete, or when the nature of excitations becomes so complicated that microscopic approximations have to include a very large number of relevant con~gurations and, therefore, cannot be applied. In this case, the system is described by a statistical density operator p^which is non-negative, hermitian and with trace equal to the unity. In terms of its eigenvectors and eigenvalues, the statistical operator p^is expressed as

(1) where the quantities

p, are statistical

weights

OGp;Gl,

that satisfy t:pi=1,

the eigenvalue pi is the probability of finding value of an observable 2 is defined by

and the entropy

0)

the system in the state ii). The average

(&=Tr{p^.ci),

(3)

S=-kTr{p^logp^}.

(4)

by

In spite of the fact that there is not a unique prescription for the election of the operator b, i.e. the statistical weights of each (pure) state in the statistical mixture (I), this is frequently done by recourse to the statistical thermodynamics principle of maximum entropy, even for systems with a not very large number of particles. Let us consider a system where the mean values a, of certain observables known. The condition of maximum entropy subject to the constraints Tr{&&j=a,, provides

the statistical

density

thermodynamics,

the energy

operator

(6)

of the system

%=(fi)=Tr(p^fi}, and the particle-number

(5)

operator ,. w [Xi AiAil ’ =Tr {exp [xi AiAi]} ’

In statistical

A, are

(7)

fi: N=Tr{p^G},

(8)

208

C. Esebbag, J. L. Egido / Number projected statistics

are, usually, represented

kept constant. by the density

In this way we are led to the state of thermal operator fi = exp [-P(A

e

where

Z, is the partition

equilibrium

-I-&I

ze

(9)

’

function Z,=Tr{exp[-/?(A--&)I},

(IO)

p is related to the absolute temperature by /3 = I/ kT, p is the chemical potential and the subscript e stands for exact, in the sense that it is the exact solution of the variational problem. The maximization of the entropy that leads, subject to the constrains (7) and (8), to eq. (9), is equivalent to minimize the thermodynamic potential F=8-pN--3-S.

(11)

In this way eq. (9) is the exact solution of minimizing the functional F. The calculation of the mean value of a one- and two-body operator A by means of the statistical density operator D,, however, is very difficult to achieve. The complexity of the hamiltonian fi, which is normally a two-body operator, makes the problem of computing the trace Tr {fi,A} a very hard task. To avoid these difficuities a simpler class of trial density operators is chosen instead of D, and the variational problem of minimizing F is solved within this class. It is usual to replace the hamiltonian fi in eqs. (9) and (10) by an effective one-body hamiltonian I;, this leads us to finite-temperature mean-field theories (as HF, BCS and HFB theories) 7’,‘277.In these approximations the statistical density operator is given by A* =

ew

(--Pfi)

Tr {exp (-@)}

(12)

’

with

6=x Here

LY~and

includes

1yj are particle

in a proper

Ep~cw,.

or quasiparti~le

operators,

(13) see eq. (20) befow,

way the term ~*_fi (see eq. (9)), to adjust

the average

and h^ particle

number.

3. Particle-number

projected statistics

3.1. DEFINITIONS

Let us consider a system described by some (general) statistical operator p^ (1) (not necessarily of the forms given in eqs. (9) or (12)), and certain observable A (for instance, the particle number operator A). Suppose we know with certainty,

C. Esebbag, IL.

perhaps

as a consequence

the observable

Egido / Number projected statistics

of an ideal measurement,

A is some specific

eigenvalue

ment, the system

has to be represented

result

in other

obtained;

eigenvalue by means

a should

words,

state

have null statistical

of the projector

that the only possible

a. Therefore,

by a statistical

every

that

209

value for

after the ideal measure-

mixture

compatible

is not an eigenstate

with the of i

with

weight (pi = 0) in eq. (1). That is achieved

ga,, which projects

eigenvalue a of k7. Thus, the statistical above can be expressed as

on the subspace

operator

satisfying

corresponding

to the

the conditions

stated

(14) that is the projection of the statistical operator p representing the statistical mixture before the measurement. Of course, we do not need any “real measurement” to define a statistical operator like eq. (14), but any density operator that provides with certainty the eigenvalue a forthe observable A, that is (2) = a and ((a-a)‘) = 0, can be written in the form (14). Therefore, we are allowed to choose any suitable ansatz for p^ that gives a good physical description of our system without worrying about the conservation of the magnitude represented by 2, because that symmetry will be restored by the projection. This is the case in projected mean-field theories, for instance, as we shall see below. In this paper we are interested in panicle-number conse~ation, the particlenumber projection operator ‘I) being given by

J

2* do

e,tifIj-N,

(15)

1

0

where fi is the particle-number operator and N is the (integer) eigenvalue defining the subspace upon we are interested to project. The density operator most frequently used in finite-temperature calculations is defined in eqs. (12) and (13), where the effective hamiltonian is of the HFB type. This kind of density operator is suitable to treat the long-range part of the interaction between particles (H F) as well as the short-range pairing correlations in a consistent way. We want number as

projected

to take

advantage

statistical

of those

properties

but within

theory, thus we will define the projected

the frame density

of a

operator

(16) where Z=Tr{@, and h^ is defined

in eq. (13)

exp(-/36)$N},

(17)

C. Esebbag, J.L. Egido / Number projected statistics

210

Before proceeding

with the calculation

of the expectation

is interesting to make a few remarks. The effective one-body particle-number projector do not commute and consequently

values

of operators

hamiltonian

[ FN, exp( -ph^)] # 0. This fact will have important

consequences

it

h and the

(18)

throughout

the developments

in this

paper and it makes the basic difference between our theory and the one by Tanabe et al. ‘), i.e. our theory allows to restore symmetries broken not only by the statistical average [as in ref. “)I but at the same time those broken by the ansatz of our quantum-mechanical approach. This symmetry breaking is strongly related to the capability of mean-field theories to describe pairing correlations. Another important point is that, although we are dealing specifically with number conservation and pairing correlations, many of the conclusions that will be drawn in this paper are more general and are characteristic for systems where the projector and the effective hamiltonian do not commute. In the numerical application of our theory we shall be mainly concerned with the pairing

hamiltonian k =C Eka:al, - C GA,a:a~a,a,, k bf--0

(19)

a particle in the state where the operators a:, uk create and destroy, respectively, k. The state &represents the time-reversal state of k, or more generally the conjugated state of k in the sense of the Bloch-Messiah theorem 23). In the BCS approach this hamiltonian is approximately diagonalized by the Bogoliubov-Valatin transformation (Yk=&,ak-v@r, .;. err= u,a,-i- vkal,, which define normalization

the quasiparticle condition

operators

(20)

(Y:, CY:, as usual

uk and

lu:I+lv;]=l.

the

(21)

The coefficients uk and vk together with the Ek in eq. (13) are variational of our ansatz and have to be determined as to minimize the functional F=(G)-

vk satisfy

TS;

the term --&? must not be included in the definition our projected density operator preserves the number

parameters

(22) of the free energy (22), because of particles as a good quantum

number. 3.2. EVALUATION

OF EXPECTATION

VALUES

In order to calculate the expectation values of operators with the density operator (16), we first have to compute the normalization 2 (17). Taking into account the

211

C. Esebbag, J.L. Egido / Number projected statistics

invariance

of the trace under

cyclic permutations,

the property

the representation (15) for pN, after some manipulations appendix A), we get

Pk = PN and using

(the details

are given in

?rr Z=Tr{@,

d0 e-‘ONZ( 0)

exp(-/?h*)}=&

,

(23)

I0 with Z(O)=

n

(24)

@P,(O)

ICY-0

and @k(0)={u~+eiZH

u:+eiH(exp(-PE,)+exp(-DE,-))

+exp(-P(E,+E,-))(uZe”‘+ni)}. We, furthermore,

define

(25)

the integrals Irr

1

Z(O) 1; .Y2......‘l,,= -2VZ (j d9 emroN eiHm I @p,,(e)@q2(e) f . . @,,w

’

(26)

where from the definition

we have I’= 1. These integrals are the natural extension projected theory at zero temdefined in ref. ‘“) in the particle-number It is easy to see that the following recursion relations are satisfied:

of those perature.

1; ,YZ...... y,‘_, = (ui,,+exp

(-P(E,,,

+&,,))uS,,)Z~.yz

+ (exp (-P&)+ev + (4,, + ut,, ev

,...,y,,_,3qI’

(-P&,,))IZYi2 (-P(E,,,

,...,q,._,.y,,

+ &,,)))I,“,T,‘2 ,.._, y,,_l,y,,.

(27)

In eqs. (26) and (27) absolute values have been taken for the subindices q and the convention that Zq= Z, for any subindex. 3.2.1. Particle-number conserving operators. Let a be an operator that does not change

the number

of particles,

then, obviously [a, 1;,]=0.

The mean

value of A is given by (f@=$Tr{@N

For the particle-hole (a:,akZ)= Similarly

(28)

eXp(-&)@Na}=iTr{?N

operator,

8k,.~.,[(G,+nZh,

for two-body

we have (see appendix

exp(-&)A}.

(29)

A)

exp (-P(E,,+Ec,)))Zf,+exp(-PE,,)Z:,l.

(30)

operators

(31)

212

if k,#k4.

C. Esebbag, J. L. Egido / Number projected statistics

For k,=k,: (aL,ah+J=

The average eqs. (30)-(32),

$~,,k~Sf,.k,(d,+4,

energy

of a system

exp (-P(&,+&,)))G,

with the pairing

hamiltonian

(32)

. (19), according

to

is given by

~~[(~Zk+~2kexp(-P(E,+E~)))lZk+exp(-PE,)Z:l

(ri,=:

-I,

G,i(d+ 4 exp (-fit% + 67)))ZZ

-k X,0GkqUkUk&pq(l

-eXp

(-P(Ek

f

EC)))

A,

(33)

x(l_eXp(-P(E,+E~)))12,,,.

3.2.2. Non-conserving operators. Any operator may be expressed as the sum of two parts:

that

does

not commute

with

fi

R=‘&+A,, where A, does not change the number of particles and Ai, that changes it. The expectation value of A, is calculated according to the equations given above (subsect. 3.2.1). The mean value of d, always vanishes (d,) = 0. As an example we will show the evaluation of (cx:(Y~):

(Y:ak can be written

in terms of the particle CY;Lyk= &:a~

+ v:a,-a;-

operators: ukuk(a:a~;+

UkUk) )

(35)

the only contribution to the expectation value comes from the first two terms on the r.h.s. on eq. (35) (the conserving part), while the contribution of the third and fourth term vanishes because ~,+~~a~~, = 0.

3.3. THE

PAIRING

The energy

GAP

(33) can be rewritten

as

(~)=C&k[(UI+UfeXp(-p(Ek+E~)))I:+eXp(-PEk)I~]

-k~,,G~k[(~~+~:ex~(-B(Ek+E,-)))I: -ulvi(l-exp x

C

k,y

-I)

(1

-eXp

(-/?(E,+E,)))‘Z~,,]

G,uL~kn,v,(l (-PtE,,+

-exp E,)))I’,.,

(-PUS+ .

E,-)))

(36)

213

C. Esebbag, J. L. Egido / Number projected statistics

Note

that

now

the third

summation

includes

the term

k= q, which

has been

subtracted from the second summation. This expression can be compared with the energy calculated in finite-temperature BCS theory, i.e. in the normal non-projected mean-field theory, using the density operator (12), where one obtains (fi)M,=;

FIMX -

+ ut(l -AA)

C G&U;&+ k>O

u;(l -fk)][u:fk

-k F>O GQWW,(~ where the subscript

MF

A, =-

-GkkpcK=

of the pairing

interaction

C G,k(aEak.MF=I\><1

the pair potential,

-h)]

-f4 -.@

,

stays for mean field. Let us denote

I’, = -G,,(a:a,->,,= the contribution

-X -&)(I

+ &I

1

by v;(l-&))

to the HF potential

,

(38)

and by

C Gqku~uk(l-fk-.h), h>O

C G+K~P=I>0

eq. (37) can be written (fi)MF=

-G&&+

(37)

(39)

as

E~P~~++~P~~+~~P~~+~~K~~,

(40)

k>O

which is the extension to finite temperature of the well-known expression for the energy in the BCS approximation. It is important to notice that eq. (37) can be obtained term by term taking the non-projection limit* of the projected energy of eq. (36). In this way, the second term in eq. (36) is analogous to the HF contribution in eqs. (37) or (40); and the last term in eq. (36) that corresponds to the A, term, may be identified with the pairing energy (without the exchange term), consequently we define within our formalism the pairing contribution to the total energy as gP= -b.if

_

o G+~u~uJ+,(~

x (1 -exp

-exp

(-P(E,+E,)J)lf,,

(-P(&

+ EC)))

.

(41)

Once that we have found an expression for the pairing energy we can define the projected gap parameter in the usual way. For the special case of a pure pairing force, i.e. Gk, = G, we have A,>=-,

(42)

if one takes the proper limits in this expression, i.e. the non-projection zero-temperature limit, one finds the well-known definitions of the pairing the corresponding theories. l

It is easy to see that it may be done by just replacing:

argument

8 = 0.

or the gap in

I for C1/3n) s,‘,”dfJ erHh and setting the

214

C. Esebbag,

3.4. THE

ENTROPY

The entropy

IN THE

as a function

J.L. Egido

PROJECTED

/ Number projected

statistics

STATISTICS

of the density

operator

is defined

by

S=-kTr{p^logp^}. This equation since

has always the right behavior

its eigenvalues

satisfy

(43)

for any acceptable

density

0 G pi s 1 (eq. (2)), and the function

operator

p*,

pi log pi is well

defined even for pi =O, where it is defined as the corresponding limit. In the simple approximations generally used, p^ has an exponential form (see eq. (I2)), if, in addition, a one-body effective hamiltonian (h) is used in the exponent, the calculation of eq. (43) becomes trivial as it is well known in the frame of finite-temperature mean-field theories. Any attempt to go beyond these simple approximations, in order to introduce further correlations and/or to restore broken symmetries, will result in replacing the one-body hamiltonian with a two-body one (this would bring us back to the difficulties of the operator D, in eq. (9)) or to change the simple exponential form of 6. In either of the two possibilities the calculation of the entropy through In our treatment for the entropy

eq. (43) is no longer we have

so easy.

S=-kTr{slogfi} I;, exp (-$)gN

=-kTr I by means equivalent

II;, exp (-/3fi)@, log

Z

of a suitable power expansion to the following simpler form: S=-kTr

3

Z

it is easy to show that this equation

F, exp (-pi)

is

rj, exp (-ph*) log

Z

(44)

Z

)I.

In those cases where the projector and the effective hamiltonian commute it is possible to take out the projector from the logarithm and proceed in the usual way: S=-kTr

FN exp (-pi) Z

In this paper we are interested in the important situation where ?N and h^do not commute; in fact, we are restoring the symmetry broken by the effective hamiltonian h: In this case, the evaluation of the entropy is a very hard task because there is not a direct prescription of how to calculate the logarithm of two operators that do not commute. Since the operator 6 (16), introduced in eq. (44), is a good statistical operator (hermitian, non-negative and with trace equal to unity) the mathematical rigor of that equation guaranties the possibility (in principle) of making an exact calculation. Thus, we are allowed to apply approximate methods in computing S. Work in this direction is now under progress. However, in this paper we are interested in the study of the projection effects. It would be desirable to be able to compare the normal mean-field theory, the projected statistics and the exact canonical one

C. Esebbag,

without

recourse

investigations fore,

to any

J. L. Egido

approximation

or generate

doubts

may be solved

that

projected

could

about the accuracy

we will work out a simple

theories

/ Number

(but non-trivial)

statistics

obscure

215

the meaning

of our

of the results achieved. soluble

model

where

There-

the three

exactly. 4. The degenerate model

We will consider an exact soluble model consisting of N (even) particles in a degenerate j-shell and the pairing hamiltonian. This model has been used often in the literature as a test of many-body theories 7*23). If we set the single-particle energies

equal to zero, this hamiltonian

reads C a:a~a,a,. !%.y-0

I?=-G There are 2M = 2j + 1 single-particle N-particle configurations is

(46)

states in the j-shell,

and the number

of possible

(47)

NC=

4.1.

EXACT

SOLUTION

AND

CANONICAL

ENSEMBLE

The energy levels are labeled by the seniority number s which, for an even number of particles N, may take any even integer value from 0 up to N: s = 0,2,4,. . . , N. The exact eigenvalues of the hamiltonian (46) are E, the degeneracy

(N)_

(48)

--;G(N-s)(2M+2-N-s),

of these levels is (49)

with eqs. (48) and functions. In the canonical

(49) it is easy to compute ensemble Z,=

and the internal (fi),=

energy,

;

d,

,=o (even,

E d,vexp

F=o

entropy

; d,E;N’eXP .s=o (WC3l)

SC=-k

the partition

exactly

function

(NJ _ E;““))

(-‘(“z:

is given by EbN'))

(-_P(EkN’-

and free energy

all the thermodynamical

,

(50)

by ,

(51)

c

exp (-p(EtN’-5

EbN’)) [-P(ESN’-

EhN’) -1s

(ZJI

,

(52)

216

C. Esebbag,

AL. Egida / dubber

projected statistics

F,=(A},-TS,=EI,N’-kTlogZ,,

(53)

respectively. The subscript c stands for canonical. The additive constant Eb”” (ground-state energy) appears in eq. (53) because of the arbitrary election of the zero of energy, if we set E0(PJ)= 0 by a shift of the energy scale, the usual expression for F is obtained. Clearly the quantities exp

Pi,N) =

(-fl(E(cN)- EbN’))

,

zc

(541

appearing in eqs. (51) and (521, are the eigenvalues of the exact density operator within the space of N-particle states. The canonical results (50)-(53) are obtained by means of the density operator (55)

and taking traces over the space of N-particle states, or equivalently using the projection operators and taking traces in the Fock space. Computing the mean values with the operator (SS), but including the term --pfi and taking traces in the Fock space, will provide the grand-canonical ensemble. 4.2. FTBCS AP~ROXl~ATlON

The finite-temperature BCS theory has been applied to this model by Goodman ‘) for a half-~lled j-shell. Since all single-particle states are completely equivalent, all the coefficients zik of the Bogoliubov transFormation (20) and the quasiparticle energies E, do not depend on the index k; therefore, we have only two variational parameters. From the particle-number condition [see ref. 7, for details] one gets IV=2 r: [v:+(l-zvZ)f;,]=2M[u’+(1-2u’)f],

156)

k-0

where f is the quasipa~icle

occupation: f

=(a;aI)NIF=

I

l+exp@E)’

For the special case of a half-Toledo-shell (M = N) one can determine the coefficient D’ to have the value 5. The BCS gap is defined as (see eq. (39)) A = G C (u~L&,,~= GMuu(l-2f) L-0

;

(58)

in the case where N = M, the quasiparticle energy satisfies E = A, and the equation to be solved for each temperature is E = ii0 tanh ($p)

,

(591

C’. Esebbag, J.L. Egido 1 Number projected statistics

217

where A,=$GM is the T =0 gap. The expectation

value of the hamiltonian

(60)

is

(fi)MF=-MG[u2f+u2(1-f)]2-GM’~2~Z(1-2f)2 =-MG[u’j”+v’(l-j-)I’-AZ/G,

(61)

where we have not neglected the HF term (the first one in the r.h.s. of eq. (61)) in order to be able to compare it with the exact results. For u2 = u3 = i it is a constant term and does not modify eq. (59). Finally, the entropy in FTBCS theory

is given by

S MF= -2M[flogf+(l

4.3. PARTICLE-NUMBER

PROJECTED

We will now apply the model hamiltonian

-f)

log (1 -f)l

(62)

.

STATISTICS

the projected statistics given by the density operator (46). According to eq. (33) the average energy is

(16) to

(A)=-GM(u’-tu2exp(-2PE))I:-GM(M-1)u”v’ x(1-exp(-2PE))‘li, where the integrals

form (1= 0, 1,2,.

. . ):

211 d0 e-iHN eiHm I0

1

I;‘=--

(26) take the simple

(63)

2rrz

x[u~+e’~H~2+2eiHexp(-PE)+exp(-2PE)(u’e”H+v~)]M~‘, since

vl, = v and El, = E for every k. The normalization

(64)

is

57 o d0 emlfJN

Z=& I

x[u~+e”Hu~+2e’Hexp(-PE)+exp(-2PE)(u~e”H+v~)]~’. Following projected

the considerations outlined Ap as

in subsect.

3.3 after eq. (40), we define

(65) the

gap parameter

3,,=GMuv(l-exp(-2PE))~,

(66)

which, obviously, goes to the BCS gap (58) when the non-projected limit is taken. According to our remarks of subsect. (3.4), in order to make reliable comparisons and to better understand the effects of the projection, we minimize the free energy (22) exactly (numerically), computing the entropy without involving any approximation.

218

C. Esebbag, J.L. Egido J Number projected statistics

The entropy

is defined

as

S=-kTr{filogfi}=-kCD;logD,, where the Q are the eigenvalues of the operator

of I% Writing

I;,,, exp (-/3h*)@,,

(67) [P exp (-ph)P],

for the eigenvalues

we have

D_

=

I

[P exp (-Ph)Pl, z

and Z=C[Pexp(-/G)P],. Thus, the entropy (67) may be computed @N exp (-&)k,. Due to the dimension task is, in principle, impossible to carry allow us to block the matrix into boxes Let us consider a basis of the Fock configurations In, i), where n represents the remaining quantum numbers, then

(69)

through the diagonalization of the operator of any interesting many-body problem, this out, but the symmetries of the model will that will be small enough to be treated. space formed by the set of many-particle the number of particles of the state and i

(n, i]PN exp (-&)P,\n’,

j)=O,

(70)

if n # N, and/or n' # N. Therefore, we only have to diagonalize the matrix within the space of N-particle configurations. There are 2M single-particle states grouped in “conjugated” pairs (m, fi) and NC (47) N-particle configurations. We shall denote a given N-particle configuration IN,_i) by ]N,j)=lk,,

kz,. . . , k,, 4,. . . ,4)=kkl,,{O,),

(71)

where I is the number of unpaired particles (a particle is unpaired when, being in the state k, the state k is empty and vice versa), n is the number of pairs of coupled particles (the number of state pairs (m, m) that are fully occupied), I and n are related

by I= N-277.

(72)

In eq. (71) we have used the letter k for unpaired particles, and I for the paired ones, each index I, means that the pair of states (li, &) is occupied (for brevity we do not write explicitly the indices &). We have also adopted the convention of The matrix element of the operator listing first k-indices and then l-indices. @N exp (-pfi)F, in the basis (71) is (see appendix B) ({k],, {Q,(exp

(-&k’],,

= exp (-ZPE)[u’(exp x[v2(exp

{U,) (-2&??)

(-2PE)-1)+l]“-N+Rd

- 1) + llVd[u’u’(exp fi c?~,,~;, i=l

(-2/3-E) - l)‘]“-“d (73)

219

C. Esebbag, J. L. Egido / Number projected statistics

where n,, is the number (I, r) in the initial

of “diagonal

pairs”,

that is pairs that occupy the same states

and final state, 9d =

It is easy to see how the matrix

5 St,,/:. i.i

of the operator

(74) @N exp (--$)l;,

is divided

into

smaller matrices. The product of Kronecker deltas (6,,,,;) indicates that only configurations with the same distribution of unpaired particles are connected, therefore a given configuration of unpaired particles k, , k?, . . . , k, defines a block of the matrix to be diagonalized. In addition, the matrices with the same number of coupled pairs n, i.e. the same number of unpaired particles Z, are identical. The number, g,,, of identical matrices with v-pairs, is equal to the number of ways the Z unpaired particles may be distributed in M-boxes, one in each box, and having two possibilities (k and E), that is (75) In this way the problem is reduced to the diagonalization of one matrix for each n value from 0 up to ;N (the case n = 0 is already diagonal). The dimension (J, x J,) of each matrix is given by the number of possible configurations obtained by the distribution of the n coupled pairs in the resting M - I pairs of states (Z, i): (76) As a numerical example for the case M = 10 and N = 10, the dimension of the whole matrix is N,x NC = (184 756)‘, however, we have to diagonalize just six matrices, where the largest one is of dimension J5 x _ZS= 252 x 252. Writing (exp (-ph)),,, for the eigenvalues corresponding ones of the statistical operator,

of each q-matrix we have

and

D+

for the

N/2 z=

D

C

7=” =

gv

i

,=,

(ew

(77)

(-Ph)),.;,

(exp (-Ph))q,i

z

‘).I

(78)

’

and for the entropy N/2

S= -k

1

7 =o

g,

2

(;;I

D,,; log D+.

Eq. (77) provides a reliable way to check the numerical procedure comparison with the result obtained for Z by means of eq. (65).

(79) through

the

220

C. Esebbag, J. L. Egido / Number projected statistics

It is remarkable that removing the matrices to be diagonalized identical

any more (the matrix

appendix

B). Thus the entropy

variational

procedure

would

the degeneration of the model, the dimension of does not increase, but the g, n-matrices are not elements

for the non-degenerate

would still be calculable present

number

more

because

of the increasing

facilities

it does not seem to be impossible.

4.4. A LOW-TEMPERATURE

difficulties

of parameters,

APPROXIMATION

case are given in

in this case. The numerical in the non-degenerate

but with the modern

case

computing

(LTA)

Looking at the difficulties arisen in the computation of the entropy one could hope such a calculation to be not really necessary. One could think that using the simple statistical operator d^ (eq. (12)) and taking traces over the N-particle Hilbert space would be good enough. We will show that it is not the case and that a full number projection is necessary in order to describe correctly the phase transition. In this approximation we use the non-projected statistical operator d^ and define the expectation values of any operator A taking traces over the N-particle space (we will write Tr, for the trace in this space in contrast with Tr that means the trace in the Fock space): (4 ~r,=~Tr”{~~}=~I:(N’ilexp(-PI;)~lN,i),

(go)

,

N

where {IN, i)} is a complete set of N-particle configurations Tr, {exp (-pi)}. It is easy to see that for number-conserving operators tion value(AJLTA is equal to the exactly projected one of eq. (29):

=iTr{gN

exp(-/3&)?,&},

and 2 = the expecta-

(81)

thus we get the same expression for the energy (Z?) (eq. (33)). However, for non-number-conserving operators the expectation values are different, and also for the entropy where we have k

s LTA=FTrN

(2 log a}

TrN (4

=$Tr,{exp

(-$)[&+log

(Tr{exp (-$)})]}

A

= k Tr

@, exp FBh)

= k log (Tr {exp (-pfi)})

[@6 + log (Tr {exp (-/3&)})] +pk C El, exp (-BE,) I,

(82)

C. Esebbag,

J.L. Egido

/ Number

projected

221

statistics

This is a very simple result that would allow us to derive exact analytical equations.

Unfortunately,

transition,

as we shall see in the next section.

values

for the relevant

temperatures peratures

4.5. THE

where

this approximation operators

the entropy

where the term -TS

LANDAU

It would temperatures

gives

poor

results

variational

after

the phase

Since we get the right expectation

but not the right entropy, does not play an important

we expect

it to work at

role, i.e. at low tem-

of F is small.

PRESCRIPTION

be desirable as well to have a microscopic complementary to the LTA, i.e. some kind

approximation at high of power expansion. A

glance to eq. (45) shows that the main problem of how to calculate the logarithm of two operators which do not commute remains and we do not gain anything. Therefore, we have to recourse to a macroscopic approach. At finite temperature we have statistical fluctuations, that means the incoherent averaging over many single-particle densities based on intrinsic mean fields with different gap parameters. According to Landau *“), the probability for a certain value A of the gap is given by p(A)xexp

(83)

(-F(A)IT),

where F(A), the free energy, is treated as a function of A which is an independent variable not constrained by T and IV. Using classical statistics for the ensemble average, we find the average gap A=

p(A)A

dA.

From this expression it is clear that a includes statistical fluctuations in a macroscopic way, in contrast to the projected gap that includes quanta1 and statistical fluctuations in a microscopic way.

5. Results In this section

we discuss

the results

of the different

calculations

mentioned

in

sect. 4. In fig. 1 we display the total energy in units of the FTBCS gap at zero temperature (see eq. (60)) as a function of the temperature, also in units of do. The exact results of eq. (51) (by exact we mean the canonical averaging with the exact eigenvalues of the hamiltonian) are represented by the dotted line. For very small values (kT/A,C 0.2) we find almost no change in the energy. From kT/A,0.25 on and until kT/A,0.5 we observe a sudden strong increase in the energy. From this point on and up to kT/A,, = 1 the energy keeps growing, although not so rapidly, and very smoothly from the latter point on. The behavior of the FTBCS energy, eq. (61), (notice that the exchange terms have not been neglected in this expression) is

222

C. Esebbag,

J. L. Egido

/ Number

projected

statistics

Oar

PrniPCfPd

Statistics:

~ --

FTBCS

-6.0

,

0.0

0.2

Fig. 1. The total energy

0.4

0.6

as a function

0.8

1.0

1 .2

of the temperature

1 4

1 .6

for several

1 .8

2 0

approximations

quite different. Now the energy starts growing very fast at kT/Ao= 0.25 and it reaches its maximum value at kT/Ao = 0.5. After this point it remains constant for all temperatures. In this theory, the whole physics takes place in a very short temperature interval. The continuous line, finally, depicts the results for the energy in the projected-statistics approximation, eq. (63). These results are in very good agreement with the exact ones. For T = 0 it is well known that in this model the particle-number projected approximation reproduces the exact results; it can be shown, however, that for T # 0 this is not the case anymore (see appendix C). We can appreciate the differences in the energies by looking at table 1 where, at the left-hand side, we have listed the energy in the three mentioned approximations as function of the temperature. The values for infinite temperature are also included. On the right-hand side of the table we have listed the entropy as given in the different approximations. Here again we can appreciate how poorly FTBCS does. The projected approximation, however, behaves very well as compared with the exact one. The characteristic parameter of a superfluid-normal phase transition is the energy gap A. In the FTBCS theory A is given by the well-known expression of eq. (39). In a projected theory A is defined as the square root of the pairing energy after neglecting the exchange terms. This expression gives the right limits: it goes to the FTBCS gap when no projection is done and it goes to zero at very high temperature. For the exact theory, however, there is no possibility to define the energy gap because one cannot identify the exchange terms. In fig. 2 we show the gap parameters in

C’. Esebbag,

J. L. Egido

/ Number

TABLE

projected

statistics

223

1

The energy (in units of A,,) and the entropy for different values of kT/A,, for several approximations: The exact canonical, projected, FTBCS and the low-temperature approximation (LTA) Entropy

Energy T

0.0 0.1 0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 2.0 2.5 3.0 a?

exact

projec.

FTBCS

LTA

~6.000 ~6.000 -5.983 -5.524 -4.052 -2.666 -1.886 -1.470 - 1.229 - 1.077 -0.9738 ~0.8999 -0.8445 -0.8017 -0.7675 -0.7398 ~0.6540 -0.6099 -0.5831 -0.4737

~6.000 ~6.000 -5.982 -5.257 -4.181 -2.602 -1.784 -1.398 -1.184 - 1.048 -0.9537 -0.8843 -0.8310 -0.7887 -0.7543 -0.7256 ~0.63 17 -0.5766 ~0.5305 -0.4737

~5.500 -5.499 -5.357 ~4.616 -3.023 -0.5000

~6.000 -5.991 -5.877 -5.312 -4.044 -2.23 1 -0.7057 -0.4737

-0.5000

-0.4737

exact 0.0000

8.18~10~” 0.0949 1.8301 6.0158 9.1359 10.571 11.217 11.540 11.720 11.829 11.900 11.948 11.983 12.008 12.027 12.077 12.097 12.107 12.126

FTBCS

LTA

0.0000 9.50x 10-h 0.0942 1.7057 5.6043 9.1545 10.664 11.264 11.552 11.713 11.812 11.879 11.925 11.959 11.985 12.004 12.059 12.084 12.101 12.126

0.0000 9.93x IO_’ 0.8527 3.7515 8.2714 13.862

0.0000 0.1414 0.8464 3.0456 6.643 1 10.687 13.483 13.862

PwjectPd

~

LTA

--_

FTBCS

Fig. 2. The energy

projec.

13.862

13.862

_-____

gap (in units of J,,) as a function of the temperature approximations.

(in

units of A,,) for several

224

C. Esebbag, J. L. Egido / Number projected statistics

units of A0 as a function

of the temperature.

In the FTBCS

approximation

(dashed

line), after a smooth behavior for kT/A ok 0.2, we find a sharp decrease to zero at kT/Ao = 0.5, i.e. the mean-field approximation predicts a sharp superfluid-normal phase Again,

transition.

The projected

gap (continuous

after a flat start it decreases

this point

on it decreases

rapidly

very smoothly

line),

however,

is quite

till A/A0 = 0.4 at kT/A,

with growing

temperature;

different

= 0.75. From for very large

temperatures it finally goes to zero. From the projected gap values which contain quanta1 as well as statistical fluctuation we conclude that the sharp phase transition predicted in the mean-field theory has been washed out, as one would expect for a finite system. Since the phase transition takes place at relatively low energy, one could hope that the low-temperature approximation discussed in subsect. 4.5 still might be a good approximation. The dash-dotted line of fig. 2 represents this approach. For 0 s kT/Ao < 0.3 we obtain same results as in the projected theory, until kT/Ao = 0.4 it is still a very good approximation. From this point on it decays steadily and at kT/Aoz 0.6 the gap becomes zero. That means the LTA provides an excellent description below the phase transition but it is not able to describe it properly. The energy and the entropy results of this approximation are also listed in table 1. There we can see that, as expected, the results for the entropy are not as good as those for the energy. It would be interesting to compare the projected gap values, with the average gap a calculated with the Landau prescription of subsect. 4.5. The average gap d is shown in fig. 3. At zero temperature, obviously, it coincides with the FTBCS value; at relatively low temperature it is smaller than the projected one and at very high temperature 2 is larger than the last one. The behavior at high temperature is clear 11 Projected ___ FTBCS Average

Fig. 3. See caption

of fig. 2

_-_---

-

-

C. Esebbag, J. L. Egido / Number projected statistics

because

& as defined

above,

0.175) while the projected well and it provides

never goes to zero (in this model

gap does. It describes

a good approximation

the phase

at relatively

225

for T+co,

transition

high energies

n/A,+

qualitatively (kT/A,>

1).

6. Conclusion We have proposed and set up the particle-number projected statistics theory as the proper theory to include quanta1 and statistical degrees of freedom in a selfconsistent way. Expressions for the expectation values of one- and two-body operators and for the pairing hamiltonian within the framework of this theory have been given. Some difficulties found in the theory for the exact calculation of the entropy in realistic calculation have been pointed out. The theory has been applied to the solution of the degenerated model. We find that the sharp phase transition predicted by the mean-field approach is washed out by the inclusion of correlations. Two further approximations have been applied to the same model, namely, a low-temperature approximation and the Landau prescription to include statistical fluctuations. We find that the last one gives a fair approximation to the projected one. We would

like to thank

J. da Providencia

Appendix CALCULATION

OF

MEAN

VALUES

WITH

THE

for interesting

discussions.

A PROJECTED

DENSITY

OPERATOR

In order to evaluate the expectation values given in eqs. (30)-(33), we shall rewrite the projection operator in terms of the quasiparticle fermion operators ai, LQ of eq. (20). Since the number operator is given by fi =C,, a:~,, the exponential term in eq. (15) can be cast as e writing

rtl.G =?e

IHrrloA=n(l+(efH-l)a:a,), h

U:U~ in terms of (Y; and LYEand replacing

eq. (A.l)

(A.l) in eq. (15), we obtain

b, = eiH- 1 + uI( 1 - e”#) , ck = uIuk(e2’H- 1) , dk = vze”‘H+ uf .

(A.3)

226

C. Esebbag, J.L. Egido / Number projected statistics

The function

2 (eq. (17)) can be written

Z = Tr {pN exp( -ph^)FN} = I&({n,]l~N c-4

as

= Tr {F,

exp( -pi)}

exp(-&)l{n,])

2?r

1

de ePieN C exp (-P

I0

297

4

(A.4)

nr,~x,)({n,)lei""l{n,)),

in,)

with I{nj})=lnr,

. . , nj, ni,.

ni,.

. .)=(a:)“l((~‘,)~‘.

* * (ctI)“~(a:)“;*.

. I-)

(A.5)

a multiquasiparticle state characterized by the occupation numbers nk, each nl; being either 0 or 1. The projection operator restricts the trace in eq. (A.4) to the N-particle subspace, independently of the unrestricted sum over all multiquasipartitle states. The unrestricted summing makes the evaluation of expectation values simpler. Substitution of eq. (A.l) in eq. (A.4) and taking into account that the terms ((Yz(Y~+ LYE

do not contribute

to the expression

(A.4) provide

271 z=& i0

‘ON{XI kFoexp (-P(n,Ek “, Y

d8e-

+ nr&))

x {(e” - l)*nLnc+bl,(nr.+nc)+d,} =-

1

2x dfI ePiHN

exp (-P(n,Ek

2rr I 0

+n,-E,-))

x [(e” - l)‘n~n~++bk(nL+n~)+d,]

, >

where we have exchanged and nc, provides

the sums and the product.

Evaluation

(A.6)

of the sums for nk

finally

jHN,!,,I u’+e”‘a~+e’“(exp(-/?E,)+exp(-PEc)) i +exp(-/3(E,+E,-))(uie’*“+v:)}. Let us now calculate

the expectation

value of the particle-hole

(A.7) operator

given in

eq. (3OL (a;a,)=$Tr

{pN exp(-P~)@,&~~,,}

=$Tr

{exp (-&)a:a,li,},

(‘4.8)

in the last step we have used the cyclic property of the trace and the commutation relation of bN and uiu,,. To calculate eq. (A.8), we will write ~:a,, in terms of the quasiparticle operators and proceed in a similar way to the calculation of Z: aiap = upvy~y~p + uyvpa~~~i+

vyvp~qa~+

u,u,a~cr,.

(A.9)

C’. Esebbag, J.L. Egido / Number Let

us

calculate

the matrix

elements

the case q = p. We proceed (In&,-a,

entering

analogously

projected

statistics

227

into the first term of eq. (A.9) and for

to eq. (A.7), obtaining

I) *~~LYk(Y~(Yk+bk((Y~LY~+(Y~~Y)+C~((Y~-(YI,+(Y~a:)+d~}({nj})

JJ, {(e”-

=~~“{(eiH-l)‘n,ni+b,(n,+n~)+d,} > k#lj - I)2+2bby+dy)~y~q+cCy(1

x ({njll((e” =c,(l-n,)(l-nB)

n

-~&,)(~+cx;~E,)~{FJ~})

{(e’H-1)2n~nh+b,,(n~+nr)+d,}.

(A.lO)

h---O Lfq

Following

the same steps leading

+Tr

{exp (-~~)cY~cx~@~} 2Ti

1

d0 eCrBN C exp(-p(@,+n,E4))cy(I-ny)(I-n4) “C,. fl’j

=-i2%- 0 x kF, kflyl replacing

to eq. (A.7), we get

C (e’“-l)‘n,nE+bk(nh+nE)+dk { n!.,n!C

the quantities

defined

in eq. (A.4) and the integrals

i Tr {exp (-&)a,~,&,} Proceeding

in a similar

,

(A.1 1)

I (26), we finally obtain

= U,U,( Z; - Z”,, .

(A.12)

way, the other terms of eq. (AX) are given by

i Tr {@, exp (-&)

(~~(~~}=exp(-PE~)[Z~+exp(-PE,-)(u~Z~-tu~Z’:)],

(A.13)

$Tr

cu:(~i}=r+u~exp(-P(E,+E,))[Zt-IO,].

(A.14)

{p,

exp (-@)

Introducing eqs. (A.12), (A.13) and (A.14) in eqs. (AX), and using relations (27) for the integrals Zr, we get (~~a~)=

the recursion

vtZi+Z:exp(-PE,)+exp(-p(E,+E,-))utZi.

It is easy to verify that the matrix element of eq. (A.10) vanishes others (CX~CX~,cried,, of eq. (A.9)), then we have (L&J

(A.15) if q #p,

so

= 8&+,).

The traces computed in eqs. (A.12), (A.13) and (A.14) are not mean values corresponding operators, for instance: (crqczy) =iTr

ZiTr{gN

do the (A.16) of the

{@, exp(-j3k)fiNaycz,}

exp(-&)a,a,},

(A.17)

228

C. Esebbag, J.L. Egido / Number projected statistics

because of the non-commutation between these operators and FN (see subsect. for the evaluation of non-conserving operators expectation values). The calculation on the same complicated

of mean values

lines

and

for two-body

operators

difficulties

are found

no further

3.2.2

(eqs. (31) and (32)) runs except

for a little

more

algebra.

Appendix B THE

MATRIX

OF THE OPERATOR

bN exp (-/3h^)fiN

Here we will show the calculation of the matrix elements of the operator 6,., exp (-/z&)?~ in the general (non-degenerated) case. As it has been stated in subsect. 4.3, this operator has a non-zero matrix element only within the Hilbert space of N-particles states. Thus we have to calculate the matrix elements between two arbitrary N-particle configurations and we do not need to consider explicitly the projector: UklI,

{rilql

=(k,,

ev

(-Pfi)l{kll~,,

{li>,f>

k2,. . . , k,, I,, . . . , /,I exp (-$)(k{,

Taking into account that the operators tial may be expanded as

= &

ki,.

(Y:CX,commute

[(exp (-PEk)

x [(exp (-PEE)

. _, k;,, I{, . . . , Z’,,). (B.l)

with each other, the exponen-

- l)aLak - l)cuLak+

+ 11 11,

03.2)

or in terms of a: and a,, : exp(-@)=

n

{ykv,u~u,a:a,-+b~a~a,+b~a~uI;+c~(u~u~+u~u~)+d~},

(B.3)

k>O

where

b;=eXp(-PE,)+v:(l-eXp(-P(E,,$-E~)))-1, C;=Ukvk(eXp(-P(E,+E,-))-l),

d;=?~~exp(-p(E~+E~))+~~. Substitution of the expansion (B.3) that each k-factor in expression change the occupation of the pair just need to analyze separately the

(B.4)

in the mentioned matrix element and considering (B.3) commutes with the others and may only of states (k, f) together but not any other k’, we effect of each factor. The resulting matrix element

C. Esebbag,

will be the product

of similar

of the pair (k, E) unaltered

J. L. Egido

factors.

/ Number

projected

As the operator

or it changes

statistics

229

leaves either the occupations

both at the same time, each particle

that

is unpaired in the initial state must be unpaired in the final state, too. Thus, the operator exp (-/I&) only connects states with the same configuration of unpaired particles

that is to say: I = I’ (both configurations

particles)

have the same number

and k, = ki, k2 = kh, . . . , k, = k;, in eq. (B.l)

of paired

for a non-vanishing

matrix

element. Each unpaired particle in a kj state gives a contribution of a factor (b;, + dh). Each pair of states (I,, I;) that is occupied in the initial and the final configurations “the diagonal pairs”: Ii = Ii) gives a factor (y,yr+ bj+ bk+ d,‘). On the other hand, let us suppose that the pair (ji,jl) is occupied in the initial configuration but not in the final one (we will use the symbol j for the non-diagonal pairs), and reciprocally the pair (ji, 7:) is occupied in the final configuration but not in the initial one, this a factor ci,c:,, . Finally, every empty pair of conjugated will give a factor d;. In this way, we obtain

situation will provide on both configurations (kr,

kz,...,k,,I,,...,l~d,jl,...,juIexp(-p~)lkl,k;

I?6,,,k;(K,+d;,)

i=l

,..-,

G,C

,...,

states

&,,j’, ,...,

j:)

; (y,,yi,+bj,+b’r,+d;,)

i=I

(B.5)

where we have written states.

Replacing

v = n - r]d for the non-diagonal

the definitions

({kiI,, {l,],I exp (-&)I{kZ,, =

ir 8h,,k; exp (-PEk,)

,=I x

pairs and

q

for non-occupied

(B.5), we obtain {U,) ;i [uf(exp

(-P(E,,

> ,=I

,v,[uZvi(exp(-P(E,, + E,;))-

+ Ei,))-

I)+

11

1)‘l

x IJ [vt(exp(-P(E,+E4))-1)+11.

(B.6)

non-occ.

For the degenerated model element (B.6) becomes

of sect. 4, where

({WI, {I,],] exp(-Pfi)l{Wr,

vk = v and E, = E for all k, the matrix

{C],)

exp (--PEI)[u’(exp(-2PE)-l)+1]~d x

[u’v’(exp

(-2/?E)

- l)‘]V-qd

x[v’(exp(-2PE)-l)+l]MPN+V~.

(B.7)

C. Esebbag, J.L. Egido / Number projected statistics

230

Appendix C ZERO-

AND

In order

INFINITY-TEMPERATURE

the T =0 (p =a)

to analyze

the expectation

values

of conserving

(A)=$Tr{FN

=i

i +

LIMITS

and

operators

T =OO (j? =0)

limits

in the following

exp(-&)A)

(-IhAl-)+; exp (-P.CJ(k(~NAlk)

C ev (-PC.% + -T,))(k Lq

ql@NA(k s>+.. . ,

where{I->, IQ, lk 4). . . .I is a set of zero, one, two, etc. quasiparticle the limit

T+O

we will express

way:

only the first term of the r.h.s. in eq. (C.l)

remains

(C.1) states. Taking since Ek >O:

(A)T=,=$(-I&AI-),

(C.2)

Z,=

(C.3)

0

where lim Z =(-\I;N[-). T-O

Therefore,

for the energy

at T = 0 we have (C.4)

giving for the degenerated model the exact result as it is well known. On the other hand since lim T-cc exp (-/3 ( Ek + * . . )) = 1, the limit (C.l) provides

T + Co of eq.

(A)T=,:=~Tr{~NA}=~I~)({ni}113NAI{nj}) Lx

I

(C.5) where Z, = hmT_l Z = N, is just the number of N-particle configurations as it can easily be shown. The same expression (C.5) would be obtained if the exact density operator (9) had been used, within the canonical ensemble, because for T =OO any density operator of the usual exponential form gives an uniform distribution. In this way, in the high-temperature limit, the expectation value of a given operator is just equal to its (normalized) trace, and only depends on the working space but not on the density operator. That is the reason why the exact and projected energies coincide at T = co. It would also happen in any model, not only in the degenerated one.

C. Esebbag,

J. L. Egido

/ Number

projected

statistics

The result (for T = 00) is not the same for the FTBCS theory because the trace is taken grand-canonical

over the Fock space,

in this case

with the exact result in the

ensemble.

We have demonstrated the results

but it coincides

231

that, for the exact canonical

and the projected

statistics,

must coincide

at the T =0 and T =oo limits. For T f0 this is not the case because the operators fi (eq. (16)) and fit (eq. (55)) are different. This has been confirmed by the numerically calculated eigenvalues of both operators which do not coincide.

References 1) P. Ring. R. Beck and H.J. Mang, Z. Phys. 231 (1970) 10 2) A. Faessler, K.R. Sandya Devy, F. Gruemmer, K.W. Schmid and R.R. Hilton, Nucl. Phys. A256 (1976) 106 3) R. Bengtsson and S. Frauendorf, Nucl. Phys. A314 (1979) 27; A327 (1979) 139 4) E.R. Marshalek, Nucl. Phys. A275 (1977) 416 5) J.L. Egido, H.J. Mang and P. Ring, Nucl. Phys. A341 (1980) 229 6) Y.R. Shimizu and K. Matsuanagi, Prog. Theor. Phys. 70 (1983) 144 7) A.L. Goodman, Nucl. Phys. A352 (1981) 30, 45 8) K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20, 45 9) J.L. Egido, P. Ring and H.J. Mang, Nucl. Phys. A451 (1986) 77 10) Y. Alhassid, S. Levit and J. Zingman, Phys. Rev. Lett. 57 ( 1986) 539 11) L.C. Moretto, Phys. Lett. B44 (1973) 494; 846 (1973) 20 12) A.L. Goodman, Phys. Rev. C29 (1984) 1887 13) J.L. Egido, P. Ring, S. lwasaki and H.J. Mang, Phys. Lett. B154 (1985) 1 14). N. Vinh Mau and D. Vautherin, Nucl. Phys. A445 (1985) 245 15) A.V. Ignatyuk, Sov. J. Nucl. Phys. 21 (1975) 10 16) J. Provindencia and C. Fiolhais, Nucl. Phys. A435 (1985) 190 17) J.L. Egido, to be published 18) G. Puddu, P.F. Bortignon and R.A. Broglia, Ann. of Phys. 206 (1991) 409 19) A.L. Goodman, Phys. Rev. C34 (1986) 1942 20) L.D. Landau and E.M. Lifshitz, Course of theoretical physics (Pergamon, Oxford, 1959); U. Fano. Rev. Mod. Phys. 29 (1957) 74 21) M. Sano and S. Yamasaki, Prog. Theor. Phys. 29 (1963) 397 22) T. Kammuri, Prog. Theor. Phys. 31 (1964) 595 23) P. Ring and P. Schuck, The nuclear many-body problem, (Springer, Berlin, 1980) 24) K. Dietrich, H.J. Mang and H. Pradal, Phys. Rev. 135 (1964) B22