The RPA strength function in the presence of thermal fluctuations

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 633 (1998) 613-639

The RPA strength function in the presence of thermal fluctuations R. R o s s i g n o l i a , l R R i n g b a Departamento de F£~ica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina b Physik Department der Technischen Universitgit Miinchen, 85747 Garching, Germany

Received 3 November 1997; accepted 24 December 1997

Abstract A fully microscopic derivation of the RPA strength function at finite temperature in the presence of thermal fluctuations is presented, starting from the path integral representation of the partition function and the ensuing static path approximation. The treatment of repulsive terms in the interaction and the strength for zero energy are also examined. Comparison with exact results in a solvable model indicates that a good prediction of the energy weighted moments of the strength function is obtained, particularly in transitional regions, where the improvement over conventional thermal RPA results is significant. @ 1998 Elsevier Science B.V. PACS: 21.10.Pc; 21.60.Jz; 24.60.Ky; 05.30.Fk Keywords: Strength functions at finite temperature; RPA correlations; Fluctuation phenomena

1. Introduction

The microscopic calculation of strength functions and other decay properties in hot nuclei [1], starting "ab initio" from a given interaction in a particular configuration space, constitutes one of the most important challenges of theoretical nuclear physics. The finite temperature (FT) RPA treatment [2-6], constructed around the self-consistent FT mean field, constitutes the basic microscopic approach, but in finite systems the effects of statistical fluctuations in the relevant order parameters of the mean field are important, particularly in transitional regions, and should be taken into account. This fact was recognized earlier in theoretical descriptions of the giant dipole resonance, where i Corresponding author. 0375-9474/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 5 - 9 4 7 4 ( 9 8 ) 0 0 8 1 5 - 4

614

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639

the effects of shape fluctuations were shown to be essential [7-9]. This was also the case in the description of collective E2 transitions along rotational bands in hot nuclei, within the microscopic treatment of Ref. [ 10]. In most cases, however, statistical fluctuations were introduced by means of semimacroscopic prescriptions [ 11,8], which led at first to ambiguities in the integration measure over the shape variables and in the thermodynamic weight. The static path approximation [ 12-16], derived from the path integral representation of the partition function obtained with the Hubbard-Stratonovich transformation [17], provided for the first time a fully microscopic and precise treatment of statistical shape and pairing fluctuations [ 13]. This method was mainly applied to the calculation of static observables such as energy level densities and expectation values, in which case it provided the exact high temperature limit. SPA results for strength functions, without the inclusion of RPA correlations, were less satisfactory [ 14]. The addition of quantal fluctuations around SPA led to the SPA÷RPA treatment [ 18-20], which was shown to provide almost exact results for static observables above a certain breakdown temperature, normally very low, in the case of simple separable attractive interactions. The SPA+RPA was recently adapted to deal with interactions containing repulsive terms [21 ], where accurate results for static observables were also obtained. The aim of this work is to present a simple yet fully microscopic derivation of the RPA strength function at finite temperature in the presence of statistical fluctuations, utilizing the SPA+RPA formalism (denoted below as correlated (C) SPA). Our treatment of the strength function is simpler than that developed in Ref. [20], but is closer to current, more phenomenological approaches [22]. It is at the same time quite general, providing simple closed expressions for the energy weighted moments of the strength and the thermal correlation function, and being rigorously applicable in both canonical or grand canonical ensembles. We also discuss the strength for zero energy, which is related to the description of low energy collective transitions at the mean field+RPA level [10,3], as well as the treatment of repulsive terms in the interaction, which is obviously important for the application of the present formalism to a quadrupole plus dipole Hamiltonian [2,3], where the dipole term is repulsive. An illustrative example is given in the context of a solvable model, where the exact FT strength function is available for comparison. The formalism is discussed in Sections 2 and 3, with additional theoretical details given in the appendices; the application is given in Section 4. Conclusions are drawn in Section 5.

2. Formalism 2.1. Strength functions at finite temperature

The strength function of an operator Q in a system described by a Hamiltonian H is defined as [2]

R. Rossignoli, P. Ring~NuclearPhysicsA 633 (1998) 613-639 S(E) =_Z PKI(g'lalg)12t~( E -- EK, + EK)

615

(1)

K,K'

+ioo = 27ril f

G(~')e~edr,

(2)

--ioo

where Ig) denotes the exact many-body eigenstates HIK ) = EKIK), PK is the statistical weight of the initial states and

G(T) = (O~(r)a) = Z PKI(K'IQIK)]2er(Ex-EK') K,K'

(3)

is the imaginary time correlation function, with Q~('r) --=CHQte -~n. The total strength is

f

S(E)dE= (QtQ) = G(0),

(4)

--00

while the energy moments of S(E) are related to the derivatives of G(~') at z = 0 (Appendix A). The previous expressions are valid for any initial distribution PK. If the system can be described initially by a canonical ensemble at temperature T = l/kfl, then

Px = Z -1 e -~EK,

Z = Z e-~eK = Tre-/3H' K

(5)

and Eq. (3) becomes

G(~') = Z -l Tr[e-(/3-~)HQ t e-~nQ].

(6)

An accurate numerical determination of S(E) from Eqs. (2) and (6) is nevertheless difficult in correlated systems. Numerical evaluations of G('r) in many-body approximations beyond the mean field+RPA level, such as CSPA or Monte Carlo calculations [ 23 ], are usually reliable for ~- E [0, fi], but may degrade considerably for the large imaginary values of 7- normally required in Eq. (2). The determination of S(E) from numerical values of G(~-) in the interval [0,/3], though possible in principle (using, for instance, maximum entropy inversion methods [23]), is an ill posed inverse problem, as large changes in S(E) may result in only small variations of G(7-) for z E [0,/3]. A different alternative is to employ the representation ¢5(x) = - ( l / o r )

lira Im[ 1/(x 4- it/)]

r/--~0 ~

and rewrite Eq. (1) in a canonical ensemble as

PK-- PK' 1 - - e - ~ ' x '-EK) I(K'IQIK)I2&(E - EK, ÷ EK) + So~(E)

s(e) = Z

(7)

K~ K'

1 l i m Im ~7" r/---*0+

If(E)R(E-[-i~) ÷ E-~--Fq]

(8)

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

616

R(E+i~)

=Z[(K'IQIK)I 2

PK--PK' E K -- EK, + E + ir1 '

K,K'

(9)

where So = ~KPxI(KIQIK)[ 2 is the diagonal term in (1) and U(E) = 1/(1 - e -#E) if E 4: 0, f ( 0 ) = 0. Eq. (9) is the thermal response function [2], which can be obtained by analytic continuation of the Fourier expansion coefficients of G(T) in the interval [0, fi], as proposed in Ref. [20]

Rn =- - / G(~') ei°'"rd7-,

w,, = 21rn/fl,

( 1O)

~t

o

=~-~I(K, IQIK)I2

PK-PK'

=R(iw.),

n 4: O.

(11)

E K -- E K ' + i w n

K,K ~

There is a unique analytic continuation iton ---+ E + i~7 complying with the correct asymptotic behavior of R ( E ) for E --, oe [20,24]. For n = 0, Eq. (10) yields

Px -- PK' ](K']Q]K)J2EK EK,

go= Z K~ K~

fl Z

pKI(KIQIK)I2 '

(12)

K

so that So can be obtained as

So = - f l - '

[Ro - limoR(E) ] .

(13)

This procedure determines S(E) completely just from the knowledge of G(~r) for r E [0, fl], but requires the analytic evaluation of the coefficients (10) for n ¢ 0. It is equivalent to the analytic summation of the Fourier series for G(~-) in the interval (0, fl) and the subsequent application of Eq. (2) to its analytic continuation (Appendix A). Note that diagonal terms do not contribute to R n for n v~ 0, so that they cannot be recovered just from the analytic continuation R ( E ) . This is also the case for transitions between strictly degenerate states, but for Q hermitian they can always be considered as diagonal terms by an appropriate choice of basis (such that (KIQ[K') = ~XK' (KIQIK) if EK = EK,). Otherwise, the limit (PK -- PK,)/(EK - EK,) --~ - f l P x is to be applied in (12) if Ex = EK,, so that these terms are directly included in So as defined by Eq. (13). Diagonal terms are essential in mean field based approaches for describing actual collective transitions between quasidegenerate states (see Sections 2.3 and 4 and Refs. [ 3,10] ) and for determining the total strength.

2.2. Evaluation of R, The analytic evaluation of R, from Eq. (10) is not straightforward in methods such as the SPA and CSPA [20]. Here we will show that R, can be conveniently expressed as

1

R,, -

022

flZ aA,,c)a,*~ ~

_

1 O21nZ _ 6,ofll(Q)12 ' fl °aooaZ ~=o

(14)

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639

617

where 2 is the trace of the imaginary time evolution operator corresponding to H plus a time-dependent periodic perturbation, Z--TrT~exp

{i -

}

dr[H+Q(r)l

,

o

Q.(r) =QF(r) +QtF*(r),

(15)

F(r) = Z /l,e-i'°",

(16)

17

with ~o17= 2rrn/13, i" the time ordering operator and a = ( . . . a17...) the set of Fourier coefficients. Eq. (14) expresses R, as a dynamic susceptibility and requires just second order perturbation theory for its evaluation, providing a direct way to evaluate R n in the SPA and CSPA. Let us prove Eq. (14). Using the interaction picture in imaginary time, Z can be expanded up to second order as

2~Tr

Tr

{ Ii e-ell

1

e -n"

drQm(r) +

1-

0

drO_H(r)

o

/drQ.(r)+ o

s i o

dr o

dr'O.H(r')

}

(17)

dr, e(~ ~)HQ(r) e (r r)MQ(r) o

with 0 H ( r ) = erHO_(r)e-~H. Any perturbation Q ( r ) can be expanded in the interval (0,13) as O(r) =

0. e It

with 017 r-independent operators. Eq. (17) then leads to the general result

e

R17= Z - l ~

Tr[e-(e-')HQ_,,e-'HQ17]ei .... dr,

(19)

4

o

where (~)o) = Z - 1 T r e-eH~)o. For the case (16), Q_.17= Q,~17+ Qta*_17, and comparison with Eq. (10) leads to Eq. (14). To obtain R17, it is actually sufficient to consider a single frequency F ( r ) = 3.17e-i .... , in which case 2 ~ Z(1 -~a17A~R17) for n 4= 0, while 2 = Tre -e(H+0°) ~ Z(1 -/3(~)o) + ½BR0) for n = 0.

2.3. Mean field expressions For later use, we give here the general form of the strength and response functions when H and Q are both one-body fermion operators, i.e.

618

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

k

k,U

with Qk, k = (k'lQIk) and Ik) the single particle (sp) eigenstates of H. In this case the states Ig) are Slater determinants and Eq. (1) becomes S(E) = ~

[Q~,kl2(nk(1 - n~,)) 6 ( E - ek, + ek) + So6(E),

(20)

k~k ~

So =

~-~Qkknk

=

I(e)2l

k

+ Z Q 1 k Q k , k,((nknk,) -- (nk)(nk,)),

(21)

k,k'

where nk = c~c~ and (O) = ~-]kPkQkk, with Pk =- (nk) = S

PK(KlnklK) K

the sp occupation probabilities in the initial distribution. Eq. (20) gives the general form of the strength in the mean field approximation. The first sum in (20) describes sp transitions. In the case of a spontaneously broken symmetry in the mean field, with (Q) # 0 in the mean field and (Q) = 0 in the exact case, the diagonal term So6(E) accounts for the collective transitions between states which are degenerate at the mean field level [3,10]. For instance, if parity is broken in the mean field and Q changes parity, this term represents the transitions between nearly degenerate states of different parity of the exact strength (see Section 4). In a canonical ensemble, the averages in (20) become thermal expectation values, with Pk - Pk' (nk(l --nk,)) = (1 -- tSk~,) 1 -- e-P~k '-~k)

(ek 4: ek,),

(22)

so that the strength for E 4 : 0 has the same form (7) in terms of sp energies and probabilities. Eqs. (9) and (10) become R ( E + i~7) = Z

IOk'kl2

k#k ~

Rn = Z

IQk'kl2

k4=k ~

Pk -- Pk' ek -- ek, + E + i~?'

P k - Pk' ek - ek, + iw,,

6noflSo,

(23)

where the limit ( P k - p k , ) / ( e k --ek,) --+ --/3(nk(1 --nk,)) is to be applied in (23) if n = 0 and ek = ek,. All previous expressions remain valid in a grand canonical (GC) ensemble with variable particle number N, where Px oc e -B(e~-uN). In this case Pk becomes a Fermi probability and Wick's theorem is applicable, leading to (nkn~k) --PkPk' = 8kk'Pk( 1 --p~) and (nk(1--nk,))=

(1--6kk,)p~(l--pk,),

Pk = [1 +e~(~k-~)] -1,

(24)

R. Rossignoli, P Ring~NuclearPhysicsA 633 (1998) 613-639

S0 = 1(0)12 + ~ IOkklZpk(1--pk).

619

(25)

k

We remark, however, that Eqs. (22) and (23) are valid in both the canonical and GC ensembles. Only the values of p~ and (nknk,) in So are different. In the canonical case, these can be obtained using number projection methods [ 16,25].

3. Strength function in the SPA+RPA We now consider a general two-body Hamiltonian

H = Ho - 1 Z

v~Q~,

(26)

P

where Ho and Q~, are Hermitian one-body operators. Within a finite configuration space, any two-body Hamiltonian can be expressed in the form (26) (non-unique) [23]. Eq. (15) can be written as 2 = Tr i? [I,~,= 1 U ( f l ( m / n ) , f l ( m - 1/n)), where U(z2, r l ) is the evolution operator from ~- = rl to r = r2 for H + Q ( z ) . Applying the HubbardStratonovich transformation [ 17] in each interval, in the limit n ~ c~, we can formally write Z as the path integral

2=fD[x]TriVexp

dT{H[x(r)] +0(7")}

-

,

(27)

where x = ( . . . x ~ . . . ) is a set of auxiliary fields and

-(x) : H o + Z

(4

)

(28)

P

is a one-body operator (in a GC ensemble, H ( x ) --~ H ( x ) - tzN). In the following we shall assume a one-body strength operator Q.

3.1. SPA In the SPA, only time independent paths x ~ ( r ) = x~ are considered in (27), i.e.

~

7

d(x)TrT~exp

--~<~

{Y

dz[H(x) + 0(~')]

-

0

}

,

(29)

where d ( x ) = 1-[~ V/fl/2zrv~ dx~. Eqs. (14) and (29) lead to

Rspa n = Sspa(E)

-'fd(x)Z(x)R.(x),

(30)

Zspa

= Zsp-a1 /

d(x) Z (x) S(E,

x),

(31)

620

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

where R,,(x) and S ( E , x ) H = H ( x ) , and Zspa = f d ( x ) Z ( x ) ,

denote the one-body expressions (23)

Z(x) = Trexp[-flH(x)],

and (20)

for

(32)

is the SPA partition function. Eqs. ( 3 1 ) - ( 3 0 ) contain just the statistical fluctuations around the mean field and correspond to the adiabatic assumption [14] in a direct evaluation of (6) (i.e. assuming the same x in the SPA expansions of both exponentials in ( 6 ) ) . The mean field expressions are just R,,(x) and S(E, x) evaluated at the maximum of Z ( x ) , determined by the self-consistent equations

--L'u/~-la In Z ( x) /c~xv = x~ - v~(Q~)x = 0.

(33)

Although equation (32) provides a good estimate of the static partition function Z at high temperatures, becoming exact for T ~ ~ in a finite configuration space, the SPA is not reliable for 2, i.e. for dynamic correlations. Even when the SPA is exact for representing e -~H (i.e. when the operators Q, and H0 commute), it is not exact for 2 if [Q, H ( x ) ] 4: 0. The present SPA treatment provides only a fair estimate of R0. The inclusion of quantal fluctuations in Z becomes essential for n v~ 0.

3.2. SPA+RPA (CSPA) Let us consider the quantal or time dependent fluctuations around SPA [ 18,20,21]. The time dependent variables X(T) in (27) can be expanded for T E [0,/3] as

xz,(T) = x~, + ~"~ xvne-i .... ,

w, = 27rn/fl,

(34)

n~O

where x~ now represent the static Fourier coefficients, giving the time average fl-i f : d ~ ' x ~ ( r ) , the SPA being recovered for x~, = 0. Eq. (27) becomes

2=

d ( x ) e -~x'~'-'x/2

d ( x , ) d ( x , )- e

-

/~x,~ x" ;

(35)

where d ( x . ) = lip X//fl/2~iv. dx~., x,, is the vector of elements x~. (xt. denotes transpose), x~ - X-n and v is the matrix 8~,v.. It is convenient to consider a complete set of operators Q~ (we can set v~ ~ 0 afterwards if necessary) such that Q and Q(T) can be expanded as

Q = ~__C,,Q,,, tl

0_(~) =

~ P,l't

O~;a.,, e - i .... ,

;~. = c ~ a . + C;a*_..

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639

621

The simplest non-trivial approximation is now to expand the logarithm of the trace (36) up to second order in both x~ and ~n, thus assuming a gaussian approximation around x, = 0 for the quantal fluctuations. Using Eqs. (18) and (23), we obtain, for a perturbation with A0 = 0, a displaced gaussian integral for the variables x~

~ fd(x)Z(x)

Hd(xn)d(x~) n>0

× exp{-fl[x~o-~x,, + (x~ - -a~)TC(iw,,,xo) (x. - A.)] }, 7-C~,(iw.,x) = ~*k' Z (Q~)kk,(Q~,)k,k e k -P~-sk, Pk' +iWn'

Wl, # O,

(37) (38)

where Eq. (38) is the response matrix for the one-body Hamiltonian H(x), with [k), ek the corresponding sp eigenstates and energies. Performing the gaussian integration, we obtain

2 ~ f d(x)Z(x)Crpa(X)exp - f l Z

~tTErpa(iWn'x)Anl

Crpa(x) = H Det[ 1 + 7E(io~,,, x)v] -1

(-On

2 + e] ( x )

n>0

(39)

(40)

17>0 a > O

sinh [ fle,~(x)/2] oJ,~(x)

(41)

= or>0 H sinh[floJ~(x)/Z]e~(x)'

T~rpa(iton, x )

=

[ 1 + TC(iw,, x)v]-l~(iw,,, x),

(42)

where Eq. (42) is the thermal RPA response matrix around H(x) (see Appendix B and Ref. [2] ) and (40) is the RPA correction factor to the partition function [21,20], with e~(x) = ek, - sk the uncorrelated pair energies and w,~(x) the thermal RPA energies determined from (see Appendix B) Det[ 1 + 7E(w~, x)v] = 0.

(43)

In (40), a runs over all pairs U > k, i.e. all frequencies of a definite sign. As ~, D-~o~(i,+1/2) I sinh-l(½flw), Eq. (41) is proportional to the ratio of the partition function of a system of independent RPA bosons of energies wa(x) to that of uncorrelated fermion pairs of energies e. (x), considered as bosons [ 21 ]. The additional factors w./e~ arise due to the exclusion of the n = 0 term in (40). Eqs. (14) and (39) yield Rcspa n =

P,-pa(X)

f d(x)Prpa(x)Rwa(ioJ.,x), --

Z(x)Crpa(x)/Zcspa,

n

~ o,

Zcspa = [d(x)Z(x)C~pa(X), d

(44) (45)

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

622

F~ (x) iw--~---~oa'

Rrpa(#a,,, x) = CtT"~rpa(iw,, x ) C = Z

(46)

o[

where F ~ ( x ) = ~ , C~F,%,(x)C~, (see Appendix B, Eq. (B.9)). Eq. (46) is the thermal RPA response function. The analytic continuation iwn ~ E + it/ yields finally Scspa(E)

-

1

-

77"1 - e -l~E

f

d(x)Prpa(X)Im[Rrpa(E + irl, x) ]

E 4: O,

(47)

for r/ ~ 0 +. Eqs. (44) and (47) are our fundamental results. Eq. (47) represents the average over all static fields of the thermal RPA strength function, in qualitative agreement with more phenomenological approaches [22]. The present derivation is, however, fully microscopic, and naturally includes the RPA corrections on the thermodynamic weight Prpa(X) through the factor C,~a(X). The SPA result (31) is recovered for C~a(X) --~ 1 and 7~a(iWn,X) ---+ ~ ( i w , , x ) , i.e. o~,~(x) ~ e~(x) and F ~ ( x ) --~ IQ~,~lZ(p~ - p k , ) in (46). For fixed x, the RPA corrections shift the strength from e , ( x ) to w ~ ( x ) , but in the CSPA the corresponding peaks will then be smoothed by the integral over x, acquiring a width. The final CSPA strength for E 4 : 0 will remain a smooth function of E for r/ --~ 0 (see Section 4). We also remark that all previous expressions are strictly valid both in a canonical or GC ensemble (Wick's theorem is not used at any step). The statistics affect only Pk in (38) and Z ( x ) . We note finally that Eq. (44) can be directly expressed in terms of auxiliary field averages. By a change of integration variables x~, = x~,, - ~ in Eq. (36), the perturbation can be shifted to the gaussian factor (35). The derivative (14) can then be evaluated exactly, leading to

Ro =

]c = Z C~v~ *-1 ( 6 ~ , - f l ( x ~ x ~ , , , ) v ~ , l ) c ~ , ,

(48)

1J, p !

where (x~x~) denotes the average with respect to the full path integral (for Q(7") = 0). Applying the same procedure in (37), we find that Eq. (42) can be written as

7~pa(iW,,, x) = v - l ( 1 - fi(x~xtn)xV-1), t (x~x,,)x=f1-1

U

[l+~(iwn,x)v]

-1,

(49) n 4: 0,

(50)

where Eq. (50) is the average for fixed values of the static coefficients x in the gaussian approximation (37). Thus, Eq. (48) extends Eq. (44) to the case of non-gaussian quantal fluctuations. If [ Q , , , H ( x ) ] = 0 Vx, we can separate the term with Q,, in the exponent of (27), and flv~ 1(x~r,x~,,,) = 8~,, both in the exact case and in the CSPA, whence R,, = 0 for Q = Q~ and n 4: 0. Eq. (48) thus measures the deviation of the fluctuation from this trivial limit.

3.3. Evaluation of Ro and G ( r ) Using the previous method for a static perturbation in (36), Eq. (14) yields

Ro = C t [ v -l ( l - fl(xx')v -1) ]C,

(5l)

R. Rossignoli,P Ring~NuclearPhysicsA 633(1998)613-639

623

which is exact if the average is taken with respect to the full path integral. In the CSPA, (51) becomes

Rcspa o _-

.i

Prpa(X) C ~ [ u -1 ( 1 - t ~ x x t)

d(x)

= fd(x)P

c -I ] C

a(x)[Ro(x) +

(52)

(53)

where Ro(x) is the one-body expression (23) for n = 0 and H = H(x), and B~pa(X) a static RPA correction (depending on the derivatives of Crpa(X) with respect to x~), which is relatively small. Eq. (52) is easier to evaluate and shows that for R0, the CSPA corrections to the SPA reduce essentially to the factor Crpa(X) in Pwa(X). Using Eqs. (13) and (44) we obtain

[R0spa + S d(x)Prpa(X)E F~(x) o)a(x)

frospa: - - f l - '

(54)

Oe

When the mean field solution maximizing Z(x) is attained at x~ v~ 0 (implying in /?cspa general a spontaneously broken symmetry in the mean field, with (Q,)x 4= 0), "'0 becomes large for Q = Q~ and Eq. (54) will give rise to a collective peak at E = 0 (see Section 4). The correlation function can now be recovered analytically using Eqs. ( A . 3 ) - ( A . 6 ) of Appendix A acspa(7")

=

S~0spa -~-

f

e_rW,(x) d(x)Prpa(X)E F~(x) 1 - e ---2~-~"(x)'

(55)

a"

which, except for the first term, is just the average of RPA correlation functions. It allows for an analytic continuation for arbitrary r. Note that Gcspa(0) = (Q~Q)cspa is the total CSPA strength. For Q = Q~, this average can be independently obtained from (45) as 2 2 O In Zcspa (Q~,)cspa - /3 0v~, _ ~spa

1l

q_ 7 a d(x)Prpa(X)

x E F,~(x) coth

flw,~(x)

(56)

OL

(where we have used Eq. (B.10)) which coincides with (55) for r = 0 and Q = Q, (Hermitian). The exact average can actually be expressed, using Eq. (36) for {)(~') = 0, as

1 [fl{x2)_l+2E(_~(x~r~x~n)_l) ]

201nZ n

- nT+

,,>o

oo = _/~--1

E en' li=-- oo

(57)

R. Rossignoli, P. Ring/Nuclear Physics A 633 (1998) 613-639

624

in agreement with Eq. (A.8). In the CSPA, the series (57) leads to (56) (Appendix A). This also ensures that the total CSPA strength integrated over real energies fulfills (4) if Rcspa(E + it/) has no poles in the upper complex plane. The CSPA strength moments (n ~> l) are

-~,.c(,.) (o~ = f d(x)e~.(x) ~ r~(x)o~'~(x) T--~ J

~¢spa

\ ~/

(58)

o,

It should be remarked that when the mean field potential - T in Z (x) exhibits symmetry breaking minima, i.e. T < Tc, the lowest RPA energies wa(x) will become imaginary or complex in regions of x where the potential is unstable [21]. Nevertheless, Cwa(X ) will cease to be positive definite only in those regions where fl]o~(x) I >~ 2~- for imaginary frequencies, which will arise below a breakdown temperature Tc* < Tc, normally very low. In these regions the gaussian approximation (37) for the dynamic coefficients no longer holds and large amplitude quantal fluctuations should be considered. Note that Eqs. (45), (55), (56) and (58) remain finite in the presence of imaginary energies provided fllo~(x)l < 2~, i.e. above the CSPA breakdown, and also in the presence of zero modes w , ( x ) ~ O, where the pole in (55) is canceled by the second term in (54).

3.4. Repulsive terms In the presence of repulsive terms v~ < 0 in (26), the integral over the corresponding variables x,, and x~n in (29) and (36) is to be performed along the imaginary axes. In this case the integral over the static repulsive variables x~ can also be quite accurately evaluated in the saddle point approximation (see Ref. [21] for more details), except for very low temperatures (normally below the CSPA breakdown). It is the attractive part of the interaction which gives rise to spontaneous symmetry breaking and phase transitions in the mean field, and hence to large amplitude statistical fluctuations in the corresponding order parameters at finite temperature. This leads to [21]

Zc.,pa= fd(x)Z(x)C~(x)Crpa(X), J

C~(x) = Det[1 + 7 ~ ( x ) v ] - 1 / 2 ,

(59)

k~k'

- f l Z (Q~)kk(Q~, )k'k' ( (nknk,) -- PkPk' ),

(60)

k,k'

and P,pa(x) = Z(x)C~(x)Cwa(x)/Zc.~p,, where the labels u, u' in (60) are restricted to the repulsive terms and the final static integration variables x in (59) correspond to the attractive terms only. For repulsive terms, the gaussian approximation should also be employed for the average of xpx~, in (52), i.e. c~-' (3,,,~, - fl(xpx~,)xV~') = {[1 + Tg~(x)v]-lTgS(x)},,,,, - fl(Q~)x(Q~')x, where v~, v~, < 0, with

Rcspa(0) the

average of (61).

(61)

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639

625

The saddle point approximation for the static repulsive variables is to be applied around the unique real solution x~ = v~(Q~)x (v~ < 0) of the mean field Eqs. (33) for the repulsive terms, obtained for fixed running values of the static attractive variables [21]. The Hamiltonian H(x) to be employed in (59) and (60) is, therefore,

n(x)=Ho+

~

\~-~v -x~Q~

+ ~

p,t~j, > 0

v~ 5(Q~)x-(Q~)xQ,

,

(62)

p,t'~, < 0

which is always Hermitian. This ensures that Crpa(X) will remain positive definite (above the breakdown temperature). If the repulsive terms do not change the mean field, i.e. if (Q~)x = 0 Vx for v,, < 0, H(x) will depend only on the attractive part. In this case the main repulsive effects will arise through the factor Cwa(X ) and the RPA matrix (42), which are to be evaluated for the full interaction. The present procedure justifies in part current phenomenological approaches where large amplitude fluctuations correspond to attractive variables only (such as quadrupole shape fluctuations) and the effects of repulsive terms on the thermodynamic weight are totally neglected [ 22]. This amounts here to neglecting the effects of C~(x) and C~pa(X) on Prpa(X).

4. Application 4.1. Model and partition function In order to illustrate the present formalism and test its reliability, we consider a solvable model consisting of single particle states Ipo-), p = 1. . . . . /2, o- = 5:1, with the quasispin Hamiltonian [26]

H = eoJz - (VxJ2x + vyJ~)/Y2

(63)

=eoJz-¼[(Vx-Vy)(J2++j2_)+(Ux+Vy)(J+J-+J-J+)]/£L l Jz : -~-~o'c,,c cmr'

j,r:Jx+iO-Jv:Z,

p,o"

p,~Cp_a,

ct

(64) (65)

p

where the operators Jg, i = x, y, z, satisfy the SU(2) commutation relations. The exact partition and strength functions in different ensembles can be obtained by diagonalization (see Appendix C). We shall consider Vx > 0 (attractive), vs ~ 0 (repulsive). After integrating over the static variable corresponding to Jy in the saddle point approximation, the CSPA partition function becomes a one-dimensional integral of the form (59), with d(x) = (flg2/41rvx) 1/2 dx and (66)

Z ( x ) = e - n ~ x 2 /4''' T r e - f l ( e ° j ~ - xJ') ,

C~(x) = [ l + [ v y l p ( x ) / e ( x ) ] -U2 Crpa(X)

w(x) sinh[ ½fie(x) ] e(x) sinh[ ½fw(x) ]

(v;.~<0),

(67) (68)

626

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

e(x) = e+(x) - e_(x) = (eo2 + x2) 1/2,

(69)

o)2(x) = e2(x) [1 - Vxp(X)e2/e3(x)][1 - vyp(x)/e(x)],

(70)

p(x) = p _ ( x ) - p + ( x ) -

20lnZ(x) ¢~a

(71)

O~(x)

Here e~:(x) = + ½ e ( x ) are the sp eigenvalues of eoJz - xJx = e(X)]z, p+(x) the corresponding occupation numbers and w(x) the RPA energy [21] (see Appendix C). These expressions are valid both in the GC ensemble for N = s2, where

Tre -/3~(J:-xJ-d = { 2 c o s h [ J f l e ( x ) ] } 2a,

p(x) = tanh[¼fle(x)],

(72)

and also in the full or restricted canonical ensembles (Appendix C), where different expressions for p (x) and Z ( x ) in terms of temperature are to be employed. The self-consistent mean field is determined by the equation

x = 2Vx(Jx)x/l2 = Vxp(X)X/e(x),

(73)

the solutions of which are the stationary points of Z(x). The present repulsive term does not modify the mean field and affects Zcspa only through (67) and (68). We may recall that for vx < eo or T > T~, the only solution of Eq. (73) is x = 0, whereas for vx > eo and T < Tca "deformed" solution x -- ±x0 arises which breaks "parity" symmetry ( [ H , e irrJ: ] = 0 Vt3x, b'y ) and leads to ( Jx) ~ 0 in the mean field. The exact energies for fixed J split into multiplets of different parity and (.Ix) = 0 in the exact case at all temperatures. The energy splitting is small for the lowest states (and large g2), which justifies the mean field, but increases gradually as the excitation energy is increased (or S2 is decreased). In the mean field this is reflected as a sharp deformed to normal transition at T = Tc, with Tc determined from the equation vxp(O) = e0. This transition becomes considerably washed out for finite /2 in the exact case and in the CSPA, which yields practically exact results for static observables above the breakdown temperature (see Ref. [21] for more details).

4.2. Response and strength functions The response matrices (38) and (42) corresponding to the operators J~ and p = x, y become

112p(x)/e(x) ( 7~(iw,,x) = - 2 7-~rpa(ito,,, x) = -

t32(X) @0) 2

e2

wneo )

--O)ne0 /32(X)

(74) '

1 ap(x)/e(x) 2 w2(x) + w~

x (e2[1-Vyp(x)/e(x)] -o~nso

o)neo e2(x) [1 - v x p ( x ) ~ / ~ 3 ( x ) ]

where w, 4: 0. The diagonal elements of (75) can be written as

) '

(75)

R. Rossignoli, P. Ring~NuclearPhysics A 633 (1998) 613-639 2w(x)Fv(x) =Fv(x) [ [7-~rpa(iWn,X)]~u=

to2(x) + (-02

1 iWn - o~(x)

627 1

]

iw, qSw(x)

, (76)

with F~(x) = -½~28o~(x)/Sv~, in agreement with (B.9) and (B.10). For Q = J~, the RPA strength at fixed x is

Srpa(E,x)

1

=

1

7r 1 - e - ~ e I m [ ~ p a ( E + irl, x ) ] ~ r~(x)

~--,o+ 1 - - e - ~ ( x ) [ 6 ( E - w ( x ) )

+e-~(x)8(E+w(x))],

(77)

while the mean field strength corresponds to (77) evaluated at vx = v v = 0, i.e. o~(x) e(x), F~(x) --+ I~(x). In addition to the energy shift, for vx > 0, vy < 0, the RPA corrections increase the intensity for Q = Jx relative to that for Q = Jr, as seen from (75). The conventional thermal RPA and mean field results are obtained when x is the solution of (73). The CSPA strength for E 4: 0, Eq. (47), can be evaluated analytically OO

o-~o+

1--- e---~(-~

dx

'

(78)

--OO

where E = +~o(x) 4 : 0 and Pwa(X) e(Z(x)C~(x)C~pa(X) is the normalized probability. It is thus a smooth function of E for E 4: 0. Away from transitional regions, the sharp RPA peaks acquire in the CSPA a width essentially determined by that of Prpa(X). The deformation of the RPA strength may, however, be considerable, particularly near critical points. Using Eqs. (52), (54) and (61), the diagonal term (13) for Q = J~ becomes f

a(x)P o(x)

[/'2 (

/382 ~)

+ 2Fx(x)] j'

(79)

p(x)/e(x) +2~1 2 1 + [vylp(x)/e(x)

=0"

(80)

Thus, for Q = Jr there is no diagonal term in the strength, while for Q = Jx, Eq. (79) will be of order ~2 in the symmetry breaking phase, giving rise to a strong collective peak at E = 0 which represents the transitions between quasi-degenerate states with different parity. In the normal phase, S0x will be small. The CSPA correlation function (55) for Q = J~ becomes O(3

G~,p.(r) = So~ + ~/ dxPrp.(X)r~(x)cosh[~o(x)( ½/3 sinh[ ½tim(x) ]

r)]

(81)

'

and fulfills G~(r) = G,,(fl - 7-) since Q is Hermitian. The total strength and energy moments are OO g,

Gcspa(0) = Sou -1- / --OO

dxPrpa(x)F~(x) coth[ ½fl~o(x)] = (J~),

(82)

R. Rossignoli,P Ring~NuclearPhysicsA 633 (1998)613-639

628

~"(7(n) (0~ = /

dx Prpa(x)l~u(x)(o"(x) coth'~[ ½fl(o(x) ]

(83)

--0(3

where n )

1 in (83), with t< = 1 (0) for n even (odd). Eq. (82) coincides with

fl-l~QOlnZcspa/OUu. It is also verified that Eq. (81) is the Laplace transform of the complete CSPA strength, with (82) and (83) equal to the explicit total strength and strength moments OQ

/EnScspa(f)df='~l /

(on(y){Scspa((o(X))-I- (--)nScspa(--(o(X))}

--OO

dw(x) x ~ dx + (3noSo,.

(84)

For T < To, (oe(0) < 0 and (o(x) becomes imaginary near the origin (approximately in the region where the potential - T l n Z ( x ) possesses a negative curvature [21]). The integrands in Eqs. (81), (82) and (83) nevertheless remain real and finite for co(x) ---, 0 (for Q = Jx, the pole in Eqs. (81) and (82) is canceled by the second term in Sox, while for Q = Jy, F~,(x)/(o(x) remains finite for (o(x) --~ 0) as well as for (o(x) imaginary provided I/3(o(x)t < 2~', i.e. for T above the CSPA breakdown, determined by (/3(o(0)/2~r) 2 = - 1 . The r.h.s, of (84) always coincides with (82) or (83), but the intervals where (o(x) is imaginary are not included in the energy integral of the 1.h.s., if restricted to the real energy axes. One should actually extend the energy integrals over the imaginary axes. For representation purposes, the contribution to the total strength of the sector with (o2(X) ~ 0 in Eq. (82) can be added to Soy (in this way Sox remains finite). Nevertheless, this sector also makes small but non-vanishing contributions to higher strength moments (not necessarily positive). The SPA results are obtained by replacing (o(x) ~ e(x), F~(x) ~ F°(x) and Prpa(X) --+ P~pa(X) o ( Z ( x ) C ~ ( x ) in the CSPA expressions. The SPA strength for E¢0is Sspa(E) = 4-2 ~ _ - - ~

dx

,

E= i e ( x ) ,

(85)

which is non-zero only for [E] > e0 in all cases. Note that the CSPA strength (78) is non-zero for IEI > coo, with too = (o(0) for T > Tc and (oo = 0 for T < T~, and thus exhibits a larger spread. In the self-consistent FT RPA, the strength moments for n /> 1 are

( --~~n(7(n) --rpax(0] ~. : -/'u (Xmf) (on (Xmf) coth" [ ½fl(o (Xmf) ] where Xmf is the solution of (73). In this case

(86)

R. Rossignoli, P. Ring/Nuclear Physics A 633 (1998) 613-639 '

20

'

'

I

'

'

'

I

'

Q=Jx

(a)

':"" i ! ,i / ~

'

'

I

'

'

20

Exact

- -

CSPA SPA RPA MFA

..... .... ......... .....

'

'

I

'

'

'

/

."I 'k

0.5

1.0

./~ "'

.t

20

I

'

0.5

: • .,~.,I. . . . 1.5

1.0 E

':~'

I

'

'

'

J

I

Q=Jx

2

(b)

.~

'

1.5

E '

I

(a)

0

'

'

Q=Jy

10

10

629

i

,

I

,

i

'

'

'

i

i

Q=Jy

i

(b)

.../,., ../ .;,

l

10 ,:

%

0 0.5

1.0

1.5

E

I '~ ' - : ~ ' J - - ~ O.5 1.0 1.5 E

Fig. 1. The strength function for the operators Jx, Jy and the Hamiltonian (63) for vx/el) = 1 and Vy = 0 (a), Vy = -Vx (b). The temperature is T = 0.35eo. Exact, CSPA and SPA results are depicted, together with those from finite temperature mean field and RPA. In all cases we have set a width rffeo = 0.1 in the S-functions. All energies in units of eo. t O 2 ( X m f ) = X m2f ( l - - O y / g x ) ,

Xmf

4= 0 ( T < T c ) ,

o)2(0) = [e0 - V x p ( 0 ) ] [e0 - v y p ( 0 ) ],

(87)

(88)

where (88) is to be employed for T > T~. At T = To, W(Xm¢) = w ( 0 ) = 0 and the moments (86) will exhibit a sharp transition, which will become smoothed out in the exact and CSPA results for finite s2. 4.3. Results

We first depict in Fig. 1 results for the weak coupling regime Vx/eO = 1, in which case the system is in the normal phase at the mean field level for all T > 0. In all cases we employ a fixed width r//eo = 0.1 in the g-functions, and set N --- s'2 = 20 particles. Numerical results correspond to the GC ensemble. The CSPA results are in good agreement with the exact results, both for Vy = 0 and for the repulsive case v,. = - V x . The statistical fluctuations flatten the RPA result, as expected, and slightly shift the strength to higher energies. The RPA correlations are seen to be essential for a basic agreement, mean field and SPA results being totally inaccurate. SPA also gives a too large estimate of the diagonal term So for Q = Jx. The effects of the repulsive term are correctly reproduced by both RPA and CSPA, shifting the strength to higher energies (it increases Ion(x)] Vx) and decreasing (increasing) the intensity for Q = Jr

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639

630 '

'

'

I

12

'

'

'

I

'

'

'

I

'

'

[\

Q=Jx

/i

/o,

-

'

'

'

1

'

'

'

1

'

'

'

1

'

'

Q=Jy

(a)

12

I

6 ~,~

-

. ,.

0.5

1.0

,.";

1.5

.

0.5

•

1.0

1.5

E

E '

'

I

'

'

'

'

'

'

1

'

'

'

Q=Jy

(b)

12

6

L

,,'

(b)

12

f'x:.

f'~

0.5

1.0 E

1.5

0

0.5

1.0

1.5

E

Fig. 2. The same quantities as for Fig. I for the strong coupling case Vx/eo = 2 and Vy = 0 (a), Vy = - - V x (b). The temperature is T ~ 1.5Tc, where Tc = 0.45e0 is the critical temperature for the deformed to normal transition in the mean field.

(Jx). Similar results are obtained for other temperatures, as in this case there are no phase transitions in the mean field and no dramatic changes in the exact strength at finite T. The RPA frequency (70) is here real for all x and T > 0. Results are less satisfactory for the strong coupling case Vx/eO = 2 (Figs. 2 and 3), where the self-consistent mean field is deformed for T < Tc ~ 0.45e0. For T > To, the deviation of the exact strength from the RPA result is large, while the CSPA result provides only qualitative agreement, although the improvement over RPA and SPA is nevertheless significant. Note that if the So contribution is omitted in the CSPA, the ensuing result would underestimate the exact strength for Q = Jx at low E, which conserves a non-negligible value in this region for T above Tc (entailing an appreciable spread of the strength). The E = 0 CSPA strength is, however, too large (though smaller than the SPA estimate) and narrow. For T < Tc (Fig. 3) the most distinguishing feature is of course the huge peak at E = 0 for Q = Jx, which, for vy = 0, is well reproduced by both CSPA and SPA. The agreement for the non-collective part ( o f order 12-1 relative to the collective peak) is only qualitative• The plain RPA and mean field results are totally inaccurate. When the repulsive term is present, the collective peak becomes smaller and broader, and the quality of CSPA is poorer. Nevertheless, So decreases in the CSPA, in agreement with the exact result. Within the CSPA, the repulsive term increases Crpa(X) near x ~ 0, increasing Pwa(x) in this region and thus reflecting enhanced tunneling effects between the mean field wells [21]. This decreases the value of So. In the exact strength, this is reflected in the larger splitting between energy levels with different

R. Rossignoli, P Ring/Nuclear Physics A 633 (1998) 613-639 '

'

'

I

'

I

'

Q=Jx

80

Q=Jx

10

!~

(a)

li\

40

.:

\',.

0 '

'

'

,

,

~

0.5

-

-

-

r

-

~

1

!!

0

i

:'.

1

2

E

E '

631

'

i ' Q=Jx

80 i

it

10

(b)

i

i

I

i

i

g

!\

i

(b)

i

"1

:\ 40

0

0.5

1

0

1

E

E i i

i

I

i

10

i ..i ~ .

j

i! ' ' I

2

i

Q=JY

:i'

10

(a)

.

Vi

:

- ~__... '

~

1

' E

Q=Jy (b)

• ',.,

/i'

0 ~

i

..

~.i

.--.;- i -

2

0

1

2

E

Fig. 3. The same as for Fig. 2 for T ,.~ 0.7Tc. For Q = Jx we show the strength on two different scales in order to observe the collective peak at E = 0 and the remaining part. parity, which gives rise to a broadening of the collective peak for Q = Jx and a larger energy distance between peaks in the non-collective part of the exact strength. For Q = J,., there is no collective peak at E = 0. The CSPA results provide only modest qualitative agreement, but nevertheless constitute an improvement over SPA and RPA. It should be remarked, however, that CSPA does provide a very good evaluation of the first strength moments in all cases, being practically exact for the total strength, as seen in Figs. 4 and 5. The sharp transitions arising in the RPA moments are smoothed out in the CSPA, in agreement with the exact results, where it is seen that the scaled first and second energy moments decrease smoothly as T crosses Tc, with no remnant of the abrupt RPA drop at T = To. Moreover, the CSPA also yields a fair estimate of the strength width, as measured by the quartic dispersion W ( 4 ) , depicted in Fig. 4 for Q = J~., which vanishes at the RPA level. For Q = Jx, the total strength is strongly

632

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639 ~'

'

'

I

'

i

J

[

,

i

i

I

i

,

,

.'

~

'

_

I

'

'

I

F

'

'

'

I

'

'

'

'\',,

b

'... ...............

0.5

1.0

, , 0.5

1.5

Tffc :..'....L..:....I'''

2 -

~ ' ' '

L::;::::i::=,:::::l:: ; ; ; " 1.0 1.5

T/Tc i

'

'

'

b ~:'''..

'

I

'

I

'

'

'

I

'

'

2

0

1

"iS...2:'2222222222222222 ..........i 0.5

1.0 Tffc

1.5

0

~----~-.--t----l---.~----,.-..~...-i.---~-.--r-...,.--4---.,..-~--..~-.-

0.5

1.0

1.5

Tffc

Fig. 4. The energy weighted moments of the strength M(n) ---- ( - ) n G ( n ) ( o ) , Eq. (A.2), as a function of temperature, for Q = .ly, Vx/eo = 2 and Uy = 0 ( a ) , Uy = --Ux (b), according to exact (solid lines), CSPA (Eq. (83), dash-dotted lines) and RPA (Eq. (86), dotted lines) results. M(0) is the total strength. The remaining quantities are in units of the energy eo. W(4) = [M(4)/M(O) - M2(2)/M2(O)] 1/4 is a measure of the width. In the RPA, M(2)/M(O) = ~ 2 ( X m f ) and W(4) = 0. affected by the diagonal term So. For T < Tc, its decrease as the repulsive term is turned on is a consequence of the increase of Crpa(X) for x ~ 0 in the CSPA, and is totally absent in the SPA or conventional mean field+RPA (where So is evaluated with Eq. ( 2 l ) ) . Note also that for Q = Jx, the total RPA strength diverges at T = Tc due to the vanishing of og(Xmf). To assess the accuracy of higher RPA moments, we have therefore plotted quantities not involving the total strength. The CSPA also provides an excellent evaluation of the imaginary time correlation function for 7- E [0, fl], as seen in Fig. 6. This clearly reflects the ill posed nature of recovering the strength from the knowledge of G(7-) in this interval. Results become less accurate only when evaluated at real times 7- = it. The improvement over the RPA results is appreciable, particularly for Q -- Jx due to the inaccurate evaluation of So at the mean field+RPA level.

5.

Conclusions

We have presented a fully microscopic derivation of the RPA strength function in the presence of large amplitude statistical fluctuations, starting from the path integral representation o f the imaginary time evolution operator. The formalism provides a complete microscopic foundation of current phenomenological treatments of strength functions

R. Rossignoli, P. Ring/Nuclear Physics A 633 (1998) 613-639 .~....h...'.

"',.

I

'

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I

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'

l

'

'

'

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'

'

I

'

633

'

'

I

'

'

'

0.4

60

i '/I 7+ ,it

40

0.2

"i,." 7" i

.,"" /' ,,."

. ......:

"

...'"

:

""

20 ,

I

,

,

~ I ~ ~ ~ I ~ ~

0.5

1.0

1.5

0•0

0.5

1.0

T/Tc '

'

'

1

'

'

'

1

'

'

'

1.5

T/Tc 1

'

'

'

'

0.8

•

'

'

I

'

'

'

I

'

'

'

I

'

'

'

~°-.

-_U2252-2. " 2.0

0.6

[~

%..

1.0

{ .-'z~........... ...........]1%21..........:"

0.4 0.5

1.0

1.5

0.0

T/Tc

0.5

1.0

1.5

Tfl'c

Fig. 5. Strength moments for Q = Jx. Same parameters and line conventions as for Fig. 4. Exact and CSPA results overlap for M(0) and M(2). which include shape fluctuations only in an "ad-hoc" manner. The present scheme nevertheless leads to more rigorous and general expressions, which naturally include the RPA corrections in the thermodynamic weight, and is strictly valid for both grandcanonical or canonical ensembles. We have also discussed the evaluation of the zero energy term in the strength function, essential for the description of collective low energy transitions, as welt as the treatment of repulsive terms in the interaction. Comparison with exact results indicates that the main asset of the present approach is the accurate evaluation o f the strength moments in transitional regions, where the improvement over conventional RPA results is noticeable. Application of the present detailed expressions to realistic interactions and inclusion of further correlations in the strength are being investigated.

Acknowledgements

This work is supported by the EEC (contract CI1"-CT93-0352). RR is a member of the Research Center o f CICPBA, Argentina.

Appendix

A.

Imaginary

time

correlation

functions

The correlation function (3) can be written as

G('r) = / S( E ) e -~'E dE, J

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

634

45

T=0.75 Tc T=0.75 Tc Q=Jx

40 a

35 30

_

,.

-

j

b

0.25

0.5

0.75

0.25

0.5

0.75

v13 :...2 . • "L

'

.

'

I

.'

'

.

'

I

.

'

.'

'

I

'

'

. %5 ".

-

T=I.1 Tc

2

11 10

T=l.3 Tc Q=Jx

9 8 0.25

0.5

0.75

0.25

v13

0.5

0.75

v[~

Fig. 6. The imaginary time correlation function, Eq. (3), for Q = Jx and Q = Jr at fixed temperatures, for vx/eo = 2 and Uy = 0 ( a ) , Uy = --Ux ( b ) . Same conventions as for Fig. 4. Exact and CSPA results practically overlap for Q = Jx. and fulfills the o b v i o u s relations I'

(Qt Q) = J

G(O)

S(e)de,

G(/3)= (QQt)

I" ES(_E)dE, Je

=

(A.1)

w h e r e in the second expression we have assumed a canonical ensemble. In this case,

e#~S(-E) is the strength for the operator Q t . The derivatives at r = 0 o r / 3 give the energy m o m e n t s

( )"G (') (0) = f S(E)E" dE, We can r e c o v e r G ( r )

G(")(/3) = f et~eS(-E)E" dE.

(A.2)

f r o m the coefficients ( 1 0 ) as Oo

G(~') = _/3-1 ~

R, e -i°~'r,

0 < 7- < / 3 .

(A.3)

n=~OO

If R,, is o f the f o r m x---

Rn= /_....,d ~ . --lOin

--

,

n 4: 0,

Oia

o¢

Eq. ( A . 3 ) can be s u m m e d using the expressions

(A.4)

R. Rossignoli,P. Ring~NuclearPhysicsA 633 (1998)613-639 ~-~ COSWnr 1~ c o s h [ ½ w ( f l - 2 r ) ] 2092 n=l o~n2+ 092 - -pro -sinh[½flw] - -- 1, 2

Soo o~nsin~onr 2~o ...~ a,~ + oj2 TM

1. 2

sinh[½w(fl-2z)] - - - - sinh[ ½/3oJ]

- - p o ~

,

635

0 ~< r ~< fl'

0
(A.5)


n=l

which yield

G(r)=-/3-'Ro+ZF~[

S0=-fl-I

Ro+

~

e-~" 1 --e-B-----~

=-fl-I

l I = S ° + - -~F a /3f-oa

S0-e~01imR(E) ,

e-~° 1 -

e -/30'-

(1.6) '

(A.7)

with R(E) the analytic continuation of Rn for iwn ~ E, in agreement with (13). Eq. (1.6) allows an analytic continuation for arbitrary r. It is immediately verified that for the coefficients ( l l ) and (12) we recover from (A.6) Eq. (3). Eq. (A.6) can be directly applied to the RPA and CSPA (Eq. (B.9)), and remains finite for w~ = 0 even if F '~ ¢ 0 (in which case the bracket in (A.6) approaches -r//3) as well as for imaginary ~,, provided I/3o, 1 < 2~-. It is thus well defined above the CSPA breakdown. For Q -# Qt, G(O) 4= G(/3), and the series (A.3) at r = 0 or r =/3 will converge to the midpoint, differing from the corresponding limits of (A.6). We obtain in general the relation oo

-/3-' Z

1 Rn = 7[G(0) + G(/3)] = ½(QtQ + QQ,).

(A.8)

n=--oo

In the case (A.4), the series in (A.8) can be summed explicitly using (A.5) for z = 0 O<3

R n = S o A c ~ Z F1

--/3--1Z 11=-oo

Ot coth[

½/3w,,]

(A.9)

og

and Eq. (A.8) is verified. Eq. (A.8) can also be derived by considering the integral of the analytic continuation R(E + it/) in the complex plane. Applying the residue theorem we obtain /3 lim ~ R ( E + i n ) + R ( - E - i n )

V'°° R n = R 0 +

7-.0+ Cj

n=-- oo

,/

dE

O<3

= -~/3

[S(E) + eBeS(-E)]dE,

(A.10)

--OO

where C is a semicircle of infinite radius in the upper complex plane, with the origin excluded by a semicircle of infinitesimal radius, and S(E) is given by Eq. (7). Eq. (A.10) holds provided R(E) has no complex poles.

R. Rossignoli, P Ring~Nuclear Physics A 633 (1998) 613-639

636

Appendix B. Finite temperature RPA around an arbitrary mean field We discuss here in more detail the thermal RPA formalism around an arbitrary onebody Hamiltonian. The unperturbed one-body response matrix (38) can be written as (we omit the variable x in the following) 7~,(oJ)=~-~Q~

P---~ Q ~ , ,

(B.1)

09 - - C a

o~

where Q,~. =_ ( k ' l Q . l k ) , p~ - p~ - pk', e~ - ek, - ek and c~ runs over all pairs k # U. Here, Ik), e~ denote the single particle eigenstates and eigenvalues of h ( x ) = 1-1o - E

(B.2)

x,,Q,.. p

The RPA energies oJ,~ are defined as the eigenvalues of the matrix A.~, = s~6.., - p. Z

Q ~ v ~ Q*, ~ ,

(B.3)

p

and appear in pairs of opposite sign (see below). Considering a complete set of operators Q~ (such that we can expand c~,ck = ~ C~Q~ Vk # k ' ) , we can formally write 7-¢(w) = Qt(~ol - e ) - l p Q ,

(B.4)

A = s-

(B.5)

p Q v Q t,

, 4 - wl = ( e - w l ) ( Q * ) - I [ I

+ 7 ~ ( o ) ) v ] Q t,

(B.6)

where e and p denote the matrices e,~6,~,~, and p,~fi,~,~,. Thus, Det[A - w l ] = Det[e - w l ] D e t [ 1 + "R.(o))v],

(B.7)

so that the non-trivial RPA energies w,, # e,~ satisfy Eq. (43). Eq. (B.7) also demonstrates Eq. (40). Since ~ , ( - o 9 ) = 7~,~(w), it is apparent that both +w,~ are roots of (B.7). The RPA response matrix can then be expressed as 7~wa(w) = [1 + ' / - ¢ ( w ) v ] - i T ~ ( w ) = - Q ~ ( A - o J 1 ) - l p Q ,

(B.8)

[ ~ ' r p a ( ( . 0 ) ]vp' = E

(B.9)

F~au'-" ' og O) - - (.04

Q ~*, ~ u ~ , ~ u 2 , ~--1 , , p ~ , , Q~,,~,, F~a~' = E t~l rTgl/

where [ H - I . A H ] ~ , = w~8,~,. For t, = ~,' F~ = -

[H_IOAH]

[ ,~v~

_ J~

Ow~ 3vu '

(B.IO)

with F ~ = - F ~ for ~oa = - o ~ . We remark that all previous expressions are valid both in a canonical or GC ensemble, the statistics (and temperature) affecting only the value of p~.

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

637

The RPA matrix (B.3) can be obtained from the linearization of the equation of motion

.dp

l-'~ = [h(x),p],

(B.11)

where Pk'k = (C~Ck,) =_ p~ is the single particle density matrix, such that (O~) = ~ O~,p~. We now set x = xo + 8x, p = p(xo) + 8p, with [h(xo), p(xo)] = 0 (for instance, in a GC ensemble p(xo) = [ I + e flh(x°) ]-1 ), and impose the constraint

6x~ = v ~ ( a ~ ) = v~ Z

Q~P~"

(B.12)

Ct

We can then write (B.11) up to first order in ~p as •

d6p

t---~- = [h(xo), ~p] + [Bh, p ( x o ) ] , which, for 6h = - ~

(B.13)

6x~Q~, becomes

•d6p~ Z ¢4~, ~pa,, t d----[--= ez t where ..4 is the matrix (B.3) corresponding to h(xo), with pk, k(Xo) = pkBk~,. The RPA energies are thus the eigenfrequencies of (B.13). The expansion point xo is here arbitrary. The conventional thermal RPA energies are recovered if xo is the self-consistent mean field, i.e. if we set x0, = v~(Q~)xo. If a small oscillating external one-body potential V e -i'°t is introduced, the linearized equations of motion (up to first order in 6p, V) become

.d~p~ = Z

'--Ti--

..4a,~,Bpa, + p,~ V,~ e -i°jt,

(B.14)

Of I

where V,~ = (k'lVIk), whence

6p,,( t ) = - Z (.A - o)l ) 2~,p,~,V,~, e -i'°t. ,~,

(n.15)

It is seen that [..4 - w l ] - l p is the RPA response matrix in the a basis. Expanding V in the operators Q,, i.e. V = y ~ V~Q~, we obtain

B(Ot,)=~-~Q*~6p,,= Og

=Z

~,

Q;~(A-o)I)~,p,,,Q,~,~,,V~,e -i'°t

(B.16)

pt,Ot,OlP

['R.rpa(W) ],~,, V~, e -i°~t,

(B.17)

p!

where Te~a(w) is precisely Eq. (B.8). Within a GC ensemble, the extension of all previous expressions to operators Q~ which include pair creation and annihilation terms is straightforward, in terms of extended matrices Q,~,, = (Q~) k,t of doubled dimension [ 2 ].

638

R. Rossignoli, P. Ring~Nuclear Physics A 633 (1998) 613-639

Appendix C. Strength functions and RPA energies in the Lipkin model The exact partition and strength functions in the Lipkin model are calculated as Z = Tr exp[ - f l H ] = Z

Y ( J ) e-/zEJM'

J,M

S , ( E ) = Z -1 ~

Y(J)e-/3e~M](JM'IJ~[JM)I2~(E - E j M , -4- E j M ) ,

J,M,M'

where EjM are the Hamiltonian eigenvalues and Y ( J ) , 0 ~< J ~< 71 0 , the multiplicity of the J-representation, which depends on the ensemble considered [21]. In particular, 1

Tr e-~(~°J~-xJ~) = ~ Y ( J ) e - ~ l x ) M = Z Y ( J) s i n h [ e ( x ) ( J + 7)] J,M J sinh[ l e ( x ) ] ,

(C.1)

with e ( x ) = ( e 2 + x 2 ) 1/2. In the GC ensemble of total dimension 22a, Y ( J ) = (a_2j2a) _ 2n ) and Eq. (C.1) coincides with (72). In the restricted canonical ensemble of ~2--2J--2 : n ~,~ - (n/2-nJ-I )'~, with a = 1 and we obtain the same dimension 2 a, Y ( J ) = ~n/2-J: expressions (72) with /3 ~ 2/3 and ~ ~ s2/2 [21]. The full canonical ensemble for N = s2, of dimension t2na k.,'2 1, corresponds to a = 2. The effective reduced RPA matrix for the Hamiltonian (63) is

.A ( x ) =

1

[

e2

"~

1

'

and its eigenvalues are + o J ( x ) . Therefore, w2(x) = - D e t [ . A ( x ) ], which leads to Eq. (70).

References [1] I2] 13l 141 [5] [6] [7] [8] 19] 1101 I111 I121 113]

J.L. Egido and E Ring, J. Phys. G 19 (1993) 1. E Ring, L.M. Robledo, J.L. Egido and M. Faber, Nucl. Phys. A 419 (1984) 261. J.L. Egido and H.A. Weidenmuller, Phys. Rev. C 39 (1989) 2398. H. Sagawa and G.F. Bertsch, Phys. Lett. B 146 (1984) 138. M. Sommermann, Ann. Phys. (New York) 151 (1983) 163. H. Hofmann, Phys. Rep. (1997). S. Levit and Y. Alhassid, Nucl. Phys. A 413 (1984) 439. M. Gallardo, M. Diebel, T. Dossing and R.A. Broglia, Nucl. Phys. A 443 (1985) 415. Y. Alhassid and B. Bush, Nucl. Phys. A 509 (1990) 461. V. Martin and J.L. Egido, Phys. Rev. C 51 (1995) 3084. A.L. Goodman, Phys. Rev. C 29 (1984) 1887; Phys. Rev. C 37 (1988) 2162. Y. Alhassid and J. Zingman, Phys. Rev. C 30 (1984) 684. P. Arve, G.F. Bertsch, B. Lauritzen and G. Puddu, Ann. Phys. (New York) 183 (1988) 309; B. Lauritzen, P. Arve and G.F. Bertsch, Phys. Rev. Lett. 61 (1988) 2835.

(C.2)

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[14] B. Lauritzen and J.W. Negele, Phys. Rev. C 44 (1991) 729. [15] Y. Alhassid and B. Bush, Nucl. Phys. A 549 (1992) 43; Nucl. Phys. A 565 (1993) 399. [16] R. Rossignoli, E Ring and N. Dang, Phys. Lett. B 297 (1992) 9; R. Rossignoli, A. Ansari and E Ring, Phys. Rev. Lett. 70 (1993) 1061. [ 17l J. Hubbard, Phys. Rev. Lett. 3 (1959) 77; R.L. Stratonovich, Dokl. Akad. Nauk. SSSR 115 (1957) 1097 [Trans. Soy. Phys. Dokl. 2 (1958) 4581. [181 G. Puddu, P.F. Bortignon and R.A. Broglia, Ann. Phys. (New York) 206 (1991) 409; Phys. Rev. C 42 (1990) 1830. 1191 B. Lauritzen, G. Puddu, R.A. Broglia and P.F. Bortignon, Phys. Lett. B 246 (1990) 329. 120] H. Attias and Y. Alhassid, Nucl. Phys. A 625 (1997) 565. [211 R. Rossignoli and N. Canosa, Phys. Lett. B 394 (1997) 242; Phys. Rev. C 56 (1997) 791. 1221 B.K. Agrawal, A. Ansari and P. Ring, Nucl. Phys. A 615 (1997) 183. 1231 G.H. Lang, C.W. Johnson, S.E. Koonin and W.E. Ormand, Phys. Rev. C 48 (1993) 1518; S.E. Koonin, D.J. Dean and K. Langanke, Phys. Rep. 278 (1997). [241 J.W. Negele and H. Orland, Quantum Many Particle Systems (Addison-Wesley, Reading, MA, 1988). ]251 R. Rossignoli, N. Canosa and J.L. Egido, Nucl. Phys. A 605 (1996) 1; Nucl. Phys. A 607 (1996) 250. 126] H.J. Lipkin, N. Meshkov and A.J. Glick, Nucl. Phys. 62 (1965) 188, 199, 211.