on the variational derivation of the thermal rpa equation

Volume 172, number 2 PHYSICS LETTERS B 15 May 1986

ON THE VARIATIONAL DERIVATION OF T H E THERMAL RPA EQUATION

K. TANABE

Department of Physics, Saitama University, Urawa, Saitama 338, Japan

and

K. SUGAWARA-TANABE

Department of Physics, University of Tokyo, Tokyo 113, Japan

Received 3 February 1986

By applying the variational principle to the grand potential including residual two-body interactions, the thermal random-phase-approximation (TRPA) equation is derived without recourse to the linearization ansatz. The relation of the TRPA equation to the stability condition of the thermal Hartree-Fock-Bogoliubov (THFB) solution is elucidated.

Much attention has been attracted to the recent measurements of giant dipole resonances (GDR) at high temperature and high spin [1] regarded as collective vibrational modes built on highly excited nuclear levels [2]. The- oretical interest is in the evolution of properties caused by thermal, rotational and state.density effect, and also in the practicable formalism appropriate to describe such a mechanism from the microscopic point of view. The sta- tistical recipe with temperature T enables us to take account of many-particle-many-hole configurations and such excitation effects on the mean field. Then the change of single-particle behaviour is described in terms of a self- consistent solution to the thermal Hartree-Fock (THF) equation [3], or the thermal Hartree-Fock-Bogoliubov (THFB) equation in the case with a pairing interaction [4-7] . The thermal extension of the RPA has been at- tempted to describe collective excitations at finite temperature [8-13]. However, no care has been given to the justification and the self-consistency of the thermal RPA (TRPA) formalism itself. The present note is devoted to show that, at first the T R P A equation can be derived from a few reasonable assumptions without touching with the so-called linearization ansatz, as adopted in refs. [8,10]; and, secondly, the concrete form of the stability matrix for the thermal HFB (THFB) solution is different from the one given in refs. [8,10,11].

Here we attempt to establish the TRPA formalism on the single-particle space provided by the self-consistent solutions of the cranked THFB equation for a general two-body hamiltonian including pairing interaction, H. As presented in ref. [5], the variational parameters E u come into our problem through the trial density matrix, whose is given by

IV = exp(--~Heff)/Tr[exp(-3Heff)] , (1)

where the effective hamiltonian H eft is anticipated to be diagonal in the quasiparticle picture, i.e.

H eff = U~) ff + E E/[IO~ ~ Oliu, (2) lu

and [3 = 1/kT, k is the Boltzmann constant. If we denote the ensemble average of an operator O by

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

129


(8) = Tr(FV6) , (3)

the approximate grand potential (or thermodynamical potential) of our problem is written as

F=(H’)-S’T~(H)-w,,,djx>-~~)-S’T, (4)

where ix is the X-component of the angular momentum operator and & IS the particle-number operator, both are constrained (i.e. w^,> = I and &) = N); and the approximate entropy S’ is given by

S’ = k Tr(W In w) . (5)

In (3) the rotational frequency arot and the chemical potential A are introduced as Lagrange multipliers. Then

the quasiparticle distribution functionfP is defined as

(~~“v)=fcIspv, fp = 1/ bp(PE,) + 11 . (6)

Further, the variational parameters A,,, ~4;~) Bkr and B& are introduced through the HFB transformation re-

lating the single-particle operators ck and ctk with quasiparticle operators (Y,, and c.$, i.e.

The cranked hamiltonian H’ can be expressed in terms of the quasiparticle operators as

H’EH- wrot& - ti

= u, + c tH&,&, +$ (H20)pva$4 + h.c. + IrV CtV )(

c (H31)rvpoa~a~~$x, t h.c. MVPO

(7)

(8)

The coefficients of the independent variations 6 fcl, &A,, , 6Bkp, 6A;, and 6B&, in the variational requirement 6F = 0 yield the cranked THFB equation [5], which also gives us

ugff = uo - 2 c (HZ&wf~fv . W

(9c)

We have seen that the combination (Hf{f),v defined by (9a) is diagonalized to give the effective quasiparticle energy EM, but (Hll)pv appearing in (8) remains non-diagonal; and the combination (H$f)pv defined by (9b) vanishes, but (H20)Pv appearing in (8) does not. Therefore H’ still contains nondiagonal bilinear forms in a,, and o: in addition to quadratic terms, even after solving the THFB equation.

Now our aim is to take into account further correlations in the residual part of the hamiltonian, AH defined by

H’=Heff+AH. (10)

We vary the grand potential F by applying the unitary transformation to the trial density matrix IV, i.e.

W + “w = exp(iR) IV exp(-iR) , (11)

130


where the hermitian operator R is assumed to be a bilinear form in a u and a S , i.e.

R QS.+Q, QS. ~ 5. 5" (Xuvotuot p Yuuotvotu) + D * = = - - Z t z v (~l.~ Otv " # > u I.Lv

(12)

In the above the sum of Xuv and Yuv terms are restricted to the extent/a >p , but the one ofZu~ , term is not. Since the entropy S' is invariant under the unitary transformation (11), the change o f f is given by

AF = Tr(WH') - kT Tr(W In W ~) - F-~ i< [H', R] ) + ½( [R, [H', R] ] ), (13)

where we have expanded W into a series of R and retained up to the second-order term. In principle we may keep higher order terms of R, and in such a case we will get a set of complicated non-linear eigenvalue equations. But here, expecting that not only the norm of each coefficient Xuv, Yu~, or Zuv is much less than unity for the collective state, but also that substantial cancellation takes place among'higher order terms, we start with an approximate form for the grand potential given by (13). As long as the TRPA operator QS. is built on the quasiparticle picture provided by the THFB solution, the first term in (13) vanishes due to (9a) and (9b). We assume a normalization condition, which is allowed for the solution to the homogeneous linear equation

( [ a , QS.])= 1, (14)

and additional requirements for the ensemble averages

<[Q, [H',Q]])=<[QS., [H', QS.]])= O. (15)

We minimize AF under the constraints (14) and (15). Therefore we vary the quantity

A F - h e ( [Q, Qt ] ) _ a([Q, [HI', Q] ] ) - b( [Q5., [H', Qt ] ] )

with respect to Xuv , Ym,, Zuv and their complex conjugates to obtain

( I - a ) ~ M X ( - ) = - - (aM - "hco)X (+) , ( I - b)l'~MX (+) = - ( a M +/~co)X ( - ) .

In (17) we have introduced the notations for the hermitian TRPA matrix l i (= II t ) ,

I -~t~vpo Cl3uvpt~" eUvpo 1

=_ c~* I~vpo * I ' ~'~tzv ,po I lavpa ~ * -- ~lauap

(16)

(17a, b)

(18)

with

"¢tuvPo -- [(Eu + Ev)[(1 -- f o - f o)] (6up6 vo -- 6 #o6vp) + 4(H22)~ovp , c~t~up ° --- -24(H40)uvpo , euvpo = 6(Hal)u~,po ,

cOuvpo ~ [(E u -E~,)[( f v --f#)] 6upSvO + 4(H22)uaup ,

for the metric matrix M which takes the diagonal form

o 1 = -(1 - f . - L ) o ,

o f . - f .

(19a)

(19b, c)

(19d)

(20)

131

Volume 172, number 2

and for the vectors X (+) and X ( - )

r F.1 x22 -r22

PHYSICS LETTERS B 15May 1986

(21)

From (20) and (2 !) we get .

X(+)¢MX ( - ) = X ( - ) t MX (+) = 0 . (22)

Multiplying X(- ) t M to (17a) and X(+)tM to (17b), we get with the help c,f (22)

( 1 - a ) ( [ Q * , [ H ' , Q ] ] ) = - ( [ Q t , [H',QtII>, ( 1 - b ) < [ Q , [ H ' , Q t ] ] ) = - ( [ Q , [H' ,Q]] ). (23a, b)

Since the RHS of (23) is requested to vanish by (15), and ( [Q, [H', Qt] ] i' is Finite except for the trivial case of ( [R, [H', R] ] ) = 0, we obtain the solution a = b = 1. Thus, (17) gives the TRPA eigenvalue equations,

ffIMX (n) =/Z¢onX (n) , I2MX (-n) = hW_n X ( - n ) , (24)

with/~o n = --~6o_ n(> 0). The orthonormality relation is proved from (24), and the normalization condition is required by (14), i.e.

([Qm, Qn*] > = X(m)tMX(") = 6ran • (25)

It is also confirmed that the RHS of (22) vanishes as long as the eigenvectors satisfy (24) and (25). Then, (13) reduces formally to

AF = ~ ( [R, [H', R] ] ) =/~6o, (26)

which shows that a parameter/Zoo determined as the eigenvalue of the TRFA equation (24) describes an effective excitation energy measured from the THFB level energy for a given angular momentum I and particle number N at finite temperature T. The positive-definiteness of &F does not mean any stability condition of the THFB solution, in contrast to the HFB theory at T = 0, but simply corresponds to th4, ~ spectral condition for the physical solution to the TRPA equation. We present here an adequate form of the stability condition, which is given by the second-order variation o f F [5], to clarify its relation to the TRPA equation. The infinitesimal variation of the HFB transformation coefficients in (7) is fully parametrized by the infinitesimal unitary transformation, i.e.

where

(27)

U = exp ~ t t * • * [~(c~% a,, - % , , a , , % ) + du,, % ~v] , (28)

with the infinitesimal parameters satisfying cur = -cvu and d*uv = --.dvu. Then, taking account of the contributions from the variations 5fu , in addition to 8Aku and 6Bku , i.e.

8Aku = (B*C*)kv. + (Ad)k u , 8Bku = (A*c*)k u + (BCOku , (29a, b)

we arrive at the following stability condition after some straightforward algebra.

62F = pt"sp > O, (30)

132


where

">* > "* f / t a ) ' p'~ _~ ~.Cta v , C tau, Cl tau,

~Atav,po

B~I) ,pff $ =

g~o,l~

Ca,ld

with

C ta u ~- Cla u = .---Cut a

Btap , pa

A ,,v,po

Eop , vta

Cpo,l,t

Etal:,pa Ctav~o

Cta,oo Ftav,po Gtau,o

~" op,ta Dta, o

A tav , po = (Eta + E v ) ( f ta + . f v - 1)($taoSva - 8 tap ~ vo ) + 4( H 2 2 ) ta v p a Oeta + .t'v - 1)(fp +fo - 1),

Btav ,po- - 24(H40)tavpoO"ta + f v - 1)(fp +.to -- 1), Ctau,o=- -6(H31) tavoo( f ta + f v - 1),

Dta,o = [kT/ f ta(1 - f t a ) l S t a o + 4(H22)taotao ' E ~ , p o --6(H31)~op(fta +fv -- 1)0co - f p ) '

Ftav,p o = (Eta - Ev ) (f~, - fta)Stap(Sz, o + 4(H22)~ovp (f. - fta) Oro - f p ) '

a t a . , o =- - f t a ) .

On the other hand, the TRPA matrix M~M is directly calculated from (18), (19) and (20) as

(31)

(32)

(33a)

(33b, c)

(33d, e)

(330

(33g)

F A..,po 8.. ,po L'..,oo]

( M D I V I ) t a u , o o = I B : u , O o A : u , p o E * t a , o o l , (34)

LCo,ta. eo.,.ta Fta.,.o_, which corresponds to the submatrix of the stability matrix $ in (32). Therefore, both matrices (32) and (34) do not coincide with each other except when there is no variation of/ta, i.e., 6fta = 0, which is the case for T = 0.

In conclusion, we have shown that the form of the TRPA equation (24) can be derived from the minimization of the grand potential (13) under the constraints (14) and (15). The equation obtained directly from the linearization ansatz

[H', Qt ] = fi~oQt

does not coincide with (24), since it does not contain any thermal effect on the RPA correlation. It gives a crude approximation to the TRPA equation and should not be confused with the present set of the TRPA equation built on the ensemble space. The Z term in (12) plays an essential role in taking account of the correlations caused by the H31 term in (8) as inferred from the form of the matrix element etavp a in (19c). In the usual formalism, in- clusion of such a Z term belonging to the higher order RPA becomes a source of theoretical inconsistency. How- ever, the present formalism is free from this difficulty since it employs only the ensemble averages like (14), (15) and (16), and avoids to refer to operator relations. Discussions of the T R P A formalism in comparison with the linear response theory [12] and the application of the theory to GDR [13] will be published elsewhere.

References

[1] J.O. Newton et at, Phys. Rex,. Lett. 46 (1981) 1383; J.J. Gaaxdh~e et aL, Phys. Rev. Lett. 53 (1984) 148; C.A. Gossett et al , Phys. Rev. Lett. 54 (1984) 1486; D. Schwalm et al., Max-Planck-Institute preprint (1984).

133


[2] D.M. Brink, Ph.D. thesis, University of Oxford (1955); NucL Phys. 4 (1957) 215; H. Morinaga, Phys. Rev. 101 (1956) 100.

[3] M. Brack and Ph. Quentin, Phys. Scr. 10A (1974) 163; U. Mosel, P.-G. Zint and K. Passler, Nucl. Phys. A236 (1974) 252.

[4] K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. B 97 (1980) 337. [5] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys.A 357(1981) 20. [6] K. Sugawara-Tanabe, K. Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 45;

K. Tanabe and K. Sugawara-Tanabe, Nucl. Phys. A 390 (1982) 385. [7] A.L. Goodman, Nucl. Phys. A 352 (1981) 30, 45. [8] J. des Cloiseau, in: Many body physics, eds. C. de Witt and R. Balian (Gordon and Breach, New York, 1968) p. 5. [9] J.L. Egido and P. Ring, Phys~ Lett. B 127 (1983) 5;

P. Ring et al., Nucl. Phys. A419 (1984) 261. [10] H.M. Sommermann, Ann. Phys. 151 (1983) 163. [11] D. Vautherin and N. Vinh Mau, Nucl. Phys. A422 (1984) 140. [12] K. Tanabe and K. Sugawara-Tanabe, to be published. [13] K. Sugawara-Tanabe and K. Tanabe, to be published.

134

on the variational derivation of the thermal rpa equation

Documents