Microscopic description of hot nuclei: the SPA+RPA approach

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A654 (1999) 719e-722e www.elsevier.nl/locate/npe

Microscopic description of hot nuclei: the SPA+RPA approach * R. Rossignoli, N. Canosa, a t and P. Ring b aDepartamento de Fisica, Universidad Nacional de La Plata, c.c. 67, 1900 La Plata, Argentina bphysik-Department der Technischen Universit~t Miinchen, D-85747 Garching, Germany We discuss the static path plus random phase approximation (SPA+RPA) to the partition function of warm finite nuclei. The method, derived from the auxiliary field path integral representation of the partition function, takes into account large amplitude statistical fluctuations around the mean field, and is able to provide an accurate evaluation of thermodynamics properties within finite configuration spaces. We present some recent improvements which include the treatment of repulsive terms in the interaction, the exact implementation in canonical and in restricted grand canonical ensembles with fixed number parity, and the evaluation of response and strength functions. 1. I n t r o d u c t i o n Small correlated quantum systems exhibit important fluctuation phenomena. At finite temperature, statistical fluctuations smooth out the sharp phase transitions that arise in the mean field approximation (MFA), which normally represents the actual behavior of the system only in the limit of infinite particle number or volume. Accordingly, in warm finite nuclei fluctuations become important. Shape fluctuations smooth out the transitions between different shapes that arise in an ordinary MFA description [1,2], the concomitant effects being essential in the microscopic description of giant resonances [3]. Similarly, gap fluctuations smooth out the superfiuid to normal transition predicted by BCS [4]. At the same time, important deviations from standard grand canonical (GC) statistics are to be expected in small systems with fixed particle number, and canonical corrections are essential for a proper description of odd-even effects. These issues have recently become relevant also in solid state physics due to the development of diverse mesoscopic structures, like for instance ultrasmall superconducting metallic grains [5], where fluctuations and odd-even effects play an essential role in small samples where the gap is comparable to the single electron level spacing [5,6]. A general and convenient microscopic framework to deal with fluctuations is provided by the path integral representation of the partition function obtained with the HubbardStratonovich transformation. In this context, the mean field approximation is seen as a stationary point of the integrand, which provides an increasingly flat minimum of the *Work supported in part by EEC (contract CI1"-CT93-0352) fRR and NC are members of CICPBA and CONICET respectively, of Argentina. 0375-9474/99/$ see front matter © 1999 ElsevierScienceB.V. All rights reserved.

R. Rossignoli et al. /Nuclear Physics A654 (1999) 719c-722c

720c

pertinent free energy potential as the relevant size parameter of the system decreases. Large amplitude static fluctuations around the mean field can be included by integrating exactly over the static components of the auxiliary fields, which leads to the static path approximation (SPA)J7-9], whereas small amplitude quantum fluctuations can be incorporated by integration over the remaining components in the gaussian approximation. This last step leads to the S P A + R P A [10-13], to be denoted for brevity as correlated (C) SPA, which yields an accurate evaluation of the full path integral for not too small temperatures and provides in principle a simple alternative to full Monte-Carlo evaluations [14]. In this contribution we give a brief general presentation of this method, suitable for restricted statistical ensembles, and briefly discuss some recent extensions. 2. G e n e r a l C S P A approach 2 For a Hamiltonian of the form H = Ho - ~1 ~ vvQ~, the Hubbard-Stratonovich transformation allows to express the partition function Z = Tr e x p [ - ~ H ] as a path integral,

Z : _fD[~]T~T _ exp{ - fj0~ dvH[x(T)]},

x~ - x~,Q~,, H ( x ) = Ho + ~ , 2v~

(1)

where x = {x~} is an auxiliary field, T denotes time ordering and H(x) is a linearized Hamiltonian, i.e., a single particle (sp) operator if H0 and Q~ are of this type. By means of a Fourier expansion in the interval (0, ~),

x~(~-) = x~ + ~ x~ e

, w. = 27~n/fl,

(2)

n#O

one retains in the CSPA the exact integration over the static coefficients x~, while the remaining variables x~ are integrated in the gaussian approximation for each value of xv. This takes into account large amplitude static fluctuations of x~ (~-), relevant in small systems and critical regions, as well as small amplitude quantum-like fluctuations. The sharp phase transitions of the "mean field" approximation, i.e., of a full saddle point evaluation of (1), become in this way smooth in a small system. After expanding the logarithm of the trace in (1) up to second order in x~#0, we obtain [12,13,15] ZCSPA

f d(x)Z(x)CRpA(z),

(3)

where Z(x) = T r e x p [ - ~ H ( x ) ] and CRpA(X) = 11 Det[(f.., + v.R..,(x, iwn)]-l= IX n:l

a>O

w~ sinh[i~A~] A~ sinh[[~wa] '

(4)

is an RPA correction, with R ~ , (x, w) the uncorrelated thermal response function around H(x) and wa(x) the finite temperature RPA energies determined from

Det[5~, + v.R~,(x, wa)] =O,

R~,(x,w) = ~ Qkk'qk'k ~ ~' )~kPk-)W + w"

(5)

k#k I

Here Q~k, -- (k]Q.lk'), with Ik) Aa, Pk the sp eigenstates, energies and occupation probabilities corresponding to H(x), and A~ = Ak -- Ak,, with a denoting all pairs k > k'. Eq.

R. Rossignoli et al. /Nuclear Physics A654 (1999) 719c-722c

721c

(3) can be applied if the determinants in (4) are positive definite Vx. The integrals over static variables associated with repulsive terms (v~ < 0) in the Hamiltonian are to be performed along the imaginary axes, and can be accurately evaluated in the saddle point approximation, at least for temperatures above the CSPA breakdown. This leads to an additional static determinant in (3) [13]. The SPA is recovered if CRpA(X) is omitted in (3) and is exact when [Q~, H(x)] = 0 Vx (in which case CRpA(x) = 1). In general, it provides the high temperature limit of (3). The stationary points of Z(x), obtained from

(6)

=

with (Q~)z = Ek Q~kPk the thermal average, determine the self-consistent mean field. The MFA partition function is just Z(x) evaluated at the fundamental solution of (6). In many fermion systems, Eqs. (3)-(6) can be applied in the GC ensemble (with H -4 H - #N), but also in any restricted ensemble defined by a projector P which commutes with H(x) Vx [16,17]. The type of statistics determines just Z(x) and the probabilities Pk in R ~ , (x, w). Thus, they are strictly applicable in a canonical ensemble for a density-like decomposition of H in which [H(x), N] = 0 [18]. In a superfluid system, a pairing-like decomposition is more convenient. In this case, H(x) is single quasiparticle operator, with Ak the single quasiparticle energies and w~ the quasiparticle RPA energies. The formalism is directly applicable in a GC ensemble with fixed number parity (NP), which is essential for correct description of odd-even effects [16]. In NP projected statistics, one obtains [16,17], for H(x) = ho(x) + Ek Ak(x)ata~, g(x)

=

1 + a'F e-~h°[H(1 + e-Z~k)] 1 + a,2'

p~ =

k [1 + e~k] -1

r = Htanh[1/~Ak],

(7)

k o'r sinh-l[j3Ak], l+a~F

(8)

where a' = aao -- 4-1, with a -- 1 ( - 1 ) for an even (odd) system and a0 the number parity of the quasiparticle vacuum [19]. Ordinary GC statistics corresponds to a' -- 0. The NP correction factor F becomes small for temperatures larger than the lowest quasiparticle energy, but its influence is essential in small superfluid systems for T < To. Let us consider finally the evaluation of strength functions. In the treatment of [15], the CSPA response function for an operator Q~ is

RCSPA(E)

=

Zc-~pn / d(x)Z(x)VnpA(X)Rut~uA(x , E),

m~n(x, E) R..,

=

{[1 + R(x, E)v]-l R(x, E ) } ~ , ,

(9) (10)

where (10) is the RPA response matrix [20] at x. The strength function is then obtained as S(E) = -~Im[RcspA(E+iy)] for ~ --4 0. Eq. (9) is just the CSPA average of RPA response functions and corresponds to an adiabatic-type approximation. The peaks of the RPA strength acquire in the CSPA a width, which remains finite for zl --+ 0, although in critical regions the deviation of the CSPA result from the RPA strength may be considerable. The total strength obtained from (9) coincides with the CSPA thermal average csPA 2 = ~0 In ZCSPA/OV~, for temperatures above the CSPA breakdown.

722c

R. Rossignoli et al./Nuclear Physics A654 (1999) 719c-722c

3. Applications The CSPA has so far been applied to simple separable interactions, for which it has provided almost exact results for thermal averages and level densities within schematic models, above a low breakdown temperature [10,12,13,18].. A considerable improvement over MFA and M F A + R P A results is obtained in all these cases, especially in transitional regions, even in the presence of repulsive terms [13]. It also provides a good evaluation of the first energy weighted moments of the strength function [15]. For superfluid systems, NP projected CSPA provides an accurate description of odd-even effects at finite temperature in small systems where the gap is of the order of the sp level spacing, where the superfluid to normal transitions predicted by NP projected BCS become considerably washed out [16]. More details will be found in [17]. Altogether, the CSPA constitutes an attractive alternative for describing strongly correlated finite quantum systems at finite temperature, at least in the context of effective interactions containing a few separable terms. The approach is also suitable for large configuration spaces, as the RPA correction can be evaluated with the first expression in (4), without requiring the explicit calculation of the RPA energies. REFERENCES 1. A.L. Goodman, Phys. Rev. C 29 (1984), 1887. 2. J.L. Egido and P. Ring, J. of Phys. G 19, (1993) 1. 3. M. Gallardo, M. Diebel, T. Dossing, R.A. Broglia, Nucl. Phys. A443 (1985), 415. Y. Alhassid, B. Bush, S. Levit Phys. Rev. Lett. 61 (1988), 1926. 4. L.G. Moretto, Phys. Lett. B 40 (1972), 1; J.L. Egido, P. Ring, S. Iwasaki, H.J. Mang, Phys. Lett. B 154 (1985), 1. 5. C.T. Black, D.C. Ralph, M. Tinkham, Phys. Rev. Lett. 74 (1995), 3241; 76 (1996), 688. 6. R. Balian, P. Flocaxd, and M. Veneroni, nucl-th/9706041; cond-mat/9802006. 7. P. Arve, G.F. Bertsch, B. Lauritzen, G. Puddu, Ann. of Phys. (N.Y.) 183 (1988), 309; B. Lauritzen P. Arve, and G.F. Bertsch, Phys. Rev. Lett. 61 (1988), 2835. 8. Y. Alhassid, B. Bush, Nucl. Phys. A549 (1992), 43. 9. R. Rossignoli, A. Ansari, and P. Ring, Phys. Rev. Left. 70 (1993), 9. R. Rossignoli and P. Ring, Ann. of Phys. (N. Y.) 235 (1994), 350. 10. G. Puddu, P.F. Bortignon, and R.A. Broglia, Ann. of Phys. (N.Y.) 206 (1991), 409; Phys. Rev. C 42 (1990), 1830. B. Lauritzen et al, Phys. Lett. B 246 (1990), 329. 11. G. Puddu, Phys. Rev. C 47 (1992), 1067. 12. H. Attias and Y. Alhassid, Nuel. Phys. A 625 (1997), 363. 13. R. Rossignoli, N. Canosa, Phys. Lett. B 394 (1997), 242; Phys. Rev. C 56 (1997) 791. 14. 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). 15. R. Rossignoli, P. Ring, Nucl. Phys. A 633 (1988) 613. 16. R. Rossignoli, N. Canosa, and P. Ring, Phys. Rev. Left. 80 (1998), 1853. 17. R. Rossignoli, N. Canosa, and P. Ring, Ann. of Phys. (N. Y.) (in press). 18. N. Canosa, R. Rossignoli, and P. Ring, Phys. Rev. C (in press). 19. P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer (NY) 1980. 20. P. Ring, L.M.. Robledo, J.L. Egido, and M. Faber, Nucl. Phys. A 419 (1984), 261.